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@ -69,7 +69,7 @@
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Acmid = {2079770},
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Acmid = {2079770},
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||||||
Doi = {10.1007/11844297_36},
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Doi = {10.1007/11844297_36},
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||||||
ISBN = {3-540-38990-3, 978-3-540-38990-3},
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ISBN = {978-3-540-38990-3},
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||||||
Location = {Reykjavik, Iceland},
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Location = {Reykjavik, Iceland},
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||||||
Numpages = {10},
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Numpages = {10},
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||||||
Url = {http://dx.doi.org/10.1007/11844297_36}
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Url = {http://dx.doi.org/10.1007/11844297_36}
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||||||
@ -86,13 +86,14 @@
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|||||||
Url = {https://www.researchgate.net/profile/Yaneer_Bar-Yam/publication/225104044_Complex_Engineered_Systems_A_New_Paradigm/links/59107f20a6fdccbfd57eb84d/Complex-Engineered-Systems-A-New-Paradigm.pdf}
|
Url = {https://www.researchgate.net/profile/Yaneer_Bar-Yam/publication/225104044_Complex_Engineered_Systems_A_New_Paradigm/links/59107f20a6fdccbfd57eb84d/Complex-Engineered-Systems-A-New-Paradigm.pdf}
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||||||
}
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}
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||||||
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||||||
@Article{anrichterEvol,
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@InProceedings{anrichterEvol,
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||||||
Title = {Evolvability as a Quality Criterion for Linear Deformation Representations in Evolutionary Optimization},
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Title = {Evolvability as a Quality Criterion for Linear Deformation Representations in Evolutionary Optimization},
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||||||
Author = {Richter, Andreas and Achenbach, Jascha and Menzel, Stefan and Botsch, Mario},
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Author = {Richter, Andreas and Achenbach, Jascha and enzel, Stefan and Botsch, Mario},
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||||||
Year = {2016},
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Year = {2016},
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||||||
Note = {\url{http://graphics.uni-bielefeld.de/publications/cec16.pdf}, \url{https://pub.uni-bielefeld.de/publication/2902698}},
|
Note = {\url{http://graphics.uni-bielefeld.de/publications/disclaimer.php?dlurl=cec16.pdf}, \url{https://pub.uni-bielefeld.de/publication/2902698}},
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Booktitle = {IEEE Congress on Evolutionary Computation},
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Booktitle = {IEEE Congress on Evolutionary Computation},
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||||||
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Pages = {901--910},
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||||||
Location = {Vancouver, Canada},
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Location = {Vancouver, Canada},
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||||||
Publisher = {IEEE}
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Publisher = {IEEE}
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||||||
}
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}
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@ -100,12 +101,12 @@
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|||||||
@InProceedings{richter2015evolvability,
|
@InProceedings{richter2015evolvability,
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||||||
Title = {Evolvability of representations in complex system engineering: a survey},
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Title = {Evolvability of representations in complex system engineering: a survey},
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||||||
Author = {Richter, Andreas and Botsch, Mario and Menzel, Stefan},
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Author = {Richter, Andreas and Botsch, Mario and Menzel, Stefan},
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||||||
Booktitle = {Evolutionary Computation (CEC), 2015 IEEE Congress on},
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Booktitle = {2015 IEEE Congress on Evolutionary Computation (CEC)},
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||||||
Year = {2015},
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Year = {2015},
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||||||
Organization = {IEEE},
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Organization = {IEEE},
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||||||
Pages = {1327--1335},
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Pages = {1327--1335},
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||||||
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||||||
Url = {http://www.graphics.uni-bielefeld.de/publications/cec15.pdf}
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Url = {http://www.graphics.uni-bielefeld.de/publications/disclaimer.php?dlurl=cec15.pdf}
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}
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}
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@InBook{Rothlauf2006,
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@InBook{Rothlauf2006,
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@ -168,8 +169,6 @@
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Year = {2012},
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Year = {2012},
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||||||
Number = {5},
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Number = {5},
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||||||
Volume = {27},
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Volume = {27},
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||||||
|
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||||||
Url = {http://jcst.ict.ac.cn:8080/jcst/EN/article/downloadArticleFile.do?attachType=PDF\&id=9543}
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||||||
}
|
}
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||||||
@article{giannelli2012thb,
|
@article{giannelli2012thb,
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||||||
title={THB-splines: The truncated basis for hierarchical splines},
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title={THB-splines: The truncated basis for hierarchical splines},
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@ -180,17 +179,17 @@
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pages={485--498},
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pages={485--498},
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year={2012},
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year={2012},
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||||||
publisher={Elsevier},
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publisher={Elsevier},
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||||||
url={https://pdfs.semanticscholar.org/a858/aa68da617ad9d41de021f6807cc422002258.pdf},
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note={\url{https://pdfs.semanticscholar.org/a858/aa68da617ad9d41de021f6807cc422002258.pdf}},
|
||||||
doi={10.1016/j.cagd.2012.03.025},
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doi={10.1016/j.cagd.2012.03.025},
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||||||
}
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}
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||||||
@article{brunet2010contributions,
|
@article{brunet2010contributions,
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||||||
title={Contributions to parametric image registration and 3d surface reconstruction},
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title={Contributions to parametric image registration and 3d surface reconstruction},
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||||||
author={Brunet, Florent},
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author={Brunet, Florent},
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||||||
journal={European Ph. D. in Computer Vision, Universit{\'e} dAuvergne, Cl{\'e}rmont-Ferrand, France, and Technische Universitat Munchen, Germany},
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journal={European Ph. D. in Computer Vision, Universit{\'e} dAuvergne, Cl{\'e}rmont-Ferrand, France, and Technische Universität München, Germany},
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year={2010},
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year={2010},
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||||||
url={http://www.brnt.eu/phd/}
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url={http://www.brnt.eu/phd/}
|
||||||
}
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}
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||||||
@article{aschenbach2015,
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@InProceedings{aschenbach2015,
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||||||
author = {Achenbach, Jascha and Zell, Eduard and Botsch, Mario},
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author = {Achenbach, Jascha and Zell, Eduard and Botsch, Mario},
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booktitle = {Vision, Modeling \& Visualization},
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booktitle = {Vision, Modeling \& Visualization},
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journal = {Proceedings of Vision, Modeling and Visualization},
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journal = {Proceedings of Vision, Modeling and Visualization},
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@ -199,7 +198,7 @@
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publisher = {Eurographics Association},
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publisher = {Eurographics Association},
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||||||
title = {Accurate Face Reconstruction through Anisotropic Fitting and Eye Correction},
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title = {Accurate Face Reconstruction through Anisotropic Fitting and Eye Correction},
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year = {2015},
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year = {2015},
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||||||
url = {http://graphics.uni-bielefeld.de/publications/disclaimer.php?dlurl=vmv15.pdf},
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note = {\url{http://graphics.uni-bielefeld.de/publications/disclaimer.php?dlurl=vmv15.pdf}},
|
||||||
ISBN = {978-3-905674-95-8},
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ISBN = {978-3-905674-95-8},
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}
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}
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@article{hauke2011comparison,
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@article{hauke2011comparison,
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@ -247,7 +246,7 @@
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publisher={IEEE},
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publisher={IEEE},
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||||||
url={https://www.researchgate.net/profile/Marc_Schoenauer/publication/223460374_Parameter_Control_in_Evolutionary_Algorithms/links/545766440cf26d5090a9b951.pdf},
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url={https://www.researchgate.net/profile/Marc_Schoenauer/publication/223460374_Parameter_Control_in_Evolutionary_Algorithms/links/545766440cf26d5090a9b951.pdf},
|
||||||
}
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}
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||||||
@article{rechenberg1973evolutionsstrategie,
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@book{rechenberg1973evolutionsstrategie,
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||||||
title={Evolutionsstrategie Optimierung technischer Systeme nach Prinzipien der biologischen Evolution},
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title={Evolutionsstrategie Optimierung technischer Systeme nach Prinzipien der biologischen Evolution},
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||||||
author={Rechenberg, Ingo},
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author={Rechenberg, Ingo},
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year={1973},
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year={1973},
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@ -44,22 +44,23 @@
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\vspace*{\stretch{4}}
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\vspace*{\stretch{4}}
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\begin{center}
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\begin{center}
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\hspace{0.99cm} {\huge\bfseries Evaluation of the Performance\\[4mm]
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\hspace{1.5cm} {\huge\bfseries Evaluation of the Performance\\[-3mm]
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\hspace{0.99cm} of Randomized FFD Control Grids}\\[28mm]
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\hspace{1.5cm} of Randomized\\[4mm]
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\hspace{0.99cm} {\LARGE Master Thesis}\\[3mm]
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\hspace{1.5cm} FFD Control Grids}\\[25mm]
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\hspace{0.99cm} {\Large {\normalsize at the}\\[4mm]
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\hspace{1.5cm} {\LARGE Master Thesis}\\
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\hspace{0.99cm} AG Computer Graphics}\\[2mm]
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\hspace{1.5cm} {\Large {\normalsize at the}\\
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\hspace{0.99cm} at the Faculty of Technology\\
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\hspace{1.5cm} AG Computer Graphics}\\[2mm]
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\hspace{0.99cm} of Bielefeld University\\[5mm]
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\hspace{1.5cm} at the Faculty of Technology\\
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\hspace{0.99cm} {\Large by}\\[5mm]
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\hspace{1.5cm} of Bielefeld University\\[3mm]
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\hspace{0.99cm} {\LARGE Stefan Dresselhaus}\\[8mm]
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\hspace{1.5cm} {\Large by}\\[3mm]
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\hspace{0.99cm} {\large \today}
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\hspace{1.5cm} {\LARGE Stefan Dresselhaus}\\[5mm]
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\hspace{1.5cm} {\large \today}
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\end{center}
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\end{center}
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\vspace*{\stretch{2}}
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\vspace*{\stretch{2}}
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\begin{center}
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\begin{center}
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\begin{tabular}{lrl}
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\begin{tabular}{lrl}
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\hspace{0.99cm} Supervisor:~&Prof.~Dr.~&Mario Botsch\\
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\hspace{1.5cm} Supervisor:~&Prof.~Dr.~&Mario Botsch\\[-5mm]
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\hspace{0.99cm} &Dipl.~Math.~&Andreas~Richter
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\hspace{1.5cm} &Dipl.~Math.~&Andreas~Richter
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\end{tabular}
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\end{tabular}
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\end{center}
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\end{center}
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\vspace*{\stretch{.2}}
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325
arbeit/ma.md
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arbeit/ma.md
@ -49,19 +49,19 @@ etc.), the translation of the problem--domain into a simple parametric
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representation (the *genome*) can be challenging.
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representation (the *genome*) can be challenging.
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This translation is often necessary as the target of the optimization may have
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This translation is often necessary as the target of the optimization may have
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too many degrees of freedom. In the example of an aerodynamic simulation of drag
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too many degrees of freedom for a reasonable computation. In the example of an
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onto an object, those object--designs tend to have a high number of vertices to
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aerodynamic simulation of drag onto an object, those object--designs tend to
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adhere to various requirements (visual, practical, physical, etc.). A simpler
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have a high number of vertices to adhere to various requirements (visual,
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representation of the same object in only a few parameters that manipulate the
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practical, physical, etc.). A simpler representation of the same object in only
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whole in a sensible matter are desirable, as this often decreases the
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a few parameters that manipulate the whole in a sensible matter are desirable,
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computation time significantly.
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as this often decreases the computation time significantly.
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Additionally one can exploit the fact, that drag in this case is especially
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Additionally one can exploit the fact, that drag in this case is especially
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sensitive to non--smooth surfaces, so that a smooth local manipulation of the
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sensitive to non--smooth surfaces, so that a smooth local manipulation of the
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surface as a whole is more advantageous than merely random manipulation of the
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surface as a whole is more advantageous than merely random manipulation of the
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vertices.
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vertices.
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The quality of such a low-dimensional representation in biological evolution is
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The quality of such a low--dimensional representation in biological evolution is
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strongly tied to the notion of *evolvability*\cite{wagner1996complex}, as the
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strongly tied to the notion of *evolvability*\cite{wagner1996complex}, as the
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parametrization of the problem has serious implications on the convergence speed
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parametrization of the problem has serious implications on the convergence speed
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and the quality of the solution\cite{Rothlauf2006}.
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and the quality of the solution\cite{Rothlauf2006}.
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@ -80,14 +80,14 @@ One example of such a general representation of an object is to generate random
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points and represent vertices of an object as distances to these points --- for
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points and represent vertices of an object as distances to these points --- for
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example via \acf{RBF}. If one (or the algorithm) would move such a point the
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example via \acf{RBF}. If one (or the algorithm) would move such a point the
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object will get deformed only locally (due to the \ac{RBF}). As this results in
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object will get deformed only locally (due to the \ac{RBF}). As this results in
|
||||||
a simple mapping from the parameter-space onto the object one can try out
|
a simple mapping from the parameter--space onto the object one can try out
|
||||||
different representations of the same object and evaluate which criteria may be
|
different representations of the same object and evaluate which criteria may be
|
||||||
suited to describe this notion of *evolvability*. This is exactly what Richter
|
suited to describe this notion of *evolvability*. This is exactly what Richter
|
||||||
et al.\cite{anrichterEvol} have done.
|
et al.\cite{anrichterEvol} have done.
|
||||||
|
|
||||||
As we transfer the results of Richter et al.\cite{anrichterEvol} from using
|
As we transfer the results of Richter et al.\cite{anrichterEvol} from using
|
||||||
\acf{RBF} as a representation to manipulate geometric objects to the use of
|
\acf{RBF} as a representation to manipulate geometric objects to the use of
|
||||||
\acf{FFD} we will use the same definition for evolvability the original author
|
\acf{FFD} we will use the same definition for *evolvability* the original author
|
||||||
used, namely *regularity*, *variability*, and *improvement potential*. We
|
used, namely *regularity*, *variability*, and *improvement potential*. We
|
||||||
introduce these term in detail in Chapter \ref{sec:intro:rvi}. In the original
|
introduce these term in detail in Chapter \ref{sec:intro:rvi}. In the original
|
||||||
publication the author could show a correlation between these
|
publication the author could show a correlation between these
|
||||||
@ -95,7 +95,7 @@ evolvability--criteria with the quality and convergence speed of such
|
|||||||
optimization.
|
optimization.
|
||||||
|
|
||||||
We will replicate the same setup on the same objects but use \acf{FFD} instead of
|
We will replicate the same setup on the same objects but use \acf{FFD} instead of
|
||||||
\acf{RBF} to create a local deformation near the control points and evaluate if
|
\acf{RBF} to create a local deformation near the control--points and evaluate if
|
||||||
the evolution--criteria still work as a predictor for *evolvability* of the
|
the evolution--criteria still work as a predictor for *evolvability* of the
|
||||||
representation given the different deformation scheme, as suspected in
|
representation given the different deformation scheme, as suspected in
|
||||||
\cite{anrichterEvol}.
|
\cite{anrichterEvol}.
|
||||||
@ -106,8 +106,8 @@ take an abstract look at the definition of \ac{FFD} for a one--dimensional line
|
|||||||
(in \ref{sec:back:ffdgood}).
|
(in \ref{sec:back:ffdgood}).
|
||||||
Then we establish some background--knowledge of evolutionary algorithms (in
|
Then we establish some background--knowledge of evolutionary algorithms (in
|
||||||
\ref{sec:back:evo}) and why this is useful in our domain (in
|
\ref{sec:back:evo}) and why this is useful in our domain (in
|
||||||
\ref{sec:back:evogood}) followed by the definition of the different evolvability
|
\ref{sec:back:evogood}) followed by the definition of the different
|
||||||
criteria established in \cite{anrichterEvol} (in \ref {sec:intro:rvi}).
|
evolvability--criteria established in \cite{anrichterEvol} (in \ref {sec:intro:rvi}).
|
||||||
|
|
||||||
In Chapter \ref{sec:impl} we take a look at our implementation of \ac{FFD} and
|
In Chapter \ref{sec:impl} we take a look at our implementation of \ac{FFD} and
|
||||||
the adaptation for 3D--meshes that were used. Next, in Chapter \ref{sec:eval},
|
the adaptation for 3D--meshes that were used. Next, in Chapter \ref{sec:eval},
|
||||||
@ -132,19 +132,20 @@ chapter \ref{3dffd}.
|
|||||||
|
|
||||||
The main idea of \ac{FFD} is to create a function $s : [0,1[^d \mapsto
|
The main idea of \ac{FFD} is to create a function $s : [0,1[^d \mapsto
|
||||||
\mathbb{R}^d$ that spans a certain part of a vector--space and is only linearly
|
\mathbb{R}^d$ that spans a certain part of a vector--space and is only linearly
|
||||||
parametrized by some special control points $p_i$ and an constant
|
parametrized by some special control--points $p_i$ and an constant
|
||||||
attribution--function $a_i(u)$, so
|
attribution--function $a_i(u)$, so
|
||||||
$$
|
$$
|
||||||
s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}
|
s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}
|
||||||
$$
|
$$
|
||||||
can be thought of a representation of the inside of the convex hull generated by
|
can be thought of a representation of the inside of the convex hull generated by
|
||||||
the control points where each point can be accessed by the right $u \in [0,1[^d$.
|
the control--points where each position inside can be accessed by the right
|
||||||
|
$u \in [0,1[^d$.
|
||||||
|
|
||||||
\begin{figure}[!ht]
|
\begin{figure}[!ht]
|
||||||
\begin{center}
|
\begin{center}
|
||||||
\includegraphics[width=0.7\textwidth]{img/B-Splines.png}
|
\includegraphics[width=0.7\textwidth]{img/B-Splines.png}
|
||||||
\end{center}
|
\end{center}
|
||||||
\caption[Example of B-Splines]{Example of a parametrization of a line with
|
\caption[Example of B--Splines]{Example of a parametrization of a line with
|
||||||
corresponding deformation to generate a deformed objet}
|
corresponding deformation to generate a deformed objet}
|
||||||
\label{fig:bspline}
|
\label{fig:bspline}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
@ -184,7 +185,7 @@ $$\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,
|
|||||||
|
|
||||||
For a B--Spline
|
For a B--Spline
|
||||||
$$s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i$$
|
$$s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i$$
|
||||||
these derivations yield $\frac{\partial^d}{\partial u} s(u) = 0$.
|
these derivations yield $\left(\frac{\partial}{\partial u}\right)^d s(u) = 0$.
|
||||||
|
|
||||||
Another interesting property of these recursive polynomials is that they are
|
Another interesting property of these recursive polynomials is that they are
|
||||||
continuous (given $d \ge 1$) as every $p_i$ gets blended in between $\tau_i$ and
|
continuous (given $d \ge 1$) as every $p_i$ gets blended in between $\tau_i$ and
|
||||||
@ -193,21 +194,21 @@ in every step of the recursion.
|
|||||||
|
|
||||||
This means that all changes are only a local linear combination between the
|
This means that all changes are only a local linear combination between the
|
||||||
control--point $p_i$ to $p_{i+d+1}$ and consequently this yields to the
|
control--point $p_i$ to $p_{i+d+1}$ and consequently this yields to the
|
||||||
convex--hull--property of B-Splines --- meaning, that no matter how we choose
|
convex--hull--property of B--Splines --- meaning, that no matter how we choose
|
||||||
our coefficients, the resulting points all have to lie inside convex--hull of
|
our coefficients, the resulting points all have to lie inside convex--hull of
|
||||||
the control--points.
|
the control--points.
|
||||||
|
|
||||||
For a given point $v_i$ we can then calculate the contributions
|
For a given point $s_i$ we can then calculate the contributions
|
||||||
$n_{i,j}~:=~N_{j,d,\tau}$ of each control point $p_j$ to get the
|
$u_{i,j}~:=~N_{j,d,\tau}$ of each control point $p_j$ to get the
|
||||||
projection from the control--point--space into the object--space:
|
projection from the control--point--space into the object--space:
|
||||||
$$
|
$$
|
||||||
v_i = \sum_j n_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p}
|
s_i = \sum_j u_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p}
|
||||||
$$
|
$$
|
||||||
or written for all points at the same time:
|
or written for all points at the same time:
|
||||||
$$
|
$$
|
||||||
\vec{v} = \vec{N} \vec{p}
|
\vec{s} = \vec{U} \vec{p}
|
||||||
$$
|
$$
|
||||||
where $\vec{N}$ is the $n \times m$ transformation--matrix (later on called
|
where $\vec{U}$ is the $n \times m$ transformation--matrix (later on called
|
||||||
**deformation matrix**) for $n$ object--space--points and $m$ control--points.
|
**deformation matrix**) for $n$ object--space--points and $m$ control--points.
|
||||||
|
|
||||||
\begin{figure}[ht]
|
\begin{figure}[ht]
|
||||||
@ -220,7 +221,7 @@ of the B--spline ($[k_0,k_4]$ on this figure), the B--Spline basis functions sum
|
|||||||
up to one (partition of unity). In this example, we use B--Splines of degree 2.
|
up to one (partition of unity). In this example, we use B--Splines of degree 2.
|
||||||
The horizontal segment below the abscissa axis represents the domain of
|
The horizontal segment below the abscissa axis represents the domain of
|
||||||
influence of the B--splines basis function, i.e. the interval on which they are
|
influence of the B--splines basis function, i.e. the interval on which they are
|
||||||
not null. At a given point, there are at most $ d+1$ non-zero B--Spline basis
|
not null. At a given point, there are at most $ d+1$ non--zero B--Spline basis
|
||||||
functions (compact support).\grqq \newline
|
functions (compact support).\grqq \newline
|
||||||
Note, that Brunet starts his index at $-d$ opposed to our definition, where we
|
Note, that Brunet starts his index at $-d$ opposed to our definition, where we
|
||||||
start at $0$.}
|
start at $0$.}
|
||||||
@ -228,8 +229,8 @@ start at $0$.}
|
|||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
Furthermore B--Spline--basis--functions form a partition of unity for all, but
|
Furthermore B--Spline--basis--functions form a partition of unity for all, but
|
||||||
the first and last $d$ control-points\cite{brunet2010contributions}. Therefore
|
the first and last $d$ control--points\cite{brunet2010contributions}. Therefore
|
||||||
we later on use the border-points $d+1$ times, such that $\sum_j n_{i,j} p_j = p_i$
|
we later on use the border--points $d+1$ times, such that $\sum_j u_{i,j} p_j = p_i$
|
||||||
for these points.
|
for these points.
|
||||||
|
|
||||||
The locality of the influence of each control--point and the partition of unity
|
The locality of the influence of each control--point and the partition of unity
|
||||||
@ -240,8 +241,8 @@ was beautifully pictured by Brunet, which we included here as figure
|
|||||||
\label{sec:back:ffdgood}
|
\label{sec:back:ffdgood}
|
||||||
|
|
||||||
The usage of \ac{FFD} as a tool for manipulating follows directly from the
|
The usage of \ac{FFD} as a tool for manipulating follows directly from the
|
||||||
properties of the polynomials and the correspondence to the control points.
|
properties of the polynomials and the correspondence to the control--points.
|
||||||
Having only a few control points gives the user a nicer high--level--interface, as
|
Having only a few control--points gives the user a nicer high--level--interface, as
|
||||||
she only needs to move these points and the model follows in an intuitive
|
she only needs to move these points and the model follows in an intuitive
|
||||||
manner. The deformation is smooth as the underlying polygon is smooth as well
|
manner. The deformation is smooth as the underlying polygon is smooth as well
|
||||||
and affects as many vertices of the model as needed. Moreover the changes are
|
and affects as many vertices of the model as needed. Moreover the changes are
|
||||||
@ -317,7 +318,7 @@ the *phenotypes* make certain behaviour observable (algorithmically through our
|
|||||||
*fitness--function*, biologically by the ability to survive and produce
|
*fitness--function*, biologically by the ability to survive and produce
|
||||||
offspring). Any individual in our algorithm thus experience a biologically
|
offspring). Any individual in our algorithm thus experience a biologically
|
||||||
motivated life cycle of inheriting genes from the parents, modified by mutations
|
motivated life cycle of inheriting genes from the parents, modified by mutations
|
||||||
occurring, performing according to a fitness--metric and generating offspring
|
occurring, performing according to a fitness--metric, and generating offspring
|
||||||
based on this. Therefore each iteration in the while--loop above is also often
|
based on this. Therefore each iteration in the while--loop above is also often
|
||||||
named generation.
|
named generation.
|
||||||
|
|
||||||
@ -346,7 +347,7 @@ The main algorithm just repeats the following steps:
|
|||||||
- **Selection** takes a selection--function $s : (I^\lambda \cup I^{\mu + \lambda},\Phi) \mapsto I^\mu$ that
|
- **Selection** takes a selection--function $s : (I^\lambda \cup I^{\mu + \lambda},\Phi) \mapsto I^\mu$ that
|
||||||
selects from the previously generated $I^\lambda$ children and optionally also
|
selects from the previously generated $I^\lambda$ children and optionally also
|
||||||
the parents (denoted by the set $Q$ in the algorithm) using the
|
the parents (denoted by the set $Q$ in the algorithm) using the
|
||||||
fitness--function $\Phi$. The result of this operation is the next Population
|
*fitness--function* $\Phi$. The result of this operation is the next Population
|
||||||
of $\mu$ individuals.
|
of $\mu$ individuals.
|
||||||
|
|
||||||
All these functions can (and mostly do) have a lot of hidden parameters that
|
All these functions can (and mostly do) have a lot of hidden parameters that
|
||||||
@ -370,7 +371,7 @@ also take ancestry, distance of genes or groups of individuals into account.
|
|||||||
\label{sec:back:evogood}
|
\label{sec:back:evogood}
|
||||||
|
|
||||||
The main advantage of evolutionary algorithms is the ability to find optima of
|
The main advantage of evolutionary algorithms is the ability to find optima of
|
||||||
general functions just with the help of a given fitness--function. Components
|
general functions just with the help of a given *fitness--function*. Components
|
||||||
and techniques for evolutionary algorithms are specifically known to
|
and techniques for evolutionary algorithms are specifically known to
|
||||||
help with different problems arising in the domain of
|
help with different problems arising in the domain of
|
||||||
optimization\cite{weise2012evolutionary}. An overview of the typical problems
|
optimization\cite{weise2012evolutionary}. An overview of the typical problems
|
||||||
@ -383,13 +384,13 @@ are shown in figure \ref{fig:probhard}.
|
|||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
Most of the advantages stem from the fact that a gradient--based procedure has
|
Most of the advantages stem from the fact that a gradient--based procedure has
|
||||||
only one point of observation from where it evaluates the next steps, whereas an
|
usually only one point of observation from where it evaluates the next steps,
|
||||||
evolutionary strategy starts with a population of guessed solutions. Because an
|
whereas an evolutionary strategy starts with a population of guessed solutions.
|
||||||
evolutionary strategy can be modified according to the problem--domain (i.e. by
|
Because an evolutionary strategy can be modified according to the
|
||||||
the ideas given above) it can also approximate very difficult problems in an
|
problem--domain (i.e. by the ideas given above) it can also approximate very
|
||||||
efficient manner and even self--tune parameters depending on the ancestry at
|
difficult problems in an efficient manner and even self--tune parameters
|
||||||
runtime^[Some examples of this are explained in detail in
|
depending on the ancestry at runtime^[Some examples of this are explained in
|
||||||
\cite{eiben1999parameter}].
|
detail in \cite{eiben1999parameter}].
|
||||||
|
|
||||||
If an analytic best solution exists and is easily computable (i.e. because the
|
If an analytic best solution exists and is easily computable (i.e. because the
|
||||||
error--function is convex) an evolutionary algorithm is not the right choice.
|
error--function is convex) an evolutionary algorithm is not the right choice.
|
||||||
@ -421,23 +422,23 @@ coordinates
|
|||||||
$$
|
$$
|
||||||
\Delta \vec{S} = \vec{U} \cdot \Delta \vec{P}
|
\Delta \vec{S} = \vec{U} \cdot \Delta \vec{P}
|
||||||
$$
|
$$
|
||||||
which is isomorphic to the former due to the linear correlation in the
|
which is isomorphic to the former due to the linearity of the deformation. One
|
||||||
deformation. One can see in this way, that the way the deformation behaves lies
|
can see in this way, that the way the deformation behaves lies solely in the
|
||||||
solely in the entries of $\vec{U}$, which is why the three criteria focus on this.
|
entries of $\vec{U}$, which is why the three criteria focus on this.
|
||||||
|
|
||||||
### Variability
|
### Variability
|
||||||
|
|
||||||
In \cite{anrichterEvol} *variability* is defined as
|
In \cite{anrichterEvol} *variability* is defined as
|
||||||
$$\mathrm{variability}(\vec{U}) := \frac{\mathrm{rank}(\vec{U})}{n},$$
|
$$\mathrm{variability}(\vec{U}) := \frac{\mathrm{rank}(\vec{U})}{n},$$
|
||||||
whereby $\vec{U}$ is the $n \times m$ deformation--Matrix used to map the $m$
|
whereby $\vec{U}$ is the $n \times m$ deformation--Matrix used to map the $m$
|
||||||
control points onto the $n$ vertices.
|
control--points onto the $n$ vertices.
|
||||||
|
|
||||||
Given $n = m$, an identical number of control--points and vertices, this
|
Given $n = m$, an identical number of control--points and vertices, this
|
||||||
quotient will be $=1$ if all control points are independent of each other and
|
quotient will be $=1$ if all control--points are independent of each other and
|
||||||
the solution is to trivially move every control--point onto a target--point.
|
the solution is to trivially move every control--point onto a target--point.
|
||||||
|
|
||||||
In praxis the value of $V(\vec{U})$ is typically $\ll 1$, because as
|
In praxis the value of $V(\vec{U})$ is typically $\ll 1$, because there are only
|
||||||
there are only few control--points for many vertices, so $m \ll n$.
|
few control--points for many vertices, so $m \ll n$.
|
||||||
|
|
||||||
This criterion should correlate to the degrees of freedom the given
|
This criterion should correlate to the degrees of freedom the given
|
||||||
parametrization has. This can be seen from the fact, that
|
parametrization has. This can be seen from the fact, that
|
||||||
@ -459,7 +460,7 @@ value of the deformation--matrix $\vec{U}$.
|
|||||||
As we deform the given Object only based on the parameters as $\vec{p} \mapsto
|
As we deform the given Object only based on the parameters as $\vec{p} \mapsto
|
||||||
f(\vec{x} + \vec{U}\vec{p})$ this makes sure that $\|\vec{Up}\| \propto
|
f(\vec{x} + \vec{U}\vec{p})$ this makes sure that $\|\vec{Up}\| \propto
|
||||||
\|\vec{p}\|$ when $\kappa(\vec{U}) \approx 1$. The inversion of $\kappa(\vec{U})$
|
\|\vec{p}\|$ when $\kappa(\vec{U}) \approx 1$. The inversion of $\kappa(\vec{U})$
|
||||||
is only performed to map the criterion--range to $[0..1]$, whereas $1$ is the
|
is only performed to map the criterion--range to $[0..1]$, where $1$ is the
|
||||||
optimal value and $0$ is the worst value.
|
optimal value and $0$ is the worst value.
|
||||||
|
|
||||||
On the one hand this criterion should be characteristic for numeric
|
On the one hand this criterion should be characteristic for numeric
|
||||||
@ -470,7 +471,7 @@ to the notion of locality\cite{weise2012evolutionary,thorhauer2014locality}.
|
|||||||
### Improvement Potential
|
### Improvement Potential
|
||||||
|
|
||||||
In contrast to the general nature of *variability* and *regularity*, which are
|
In contrast to the general nature of *variability* and *regularity*, which are
|
||||||
agnostic of the fitness--function at hand, the third criterion should reflect a
|
agnostic of the *fitness--function* at hand, the third criterion should reflect a
|
||||||
notion of the potential for optimization, taking a guess into account.
|
notion of the potential for optimization, taking a guess into account.
|
||||||
|
|
||||||
Most of the times some kind of gradient $g$ is available to suggest a
|
Most of the times some kind of gradient $g$ is available to suggest a
|
||||||
@ -509,7 +510,7 @@ As we have established in Chapter \ref{sec:back:ffd} we can define an
|
|||||||
\Delta_x(u) = \sum_i N_{i,d,\tau_i}(u) \Delta_x c_i
|
\Delta_x(u) = \sum_i N_{i,d,\tau_i}(u) \Delta_x c_i
|
||||||
\end{equation}
|
\end{equation}
|
||||||
|
|
||||||
Note that we only sum up the $\Delta$--displacements in the control points $c_i$ to get
|
Note that we only sum up the $\Delta$--displacements in the control--points $c_i$ to get
|
||||||
the change in position of the point we are interested in.
|
the change in position of the point we are interested in.
|
||||||
|
|
||||||
In this way every deformed vertex is defined by
|
In this way every deformed vertex is defined by
|
||||||
@ -539,8 +540,8 @@ and do a gradient--descend to approximate the value of $u$ up to an $\epsilon$ o
|
|||||||
|
|
||||||
For this we employ the Gauss--Newton algorithm\cite{gaussNewton}, which
|
For this we employ the Gauss--Newton algorithm\cite{gaussNewton}, which
|
||||||
converges into the least--squares solution. An exact solution of this problem is
|
converges into the least--squares solution. An exact solution of this problem is
|
||||||
impossible most of the times, because we usually have way more vertices
|
impossible most of the time, because we usually have way more vertices
|
||||||
than control points ($\#v~\gg~\#c$).
|
than control--points ($\#v~\gg~\#c$).
|
||||||
|
|
||||||
## Adaption of \ac{FFD} for a 3D--Mesh
|
## Adaption of \ac{FFD} for a 3D--Mesh
|
||||||
\label{3dffd}
|
\label{3dffd}
|
||||||
@ -550,7 +551,7 @@ chapter. But this time things get a bit more complicated. As we have a
|
|||||||
3--dimensional grid we may have a different amount of control--points in each
|
3--dimensional grid we may have a different amount of control--points in each
|
||||||
direction.
|
direction.
|
||||||
|
|
||||||
Given $n,m,o$ control points in $x,y,z$--direction each Point on the curve is
|
Given $n,m,o$ control--points in $x,y,z$--direction each Point on the curve is
|
||||||
defined by
|
defined by
|
||||||
$$V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.$$
|
$$V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.$$
|
||||||
|
|
||||||
@ -624,13 +625,13 @@ beneficial for a good behaviour of the evolutionary algorithm.
|
|||||||
|
|
||||||
As mentioned in chapter \ref{sec:back:evo}, the way of choosing the
|
As mentioned in chapter \ref{sec:back:evo}, the way of choosing the
|
||||||
representation to map the general problem (mesh--fitting/optimization in our
|
representation to map the general problem (mesh--fitting/optimization in our
|
||||||
case) into a parameter-space is very important for the quality and runtime of
|
case) into a parameter--space is very important for the quality and runtime of
|
||||||
evolutionary algorithms\cite{Rothlauf2006}.
|
evolutionary algorithms\cite{Rothlauf2006}.
|
||||||
|
|
||||||
Because our control--points are arranged in a grid, we can accurately represent
|
Because our control--points are arranged in a grid, we can accurately represent
|
||||||
each vertex--point inside the grids volume with proper B--Spline--coefficients
|
each vertex--point inside the grids volume with proper B--Spline--coefficients
|
||||||
between $[0,1[$ and --- as a consequence --- we have to embed our object into it
|
between $[0,1[$ and --- as a consequence --- we have to embed our object into it
|
||||||
(or create constant "dummy"-points outside).
|
(or create constant "dummy"--points outside).
|
||||||
|
|
||||||
The great advantage of B--Splines is the local, direct impact of each
|
The great advantage of B--Splines is the local, direct impact of each
|
||||||
control point without having a $1:1$--correlation, and a smooth deformation.
|
control point without having a $1:1$--correlation, and a smooth deformation.
|
||||||
@ -651,20 +652,20 @@ control--points.}
|
|||||||
One would normally think, that the more control--points you add, the better the
|
One would normally think, that the more control--points you add, the better the
|
||||||
result will be, but this is not the case for our B--Splines. Given any point
|
result will be, but this is not the case for our B--Splines. Given any point
|
||||||
$\vec{p}$ only the $2 \cdot (d-1)$ control--points contribute to the parametrization of
|
$\vec{p}$ only the $2 \cdot (d-1)$ control--points contribute to the parametrization of
|
||||||
that point^[Normally these are $d-1$ to each side, but at the boundaries the
|
that point^[Normally these are $d-1$ to each side, but at the boundaries border
|
||||||
number gets increased to the inside to meet the required smoothness].
|
points get used multiple times to meet the number of points required].
|
||||||
This means, that a high resolution can have many control-points that are not
|
This means, that a high resolution can have many control--points that are not
|
||||||
contributing to any point on the surface and are thus completely irrelevant to
|
contributing to any point on the surface and are thus completely irrelevant to
|
||||||
the solution.
|
the solution.
|
||||||
|
|
||||||
We illustrate this phenomenon in figure \ref{fig:enoughCP}, where the four red
|
We illustrate this phenomenon in figure \ref{fig:enoughCP}, where the red
|
||||||
central points are not relevant for the parametrization of the circle. This
|
central points are not relevant for the parametrization of the circle. This
|
||||||
leads to artefacts in the deformation--matrix $\vec{U}$, as the columns
|
leads to artefacts in the deformation--matrix $\vec{U}$, as the columns
|
||||||
corresponding to those control--points are $0$.
|
corresponding to those control--points are $0$.
|
||||||
|
|
||||||
This leads to useless increased complexity, as the parameters corresponding to
|
This also leads to useless increased complexity, as the parameters corresponding
|
||||||
those points will never have any effect, but a naive algorithm will still try to
|
to those points will never have any effect, but a naive algorithm will still try
|
||||||
optimize them yielding numeric artefacts in the best and non--terminating or
|
to optimize them yielding numeric artefacts in the best and non--terminating or
|
||||||
ill--defined solutions^[One example would be, when parts of an algorithm depend
|
ill--defined solutions^[One example would be, when parts of an algorithm depend
|
||||||
on the inverse of the minimal right singular value leading to a division by $0$.]
|
on the inverse of the minimal right singular value leading to a division by $0$.]
|
||||||
at worst.
|
at worst.
|
||||||
@ -674,18 +675,18 @@ but this raises the question why they were introduced in the first place. We
|
|||||||
will address this in a special scenario in \ref{sec:res:3d:var}.
|
will address this in a special scenario in \ref{sec:res:3d:var}.
|
||||||
|
|
||||||
For our tests we chose different uniformly sized grids and added noise
|
For our tests we chose different uniformly sized grids and added noise
|
||||||
onto each control-point^[For the special case of the outer layer we only applied
|
onto each control--point^[For the special case of the outer layer we only applied
|
||||||
noise away from the object, so the object is still confined in the convex hull
|
noise away from the object, so the object is still confined in the convex hull
|
||||||
of the control--points.] to simulate different starting-conditions.
|
of the control--points.] to simulate different starting--conditions.
|
||||||
|
|
||||||
# Scenarios for testing evolvability criteria using \ac{FFD}
|
# Scenarios for testing evolvability--criteria using \ac{FFD}
|
||||||
\label{sec:eval}
|
\label{sec:eval}
|
||||||
|
|
||||||
In our experiments we use the same two testing--scenarios, that were also used
|
In our experiments we use the same two testing--scenarios, that were also used
|
||||||
by \cite{anrichterEvol}. The first scenario deforms a plane into a shape
|
by Richter et al.\cite{anrichterEvol} The first scenario deforms a plane into a shape
|
||||||
originally defined in \cite{giannelli2012thb}, where we setup control-points in
|
originally defined by Giannelli et al.\cite{giannelli2012thb}, where we setup
|
||||||
a 2--dimensional manner and merely deform in the height--coordinate to get the
|
control--points in a 2--dimensional manner and merely deform in the
|
||||||
resulting shape.
|
height--coordinate to get the resulting shape.
|
||||||
|
|
||||||
In the second scenario we increase the degrees of freedom significantly by using
|
In the second scenario we increase the degrees of freedom significantly by using
|
||||||
a 3--dimensional control--grid to deform a sphere into a face, so each control
|
a 3--dimensional control--grid to deform a sphere into a face, so each control
|
||||||
@ -717,7 +718,7 @@ including a wireframe--overlay of the vertices.}
|
|||||||
\label{fig:1dtarget}
|
\label{fig:1dtarget}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
As the starting-plane we used the same shape, but set all
|
As the starting--plane we used the same shape, but set all
|
||||||
$z$--coordinates to $0$, yielding a flat plane, which is partially already
|
$z$--coordinates to $0$, yielding a flat plane, which is partially already
|
||||||
correct.
|
correct.
|
||||||
|
|
||||||
@ -728,10 +729,10 @@ of calculating the squared distances for each corresponding vertex
|
|||||||
\end{equation}
|
\end{equation}
|
||||||
where $t_i$ are the respective target--vertices to the parametrized
|
where $t_i$ are the respective target--vertices to the parametrized
|
||||||
source--vertices^[The parametrization is encoded in $\vec{U}$ and the initial
|
source--vertices^[The parametrization is encoded in $\vec{U}$ and the initial
|
||||||
position of the control points. See \ref{sec:ffd:adapt}] with the current
|
position of the control--points. See \ref{sec:ffd:adapt}] with the current
|
||||||
deformation--parameters $\vec{p} = (p_1,\dots, p_m)$. We can do this
|
deformation--parameters $\vec{p} = (p_1,\dots, p_m)$. We can do this
|
||||||
one--to--one--correspondence because we have exactly the same number of
|
one--to--one--correspondence because we have exactly the same number of
|
||||||
source and target-vertices do to our setup of just flattening the object.
|
source and target--vertices do to our setup of just flattening the object.
|
||||||
|
|
||||||
This formula is also the least--squares approximation error for which we
|
This formula is also the least--squares approximation error for which we
|
||||||
can compute the analytic solution $\vec{p^{*}} = \vec{U^+}\vec{t}$, yielding us
|
can compute the analytic solution $\vec{p^{*}} = \vec{U^+}\vec{t}$, yielding us
|
||||||
@ -762,16 +763,16 @@ these Models can be seen in figure \ref{fig:3dtarget}.
|
|||||||
Opposed to the 1D--case we cannot map the source and target--vertices in a
|
Opposed to the 1D--case we cannot map the source and target--vertices in a
|
||||||
one--to--one--correspondence, which we especially need for the approximation of
|
one--to--one--correspondence, which we especially need for the approximation of
|
||||||
the fitting--error. Hence we state that the error of one vertex is the distance
|
the fitting--error. Hence we state that the error of one vertex is the distance
|
||||||
to the closest vertex of the other model and sum up the error from the
|
to the closest vertex of the respective other model and sum up the error from
|
||||||
respective source and target.
|
the source and target.
|
||||||
|
|
||||||
We therefore define the *fitness--function* to be:
|
We therefore define the *fitness--function* to be:
|
||||||
|
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\mathrm{f}(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} -
|
\mathrm{f}(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} -
|
||||||
\vec{s_i}\|_2^2}_{\textrm{source-to-target--distance}}
|
\vec{s_i}\|_2^2}_{\textrm{source--to--target--distance}}
|
||||||
+ \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} -
|
+ \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} -
|
||||||
\vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}}
|
\vec{t_i}\|_2^2}_{\textrm{target--to--source--distance}}
|
||||||
+ \lambda \cdot \textrm{regularization}(\vec{P})
|
+ \lambda \cdot \textrm{regularization}(\vec{P})
|
||||||
\label{eq:fit3d}
|
\label{eq:fit3d}
|
||||||
\end{equation}
|
\end{equation}
|
||||||
@ -787,7 +788,7 @@ $n \times m$--matrix of calculated coefficients for the \ac{FFD} --- analog to
|
|||||||
the 1D case --- and finally $\vec{P}$ being the $m \times 3$--matrix of the
|
the 1D case --- and finally $\vec{P}$ being the $m \times 3$--matrix of the
|
||||||
control--grid defining the whole deformation.
|
control--grid defining the whole deformation.
|
||||||
|
|
||||||
As regularization-term we add a weighted Laplacian of the deformation that has
|
As regularization--term we add a weighted Laplacian of the deformation that has
|
||||||
been used before by Aschenbach et al.\cite[Section 3.2]{aschenbach2015} on
|
been used before by Aschenbach et al.\cite[Section 3.2]{aschenbach2015} on
|
||||||
similar models and was shown to lead to a more precise fit. The Laplacian
|
similar models and was shown to lead to a more precise fit. The Laplacian
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
@ -812,7 +813,7 @@ ill--defined grids mentioned in section \ref{sec:impl:grid}.
|
|||||||
To compare our results to the ones given by Richter et al.\cite{anrichterEvol},
|
To compare our results to the ones given by Richter et al.\cite{anrichterEvol},
|
||||||
we also use Spearman's rank correlation coefficient. Opposed to other popular
|
we also use Spearman's rank correlation coefficient. Opposed to other popular
|
||||||
coefficients, like the Pearson correlation coefficient, which measures a linear
|
coefficients, like the Pearson correlation coefficient, which measures a linear
|
||||||
relationship between variables, the Spearmans's coefficient assesses \glqq how
|
relationship between variables, the Spearman's coefficient assesses \glqq how
|
||||||
well an arbitrary monotonic function can describe the relationship between two
|
well an arbitrary monotonic function can describe the relationship between two
|
||||||
variables, without making any assumptions about the frequency distribution of
|
variables, without making any assumptions about the frequency distribution of
|
||||||
the variables\grqq\cite{hauke2011comparison}.
|
the variables\grqq\cite{hauke2011comparison}.
|
||||||
@ -846,18 +847,19 @@ well. We leave the parameters at their sensible defaults as further explained in
|
|||||||
\label{sec:proc:1d}
|
\label{sec:proc:1d}
|
||||||
|
|
||||||
For our setup we first compute the coefficients of the deformation--matrix and
|
For our setup we first compute the coefficients of the deformation--matrix and
|
||||||
use then the formulas for *variability* and *regularity* to get our predictions.
|
use the formulas for *variability* and *regularity* to get our predictions.
|
||||||
Afterwards we solve the problem analytically to get the (normalized) correct
|
Afterwards we solve the problem analytically to get the (normalized) correct
|
||||||
gradient that we use as guess for the *improvement potential*. To check we also
|
gradient that we use as guess for the *improvement potential*. To further test
|
||||||
consider a distorted gradient $\vec{g}_{\mathrm{d}}$
|
the *improvement potential* we also consider a distorted gradient
|
||||||
|
$\vec{g}_{\mathrm{d}}$:
|
||||||
$$
|
$$
|
||||||
\vec{g}_{\mathrm{d}} = \frac{\mu \vec{g}_{\mathrm{c}} + (1-\mu)\mathbb{1}}{\|\mu \vec{g}_{\mathrm{c}} + (1-\mu) \mathbb{1}\|}
|
\vec{g}_{\mathrm{d}} = \frac{\mu \vec{g}_{\mathrm{c}} + (1-\mu)\mathbb{1}}{\|\mu \vec{g}_{\mathrm{c}} + (1-\mu) \mathbb{1}\|}
|
||||||
$$
|
$$
|
||||||
where $\mathbb{1}$ is the vector consisting of $1$ in every dimension,
|
where $\mathbb{1}$ is the vector consisting of $1$ in every dimension,
|
||||||
$\vec{g}_\mathrm{c} = \vec{p^{*}} - \vec{p}$ is the calculated correct gradient,
|
$\vec{g}_\mathrm{c} = \vec{p^{*}} - \vec{p}$ is the calculated correct gradient,
|
||||||
and $\mu$ is used to blend between $\vec{g}_\mathrm{c}$ and $\mathbb{1}$. As
|
and $\mu$ is used to blend between $\vec{g}_\mathrm{c}$ and $\mathbb{1}$. As
|
||||||
we always start with a gradient of $p = \mathbb{0}$ this means shortens
|
we always start with a gradient of $p = \mathbb{0}$ this means we can shorten
|
||||||
$\vec{g}_\mathrm{c} = \vec{p^{*}}$.
|
the definition of $\vec{g}_\mathrm{c}$ to $\vec{g}_\mathrm{c} = \vec{p^{*}}$.
|
||||||
|
|
||||||
\begin{figure}[ht]
|
\begin{figure}[ht]
|
||||||
\begin{center}
|
\begin{center}
|
||||||
@ -870,9 +872,9 @@ random distortion to generate a testcase.}
|
|||||||
|
|
||||||
We then set up a regular 2--dimensional grid around the object with the desired
|
We then set up a regular 2--dimensional grid around the object with the desired
|
||||||
grid resolutions. To generate a testcase we then move the grid--vertices
|
grid resolutions. To generate a testcase we then move the grid--vertices
|
||||||
randomly inside the x--y--plane. As self-intersecting grids get tricky to solve
|
randomly inside the x--y--plane. As self--intersecting grids get tricky to solve
|
||||||
with our implemented newtons--method we avoid the generation of such
|
with our implemented newtons--method (see section \ref{3dffd}) we avoid the
|
||||||
self--intersecting grids for our testcases (see section \ref{3dffd}).
|
generation of such self--intersecting grids for our testcases.
|
||||||
|
|
||||||
To achieve that we generated a gaussian distributed number with $\mu = 0, \sigma=0.25$
|
To achieve that we generated a gaussian distributed number with $\mu = 0, \sigma=0.25$
|
||||||
and clamped it to the range $[-0.25,0.25]$. We chose such an $r \in [-0.25,0.25]$
|
and clamped it to the range $[-0.25,0.25]$. We chose such an $r \in [-0.25,0.25]$
|
||||||
@ -899,11 +901,11 @@ In the case of our 1D--Optimization--problem, we have the luxury of knowing the
|
|||||||
analytical solution to the given problem--set. We use this to experimentally
|
analytical solution to the given problem--set. We use this to experimentally
|
||||||
evaluate the quality criteria we introduced before. As an evolutional
|
evaluate the quality criteria we introduced before. As an evolutional
|
||||||
optimization is partially a random process, we use the analytical solution as a
|
optimization is partially a random process, we use the analytical solution as a
|
||||||
stopping-criteria. We measure the convergence speed as number of iterations the
|
stopping--criteria. We measure the convergence speed as number of iterations the
|
||||||
evolutional algorithm needed to get within $1.05 \times$ of the optimal solution.
|
evolutional algorithm needed to get within $1.05 \times$ of the optimal solution.
|
||||||
|
|
||||||
We used different regular grids that we manipulated as explained in Section
|
We used different regular grids that we manipulated as explained in Section
|
||||||
\ref{sec:proc:1d} with a different number of control points. As our grids have
|
\ref{sec:proc:1d} with a different number of control--points. As our grids have
|
||||||
to be the product of two integers, we compared a $5 \times 5$--grid with $25$
|
to be the product of two integers, we compared a $5 \times 5$--grid with $25$
|
||||||
control--points to a $4 \times 7$ and $7 \times 4$--grid with $28$
|
control--points to a $4 \times 7$ and $7 \times 4$--grid with $28$
|
||||||
control--points. This was done to measure the impact an \glqq improper\grqq \
|
control--points. This was done to measure the impact an \glqq improper\grqq \
|
||||||
@ -924,7 +926,7 @@ Note that $7 \times 4$ and $4 \times 7$ have the same number of control--points.
|
|||||||
\label{fig:1dvar}
|
\label{fig:1dvar}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
Variability should characterize the potential for design space exploration and
|
*Variability* should characterize the potential for design space exploration and
|
||||||
is defined in terms of the normalized rank of the deformation matrix $\vec{U}$:
|
is defined in terms of the normalized rank of the deformation matrix $\vec{U}$:
|
||||||
$V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n}$, whereby $n$ is the number of
|
$V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n}$, whereby $n$ is the number of
|
||||||
vertices.
|
vertices.
|
||||||
@ -933,27 +935,27 @@ grid), we have merely plotted the errors in the box plot in figure
|
|||||||
\ref{fig:1dvar}
|
\ref{fig:1dvar}
|
||||||
|
|
||||||
It is also noticeable, that although the $7 \times 4$ and $4 \times 7$ grids
|
It is also noticeable, that although the $7 \times 4$ and $4 \times 7$ grids
|
||||||
have a higher variability, they perform not better than the $5 \times 5$ grid.
|
have a higher *variability*, they perform not better than the $5 \times 5$ grid.
|
||||||
Also the $7 \times 4$ and $4 \times 7$ grids differ distinctly from each other
|
Also the $7 \times 4$ and $4 \times 7$ grids differ distinctly from each other
|
||||||
with a mean$\pm$sigma of $233.09 \pm 12.32$ for the former and $286.32 \pm 22.36$ for the
|
with a mean$\pm$sigma of $233.09 \pm 12.32$ for the former and $286.32 \pm 22.36$ for the
|
||||||
latter, although they have the same number of control--points. This is an
|
latter, although they have the same number of control--points. This is an
|
||||||
indication of an impact a proper or improper grid--setup can have. We do not
|
indication of an impact a proper or improper grid--setup can have. We do not
|
||||||
draw scientific conclusions from these findings, as more research on non-squared
|
draw scientific conclusions from these findings, as more research on non--squared
|
||||||
grids seem necessary.
|
grids seem necessary.
|
||||||
|
|
||||||
Leaving the issue of the grid--layout aside we focused on grids having the same
|
Leaving the issue of the grid--layout aside we focused on grids having the same
|
||||||
number of prototypes in every dimension. For the $5 \times 5$, $7 \times 7$ and
|
number of prototypes in every dimension. For the $5 \times 5$, $7 \times 7$ and
|
||||||
$10 \times 10$ grids we found a *very strong* correlation ($-r_S = 0.94, p = 0$)
|
$10 \times 10$ grids we found a *very strong* correlation ($-r_S = 0.94, p = 0$)
|
||||||
between the variability and the evolutionary error.
|
between the *variability* and the evolutionary error.
|
||||||
|
|
||||||
### Regularity
|
### Regularity
|
||||||
|
|
||||||
\begin{figure}[tbh]
|
\begin{figure}[tbh]
|
||||||
\centering
|
\centering
|
||||||
\includegraphics[width=\textwidth]{img/evolution1d/55_to_1010_steps.png}
|
\includegraphics[width=\textwidth]{img/evolution1d/55_to_1010_steps.png}
|
||||||
\caption[Improvement potential and regularity vs. steps]{\newline
|
\caption[Improvement potential and regularity against iterations]{\newline
|
||||||
Left: Improvement potential against steps until convergence\newline
|
Left: *Improvement potential* against number of iterations until convergence\newline
|
||||||
Right: Regularity against steps until convergence\newline
|
Right: *Regularity* against number of iterations until convergence\newline
|
||||||
Coloured by their grid--resolution, both with a linear fit over the whole
|
Coloured by their grid--resolution, both with a linear fit over the whole
|
||||||
dataset.}
|
dataset.}
|
||||||
\label{fig:1dreg}
|
\label{fig:1dreg}
|
||||||
@ -966,15 +968,15 @@ $5 \times 5$ & $7 \times 4$ & $4 \times 7$ & $7 \times 7$ & $10 \times 10$\\
|
|||||||
\hline
|
\hline
|
||||||
$0.28$ ($0.0045$) & \textcolor{red}{$0.21$} ($0.0396$) & \textcolor{red}{$0.1$} ($0.3019$) & \textcolor{red}{$0.01$} ($0.9216$) & \textcolor{red}{$0.01$} ($0.9185$)
|
$0.28$ ($0.0045$) & \textcolor{red}{$0.21$} ($0.0396$) & \textcolor{red}{$0.1$} ($0.3019$) & \textcolor{red}{$0.01$} ($0.9216$) & \textcolor{red}{$0.01$} ($0.9185$)
|
||||||
\end{tabular}
|
\end{tabular}
|
||||||
\caption[Correlation 1D Regularity/Steps]{Spearman's correlation (and p-values)
|
\caption[Correlation 1D *regularity* against iterations]{Inverted Spearman's correlation (and p--values)
|
||||||
between regularity and convergence speed for the 1D function approximation
|
between *regularity* and number of iterations for the 1D function approximation
|
||||||
problem.
|
problem.
|
||||||
\newline Note: Not significant results are marked in \textcolor{red}{red}.
|
\newline Note: Not significant results are marked in \textcolor{red}{red}.
|
||||||
}
|
}
|
||||||
\label{tab:1dreg}
|
\label{tab:1dreg}
|
||||||
\end{table}
|
\end{table}
|
||||||
|
|
||||||
Regularity should correspond to the convergence speed (measured in
|
*Regularity* should correspond to the convergence speed (measured in
|
||||||
iteration--steps of the evolutionary algorithm), and is computed as inverse
|
iteration--steps of the evolutionary algorithm), and is computed as inverse
|
||||||
condition number $\kappa(\vec{U})$ of the deformation--matrix.
|
condition number $\kappa(\vec{U})$ of the deformation--matrix.
|
||||||
|
|
||||||
@ -986,9 +988,9 @@ correlation of $- r_S = -0.72, p = 0$, that is opposed to our expectations.
|
|||||||
|
|
||||||
To explain this discrepancy we took a closer look at what caused these high number
|
To explain this discrepancy we took a closer look at what caused these high number
|
||||||
of iterations. In figure \ref{fig:1dreg} we also plotted the
|
of iterations. In figure \ref{fig:1dreg} we also plotted the
|
||||||
improvement-potential against the steps next to the regularity--plot. Our theory
|
*improvement potential* against the steps next to the *regularity*--plot. Our theory
|
||||||
is that the *very strong* correlation ($-r_S = -0.82, p=0$) between
|
is that the *very strong* correlation ($-r_S = -0.82, p=0$) between
|
||||||
improvement--potential and number of iterations hints that the employed
|
*improvement potential* and number of iterations hints that the employed
|
||||||
algorithm simply takes longer to converge on a better solution (as seen in
|
algorithm simply takes longer to converge on a better solution (as seen in
|
||||||
figure \ref{fig:1dvar} and \ref{fig:1dimp}) offsetting any gain the
|
figure \ref{fig:1dvar} and \ref{fig:1dimp}) offsetting any gain the
|
||||||
regularity--measurement could achieve.
|
regularity--measurement could achieve.
|
||||||
@ -998,14 +1000,14 @@ regularity--measurement could achieve.
|
|||||||
\begin{figure}[ht]
|
\begin{figure}[ht]
|
||||||
\centering
|
\centering
|
||||||
\includegraphics[width=0.8\textwidth]{img/evolution1d/55_to_1010_improvement-vs-evo-error.png}
|
\includegraphics[width=0.8\textwidth]{img/evolution1d/55_to_1010_improvement-vs-evo-error.png}
|
||||||
\caption[Correlation 1D Improvement vs. Error]{Improvement potential plotted
|
\caption[Correlation 1D Improvement vs. Error]{*Improvement potential* plotted
|
||||||
against the error yielded by the evolutionary optimization for different
|
against the error yielded by the evolutionary optimization for different
|
||||||
grid--resolutions}
|
grid--resolutions}
|
||||||
\label{fig:1dimp}
|
\label{fig:1dimp}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
The improvement potential should correlate to the quality of the
|
The *improvement potential* should correlate to the quality of the
|
||||||
fitting--result. We plotted the results for the tested grid-sizes $5 \times 5$,
|
fitting--result. We plotted the results for the tested grid--sizes $5 \times 5$,
|
||||||
$7 \times 7$ and $10 \times 10$ in figure \ref{fig:1dimp}. We tested the
|
$7 \times 7$ and $10 \times 10$ in figure \ref{fig:1dimp}. We tested the
|
||||||
$4 \times 7$ and $7 \times 4$ grids as well, but omitted them from the plot.
|
$4 \times 7$ and $7 \times 4$ grids as well, but omitted them from the plot.
|
||||||
|
|
||||||
@ -1035,7 +1037,7 @@ Initially we set up the correspondences $\vec{c_T(\dots)}$ and $\vec{c_S(\dots)}
|
|||||||
the respectively closest vertices of the other model. We then calculate the
|
the respectively closest vertices of the other model. We then calculate the
|
||||||
analytical solution given these correspondences via $\vec{P^{*}} = \vec{U^+}\vec{T}$,
|
analytical solution given these correspondences via $\vec{P^{*}} = \vec{U^+}\vec{T}$,
|
||||||
and also use the first solution as guessed gradient for the calculation of the
|
and also use the first solution as guessed gradient for the calculation of the
|
||||||
*improvement--potential*, as the optimal solution is not known.
|
*improvement potential*, as the optimal solution is not known.
|
||||||
We then let the evolutionary algorithm run up within $1.05$ times the error of
|
We then let the evolutionary algorithm run up within $1.05$ times the error of
|
||||||
this solution and afterwards recalculate the correspondences $\vec{c_T(\dots)}$
|
this solution and afterwards recalculate the correspondences $\vec{c_T(\dots)}$
|
||||||
and $\vec{c_S(\dots)}$.
|
and $\vec{c_S(\dots)}$.
|
||||||
@ -1063,11 +1065,11 @@ iterations until the regularization--effect wears off.
|
|||||||
|
|
||||||
The grid we use for our experiments is just very coarse due to computational
|
The grid we use for our experiments is just very coarse due to computational
|
||||||
limitations. We are not interested in a good reconstruction, but an estimate if
|
limitations. We are not interested in a good reconstruction, but an estimate if
|
||||||
the mentioned evolvability criteria are good.
|
the mentioned evolvability--criteria are good.
|
||||||
|
|
||||||
In figure \ref{fig:setup3d} we show an example setup of the scene with a
|
In figure \ref{fig:setup3d} we show an example setup of the scene with a
|
||||||
$4\times 4\times 4$--grid. Identical to the 1--dimensional scenario before, we create a
|
$4\times 4\times 4$--grid. Identical to the 1--dimensional scenario before, we create a
|
||||||
regular grid and move the control-points in the exact same random manner between
|
regular grid and move the control--points in the exact same random manner between
|
||||||
their neighbours as described in section \ref{sec:proc:1d}, but in three instead
|
their neighbours as described in section \ref{sec:proc:1d}, but in three instead
|
||||||
of two dimensions^[Again, we flip the signs for the edges, if necessary to have
|
of two dimensions^[Again, we flip the signs for the edges, if necessary to have
|
||||||
the object still in the convex hull.].
|
the object still in the convex hull.].
|
||||||
@ -1083,16 +1085,16 @@ Right: A $4 \times 4 \times 7$ grid that we expect to perform worse.}
|
|||||||
|
|
||||||
As is clearly visible from figure \ref{fig:3dgridres}, the target--model has many
|
As is clearly visible from figure \ref{fig:3dgridres}, the target--model has many
|
||||||
vertices in the facial area, at the ears and in the neck--region. Therefore we
|
vertices in the facial area, at the ears and in the neck--region. Therefore we
|
||||||
chose to increase the grid-resolutions for our tests in two different dimensions
|
chose to increase the grid--resolutions for our tests in two different dimensions
|
||||||
and see how well the criteria predict a suboptimal placement of these
|
and see how well the criteria predict a suboptimal placement of these
|
||||||
control-points.
|
control--points.
|
||||||
|
|
||||||
## Results of 3D Function Approximation
|
## Results of 3D Function Approximation
|
||||||
|
|
||||||
In the 3D--Approximation we tried to evaluate further on the impact of the
|
In the 3D--Approximation we tried to evaluate further on the impact of the
|
||||||
grid--layout to the overall criteria. As the target--model has many vertices in
|
grid--layout to the overall criteria. As the target--model has many vertices in
|
||||||
concentrated in the facial area we start from a $4 \times 4 \times 4$ grid and
|
concentrated in the facial area we start from a $4 \times 4 \times 4$ grid and
|
||||||
only increase the number of control points in one dimension, yielding a
|
only increase the number of control--points in one dimension, yielding a
|
||||||
resolution of $7 \times 4 \times 4$ and $4 \times 4 \times 7$ respectively. We
|
resolution of $7 \times 4 \times 4$ and $4 \times 4 \times 7$ respectively. We
|
||||||
visualized those two grids in figure \ref{fig:3dgridres}.
|
visualized those two grids in figure \ref{fig:3dgridres}.
|
||||||
|
|
||||||
@ -1121,10 +1123,10 @@ $4 \times 4 \times \mathrm{X}$ & $\mathrm{X} \times 4 \times 4$ & $\mathrm{Y} \t
|
|||||||
\hline
|
\hline
|
||||||
0.89 (0) & 0.9 (0) & 0.91 (0) & 0.94 (0)
|
0.89 (0) & 0.9 (0) & 0.91 (0) & 0.94 (0)
|
||||||
\end{tabular}
|
\end{tabular}
|
||||||
\caption[Correlation between variability and fitting error for 3D]{Correlation
|
\caption[Correlation between *variability* and fitting error for 3D]{Correlation
|
||||||
between variability and fitting error for the 3D fitting scenario.\newline
|
between *variability* and fitting error for the 3D fitting scenario.\newline
|
||||||
Displayed are the negated Spearman coefficients with the corresponding p-values
|
Displayed are the negated Spearman coefficients with the corresponding p--values
|
||||||
in brackets for three cases of increasing variability ($\mathrm{X} \in [4,5,7],
|
in brackets for three cases of increasing *variability* ($\mathrm{X} \in [4,5,7],
|
||||||
\mathrm{Y} \in [4,5,6]$).
|
\mathrm{Y} \in [4,5,6]$).
|
||||||
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
|
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
|
||||||
\label{tab:3dvar}
|
\label{tab:3dvar}
|
||||||
@ -1143,37 +1145,37 @@ Interestingly both variants end up closer in terms of fitting error than we
|
|||||||
anticipated, which shows that the evolutionary algorithm we employed is capable
|
anticipated, which shows that the evolutionary algorithm we employed is capable
|
||||||
of correcting a purposefully created \glqq bad\grqq \ grid. Also this confirms,
|
of correcting a purposefully created \glqq bad\grqq \ grid. Also this confirms,
|
||||||
that in our cases the number of control--points is more important for quality
|
that in our cases the number of control--points is more important for quality
|
||||||
than their placement, which is captured by the variability via the rank of the
|
than their placement, which is captured by the *variability* via the rank of the
|
||||||
deformation--matrix.
|
deformation--matrix.
|
||||||
|
|
||||||
|
Overall the correlation between *variability* and fitness--error were
|
||||||
|
*significant* and showed a *very strong* correlation in all our tests.
|
||||||
|
The detailed correlation--coefficients are given in table \ref{tab:3dvar}
|
||||||
|
alongside their p--values.
|
||||||
|
|
||||||
|
As introduces in section \ref{sec:impl:grid} and visualized in figure
|
||||||
|
\ref{fig:enoughCP}, we know, that not all control--points have to necessarily
|
||||||
|
contribute to the parametrization of our 3D--model. Because we are starting from
|
||||||
|
a sphere, some control--points are too far away from the surface to contribute
|
||||||
|
to the deformation at all.
|
||||||
|
|
||||||
|
One can already see in 2D in figure \ref{fig:enoughCP}, that this effect
|
||||||
|
starts with a regular $9 \times 9$ grid on a perfect circle. To make sure we
|
||||||
|
observe this, we evaluated the *variability* for 100 randomly moved $10 \times 10 \times 10$
|
||||||
|
grids on the sphere we start out with.
|
||||||
|
|
||||||
\begin{figure}[hbt]
|
\begin{figure}[hbt]
|
||||||
\centering
|
\centering
|
||||||
\includegraphics[width=0.8\textwidth]{img/evolution3d/variability2_boxplot.png}
|
\includegraphics[width=0.8\textwidth]{img/evolution3d/variability2_boxplot.png}
|
||||||
\caption[Histogram of ranks of high--resolution deformation--matrices]{
|
\caption[Histogram of ranks of high--resolution deformation--matrices]{
|
||||||
Histogram of ranks of various $10 \times 10 \times 10$ grids with $1000$
|
Histogram of ranks of various $10 \times 10 \times 10$ grids with $1000$
|
||||||
control--points each showing in this case how many control points are actually
|
control--points each showing in this case how many control--points are actually
|
||||||
used in the calculations.
|
used in the calculations.
|
||||||
}
|
}
|
||||||
\label{fig:histrank3d}
|
\label{fig:histrank3d}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
Overall the correlation between variability and fitness--error were
|
As the *variability* is defined by $\frac{\mathrm{rank}(\vec{U})}{n}$ we can
|
||||||
*significant* and showed a *very strong* correlation in all our tests.
|
|
||||||
The detailed correlation--coefficients are given in table \ref{tab:3dvar}
|
|
||||||
alongside their p--values.
|
|
||||||
|
|
||||||
As introduces in section \ref{sec:impl:grid} and visualized in figure
|
|
||||||
\ref{fig:enoughCP}, we know, that not all control points have to necessarily
|
|
||||||
contribute to the parametrization of our 3D--model. Because we are starting from
|
|
||||||
a sphere, some control-points are too far away from the surface to contribute
|
|
||||||
to the deformation at all.
|
|
||||||
|
|
||||||
One can already see in 2D in figure \ref{fig:enoughCP}, that this effect
|
|
||||||
starts with a regular $9 \times 9$ grid on a perfect circle. To make sure we
|
|
||||||
observe this, we evaluated the variability for 100 randomly moved $10 \times 10 \times 10$
|
|
||||||
grids on the sphere we start out with.
|
|
||||||
|
|
||||||
As the variability is defined by $\frac{\mathrm{rank}(\vec{U})}{n}$ we can
|
|
||||||
easily recover the rank of the deformation--matrix $\vec{U}$. The results are
|
easily recover the rank of the deformation--matrix $\vec{U}$. The results are
|
||||||
shown in the histogram in figure \ref{fig:histrank3d}. Especially in the centre
|
shown in the histogram in figure \ref{fig:histrank3d}. Especially in the centre
|
||||||
of the sphere and in the corners of our grid we effectively loose
|
of the sphere and in the corners of our grid we effectively loose
|
||||||
@ -1184,7 +1186,7 @@ to use and one should expect a loss in quality evident by a higher
|
|||||||
reconstruction--error opposed to a grid where they are used. Sadly we could not
|
reconstruction--error opposed to a grid where they are used. Sadly we could not
|
||||||
run a in--depth test on this due to computational limitations.
|
run a in--depth test on this due to computational limitations.
|
||||||
|
|
||||||
Nevertheless this hints at the notion, that variability is a good measure for
|
Nevertheless this hints at the notion, that *variability* is a good measure for
|
||||||
the overall quality of a fit.
|
the overall quality of a fit.
|
||||||
|
|
||||||
### Regularity
|
### Regularity
|
||||||
@ -1212,19 +1214,19 @@ $4 \times 4 \times 4$ & $4 \times 4 \times 5$ & $4 \times 4 \times 7$ & $4 \time
|
|||||||
\cline{2-4}
|
\cline{2-4}
|
||||||
\multicolumn{3}{c}{} & all: 0.15 (0) \T
|
\multicolumn{3}{c}{} & all: 0.15 (0) \T
|
||||||
\end{tabular}
|
\end{tabular}
|
||||||
\caption[Correlation between regularity and iterations for 3D]{Correlation
|
\caption[Correlation between *regularity* and iterations for 3D]{Correlation
|
||||||
between regularity and number of iterations for the 3D fitting scenario.
|
between *regularity* and number of iterations for the 3D fitting scenario.
|
||||||
Displayed are the negated Spearman coefficients with the corresponding p--values
|
Displayed are the negated Spearman coefficients with the corresponding p--values
|
||||||
in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
|
in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
|
||||||
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
|
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
|
||||||
\label{tab:3dreg}
|
\label{tab:3dreg}
|
||||||
\end{table}
|
\end{table}
|
||||||
|
|
||||||
Opposed to the predictions of variability our test on regularity gave a mixed
|
Opposed to the predictions of *variability* our test on *regularity* gave a mixed
|
||||||
result --- similar to the 1D--case.
|
result --- similar to the 1D--case.
|
||||||
|
|
||||||
In roughly half of the scenarios we have a *significant*, but *weak* to *moderate*
|
In roughly half of the scenarios we have a *significant*, but *weak* to *moderate*
|
||||||
correlation between regularity and number of iterations. On the other hand in
|
correlation between *regularity* and number of iterations. On the other hand in
|
||||||
the scenarios where we increased the number of control--points, namely $125$ for
|
the scenarios where we increased the number of control--points, namely $125$ for
|
||||||
the $5 \times 5 \times 5$ grid and $216$ for the $6 \times 6 \times 6$ grid we found
|
the $5 \times 5 \times 5$ grid and $216$ for the $6 \times 6 \times 6$ grid we found
|
||||||
a *significant*, but *weak* **anti**--correlation when taking all three tests into
|
a *significant*, but *weak* **anti**--correlation when taking all three tests into
|
||||||
@ -1233,14 +1235,14 @@ findings/trends for the sets with $64$, $80$, and $112$ control--points
|
|||||||
(first two rows of table \ref{tab:3dreg}).
|
(first two rows of table \ref{tab:3dreg}).
|
||||||
|
|
||||||
Taking all results together we only find a *very weak*, but *significant* link
|
Taking all results together we only find a *very weak*, but *significant* link
|
||||||
between regularity and the number of iterations needed for the algorithm to
|
between *regularity* and the number of iterations needed for the algorithm to
|
||||||
converge.
|
converge.
|
||||||
|
|
||||||
\begin{figure}[!htb]
|
\begin{figure}[!htb]
|
||||||
\centering
|
\centering
|
||||||
\includegraphics[width=\textwidth]{img/evolution3d/regularity_montage.png}
|
\includegraphics[width=\textwidth]{img/evolution3d/regularity_montage.png}
|
||||||
\caption[Regularity for different 3D--grids]{
|
\caption[Regularity for different 3D--grids]{
|
||||||
Plots of regularity against number of iterations for various scenarios together
|
Plots of *regularity* against number of iterations for various scenarios together
|
||||||
with a linear fit to indicate trends.}
|
with a linear fit to indicate trends.}
|
||||||
\label{fig:resreg3d}
|
\label{fig:resreg3d}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
@ -1250,10 +1252,10 @@ the number of control--points helps the convergence--speeds. The
|
|||||||
regularity--criterion first behaves as we would like to, but then switches to
|
regularity--criterion first behaves as we would like to, but then switches to
|
||||||
behave exactly opposite to our expectations, as can be seen in the first three
|
behave exactly opposite to our expectations, as can be seen in the first three
|
||||||
plots. While the number of control--points increases from red to green to blue
|
plots. While the number of control--points increases from red to green to blue
|
||||||
and the number of iterations decreases, the regularity seems to increase at
|
and the number of iterations decreases, the *regularity* seems to increase at
|
||||||
first, but then decreases again on higher grid--resolutions.
|
first, but then decreases again on higher grid--resolutions.
|
||||||
|
|
||||||
This can be an artefact of the definition of regularity, as it is defined by the
|
This can be an artefact of the definition of *regularity*, as it is defined by the
|
||||||
inverse condition--number of the deformation--matrix $\vec{U}$, being the
|
inverse condition--number of the deformation--matrix $\vec{U}$, being the
|
||||||
fraction $\frac{\sigma_{\mathrm{min}}}{\sigma_{\mathrm{max}}}$ between the
|
fraction $\frac{\sigma_{\mathrm{min}}}{\sigma_{\mathrm{max}}}$ between the
|
||||||
least and greatest right singular value.
|
least and greatest right singular value.
|
||||||
@ -1264,9 +1266,9 @@ and so a small minimal right singular value occurring on higher
|
|||||||
grid--resolutions seems likely the problem.
|
grid--resolutions seems likely the problem.
|
||||||
|
|
||||||
Adding to this we also noted, that in the case of the $10 \times 10 \times
|
Adding to this we also noted, that in the case of the $10 \times 10 \times
|
||||||
10$--grid the regularity was always $0$, as a non--contributing control-point
|
10$--grid the *regularity* was always $0$, as a non--contributing control--point
|
||||||
yields a $0$--column in the deformation--matrix, thus letting
|
yields a $0$--column in the deformation--matrix, thus letting
|
||||||
$\sigma_\mathrm{min} = 0$. A better definition for regularity (i.e. using the
|
$\sigma_\mathrm{min} = 0$. A better definition for *regularity* (i.e. using the
|
||||||
smallest non--zero right singular value) could solve this particular issue, but
|
smallest non--zero right singular value) could solve this particular issue, but
|
||||||
not fix the trend we noticed above.
|
not fix the trend we noticed above.
|
||||||
|
|
||||||
@ -1295,8 +1297,8 @@ $4 \times 4 \times 4$ & $4 \times 4 \times 5$ & $4 \times 4 \times 7$ & $4 \time
|
|||||||
\cline{2-4}
|
\cline{2-4}
|
||||||
\multicolumn{3}{c}{} & all: 0.95 (0) \T
|
\multicolumn{3}{c}{} & all: 0.95 (0) \T
|
||||||
\end{tabular}
|
\end{tabular}
|
||||||
\caption[Correlation between improvement--potential and fitting--error for 3D]{Correlation
|
\caption[Correlation between *improvement potential* and fitting--error for 3D]{Correlation
|
||||||
between improvement--potential and fitting--error for the 3D fitting scenario.
|
between *improvement potential* and fitting--error for the 3D fitting scenario.
|
||||||
Displayed are the negated Spearman coefficients with the corresponding p--values
|
Displayed are the negated Spearman coefficients with the corresponding p--values
|
||||||
in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
|
in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
|
||||||
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
|
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
|
||||||
@ -1314,20 +1316,20 @@ quality of such gradients anyway.
|
|||||||
\centering
|
\centering
|
||||||
\includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png}
|
\includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png}
|
||||||
\caption[Improvement potential for different 3D--grids]{
|
\caption[Improvement potential for different 3D--grids]{
|
||||||
Plots of improvement potential against error given by our fitness--function
|
Plots of *improvement potential* against error given by our *fitness--function*
|
||||||
after convergence together with a linear fit of each of the plotted data to
|
after convergence together with a linear fit of each of the plotted data to
|
||||||
indicate trends.}
|
indicate trends.}
|
||||||
\label{fig:resimp3d}
|
\label{fig:resimp3d}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
We plotted our findings on the improvement potential in a similar way as we did
|
We plotted our findings on the *improvement potential* in a similar way as we did
|
||||||
before with the regularity. In figure \ref{fig:resimp3d} one can clearly see the
|
before with the *regularity*. In figure \ref{fig:resimp3d} one can clearly see the
|
||||||
correlation and the spread within each setup and the behaviour when we increase
|
correlation and the spread within each setup and the behaviour when we increase
|
||||||
the number of control--points.
|
the number of control--points.
|
||||||
|
|
||||||
Along with this we also give the Spearman--coefficients along with their
|
Along with this we also give the Spearman--coefficients along with their
|
||||||
p--values in table \ref{tab:3dimp}. Within one scenario we only find a *weak* to
|
p--values in table \ref{tab:3dimp}. Within one scenario we only find a *weak* to
|
||||||
*moderate* correlation between the improvement potential and the fitting error,
|
*moderate* correlation between the *improvement potential* and the fitting error,
|
||||||
but all findings (except for $7 \times 4 \times 4$ and $6 \times 6 \times 6$)
|
but all findings (except for $7 \times 4 \times 4$ and $6 \times 6 \times 6$)
|
||||||
are significant.
|
are significant.
|
||||||
|
|
||||||
@ -1335,14 +1337,14 @@ If we take multiple datasets into account the correlation is *very strong* and
|
|||||||
*significant*, which is good, as this functions as a litmus--test, because the
|
*significant*, which is good, as this functions as a litmus--test, because the
|
||||||
quality is naturally tied to the number of control--points.
|
quality is naturally tied to the number of control--points.
|
||||||
|
|
||||||
All in all the improvement potential seems to be a good and sensible measure of
|
All in all the *improvement potential* seems to be a good and sensible measure of
|
||||||
quality, even given gradients of varying quality.
|
quality, even given gradients of varying quality.
|
||||||
|
|
||||||
Lastly, a small note on the behaviour of improvement potential and convergence
|
Lastly, a small note on the behaviour of *improvement potential* and convergence
|
||||||
speed, as we used this in the 1D case to argue, why the *regularity* defied our
|
speed, as we used this in the 1D case to argue, why the *regularity* defied our
|
||||||
expectations. As a contrast we wanted to show, that improvement potential cannot
|
expectations. As a contrast we wanted to show, that *improvement potential* cannot
|
||||||
serve for good predictions of the convergence speed. In figure
|
serve for good predictions of the convergence speed. In figure
|
||||||
\ref{fig:imp1d3d} we show improvement potential against number of iterations
|
\ref{fig:imp1d3d} we show *improvement potential* against number of iterations
|
||||||
for both scenarios. As one can see, in the 1D scenario we have a *strong*
|
for both scenarios. As one can see, in the 1D scenario we have a *strong*
|
||||||
and *significant* correlation (with $-r_S = -0.72$, $p = 0$), whereas in the 3D
|
and *significant* correlation (with $-r_S = -0.72$, $p = 0$), whereas in the 3D
|
||||||
scenario we have the opposite *significant* and *strong* effect (with
|
scenario we have the opposite *significant* and *strong* effect (with
|
||||||
@ -1352,11 +1354,11 @@ scenario and are not suited for generalization.
|
|||||||
\begin{figure}[hbt]
|
\begin{figure}[hbt]
|
||||||
\centering
|
\centering
|
||||||
\includegraphics[width=\textwidth]{img/imp1d3d.png}
|
\includegraphics[width=\textwidth]{img/imp1d3d.png}
|
||||||
\caption[Improvement potential and convergence speed for 1D and 3D--scenarios]{
|
\caption[Improvement potential and convergence speed\newline for 1D and 3D--scenarios]{
|
||||||
\newline
|
\newline
|
||||||
Left: Improvement potential against convergence speed for the
|
Left: *Improvement potential* against convergence speed for the
|
||||||
1D--scenario\newline
|
1D--scenario\newline
|
||||||
Right: Improvement potential against convergence speed for the 3D--scnario
|
Right: *Improvement potential* against convergence speed for the 3D--scnario
|
||||||
}
|
}
|
||||||
\label{fig:imp1d3d}
|
\label{fig:imp1d3d}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
@ -1364,23 +1366,23 @@ Right: Improvement potential against convergence speed for the 3D--scnario
|
|||||||
# Discussion and outlook
|
# Discussion and outlook
|
||||||
\label{sec:dis}
|
\label{sec:dis}
|
||||||
|
|
||||||
In this thesis we took a look at the different criteria for evolvability as
|
In this thesis we took a look at the different criteria for *evolvability* as
|
||||||
introduced by Richter et al.\cite{anrichterEvol}, namely *variability*,
|
introduced by Richter et al.\cite{anrichterEvol}, namely *variability*,
|
||||||
*regularity* and *improvement potential* under different setup--conditions.
|
*regularity* and *improvement potential* under different setup--conditions.
|
||||||
Where Richter et al. used \acf{RBF}, we employed \acf{FFD} to set up a
|
Where Richter et al. used \acf{RBF}, we employed \acf{FFD} to set up a
|
||||||
low--complexity parametrization of a more complex vertex--mesh.
|
low--complexity parametrization of a more complex vertex--mesh.
|
||||||
|
|
||||||
In our findings we could show in the 1D--scenario, that there were statistically
|
In our findings we could show in the 1D--scenario, that there were statistically
|
||||||
significant very strong correlations between *variability and fitting error*
|
*significant* *very strong* correlations between *variability and fitting error*
|
||||||
($0.94$) and *improvement--potential and fitting error* ($1.0$) with
|
($0.94$) and *improvement potential and fitting error* ($1.0$) with
|
||||||
comparable results than Richter et al. (with $0.31$ to $0.88$
|
comparable results than Richter et al. (with $0.31$ to $0.88$
|
||||||
for the former and $0.75$ to $0.99$ for the latter), whereas we found
|
for the former and $0.75$ to $0.99$ for the latter), whereas we found
|
||||||
only weak correlations for *regularity and convergence--speed* ($0.28$)
|
only *weak* correlations for *regularity and convergence--speed* ($0.28$)
|
||||||
opposed to Richter et al. with $0.39$ to $0.91$.^[We only took statistically
|
opposed to Richter et al. with $0.39$ to $0.91$.^[We only took statistically
|
||||||
*significant* results into consideration when compiling these numbers. Details
|
*significant* results into consideration when compiling these numbers. Details
|
||||||
are given in the respective chapters.]
|
are given in the respective chapters.]
|
||||||
|
|
||||||
For the 3D--scenario our results show a very strong, significant correlation
|
For the 3D--scenario our results show a *very strong*, *significant* correlation
|
||||||
between *variability and fitting error* with $0.89$ to $0.94$, which are pretty
|
between *variability and fitting error* with $0.89$ to $0.94$, which are pretty
|
||||||
much in line with the findings of Richter et al. ($0.65$ to $0.95$). The
|
much in line with the findings of Richter et al. ($0.65$ to $0.95$). The
|
||||||
correlation between *improvement potential and fitting error* behave similar,
|
correlation between *improvement potential and fitting error* behave similar,
|
||||||
@ -1410,10 +1412,7 @@ in \cite{anrichterEvol}, whereas we merely used an indirect \ac{FFD}--approach.
|
|||||||
As direct manipulations tend to perform better than indirect manipulations, the
|
As direct manipulations tend to perform better than indirect manipulations, the
|
||||||
usage of \acf{DM--FFD} could also work better with the criteria we examined.
|
usage of \acf{DM--FFD} could also work better with the criteria we examined.
|
||||||
This can also solve the problem of bad singular values for the *regularity* as
|
This can also solve the problem of bad singular values for the *regularity* as
|
||||||
the incorporation of the parametrization of the points on the surface, which are
|
the incorporation of the parametrization of the points on the surface --- which
|
||||||
the essential part of a direct--manipulation, could cancel out a bad
|
are the essential part of a direct--manipulation --- could cancel out a bad
|
||||||
control--grid as the bad control--points are never or negligibly used to
|
control--grid as the bad control--points are never or negligibly used to
|
||||||
parametrize those surface--points.
|
parametrize those surface--points.
|
||||||
|
|
||||||
\improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt
|
|
||||||
Direktlinks des Autors.}
|
|
||||||
|
|||||||
BIN
arbeit/ma.pdf
BIN
arbeit/ma.pdf
Binary file not shown.
503
arbeit/ma.tex
503
arbeit/ma.tex
@ -197,20 +197,20 @@ the translation of the problem--domain into a simple parametric
|
|||||||
representation (the \emph{genome}) can be challenging.
|
representation (the \emph{genome}) can be challenging.
|
||||||
|
|
||||||
This translation is often necessary as the target of the optimization
|
This translation is often necessary as the target of the optimization
|
||||||
may have too many degrees of freedom. In the example of an aerodynamic
|
may have too many degrees of freedom for a reasonable computation. In
|
||||||
simulation of drag onto an object, those object--designs tend to have a
|
the example of an aerodynamic simulation of drag onto an object, those
|
||||||
high number of vertices to adhere to various requirements (visual,
|
object--designs tend to have a high number of vertices to adhere to
|
||||||
practical, physical, etc.). A simpler representation of the same object
|
various requirements (visual, practical, physical, etc.). A simpler
|
||||||
in only a few parameters that manipulate the whole in a sensible matter
|
representation of the same object in only a few parameters that
|
||||||
are desirable, as this often decreases the computation time
|
manipulate the whole in a sensible matter are desirable, as this often
|
||||||
significantly.
|
decreases the computation time significantly.
|
||||||
|
|
||||||
Additionally one can exploit the fact, that drag in this case is
|
Additionally one can exploit the fact, that drag in this case is
|
||||||
especially sensitive to non--smooth surfaces, so that a smooth local
|
especially sensitive to non--smooth surfaces, so that a smooth local
|
||||||
manipulation of the surface as a whole is more advantageous than merely
|
manipulation of the surface as a whole is more advantageous than merely
|
||||||
random manipulation of the vertices.
|
random manipulation of the vertices.
|
||||||
|
|
||||||
The quality of such a low-dimensional representation in biological
|
The quality of such a low--dimensional representation in biological
|
||||||
evolution is strongly tied to the notion of
|
evolution is strongly tied to the notion of
|
||||||
\emph{evolvability}\cite{wagner1996complex}, as the parametrization of
|
\emph{evolvability}\cite{wagner1996complex}, as the parametrization of
|
||||||
the problem has serious implications on the convergence speed and the
|
the problem has serious implications on the convergence speed and the
|
||||||
@ -230,7 +230,7 @@ One example of such a general representation of an object is to generate
|
|||||||
random points and represent vertices of an object as distances to these
|
random points and represent vertices of an object as distances to these
|
||||||
points --- for example via \acf{RBF}. If one (or the algorithm) would
|
points --- for example via \acf{RBF}. If one (or the algorithm) would
|
||||||
move such a point the object will get deformed only locally (due to the
|
move such a point the object will get deformed only locally (due to the
|
||||||
\ac{RBF}). As this results in a simple mapping from the parameter-space
|
\ac{RBF}). As this results in a simple mapping from the parameter--space
|
||||||
onto the object one can try out different representations of the same
|
onto the object one can try out different representations of the same
|
||||||
object and evaluate which criteria may be suited to describe this notion
|
object and evaluate which criteria may be suited to describe this notion
|
||||||
of \emph{evolvability}. This is exactly what Richter et
|
of \emph{evolvability}. This is exactly what Richter et
|
||||||
@ -238,18 +238,19 @@ al.\cite{anrichterEvol} have done.
|
|||||||
|
|
||||||
As we transfer the results of Richter et al.\cite{anrichterEvol} from
|
As we transfer the results of Richter et al.\cite{anrichterEvol} from
|
||||||
using \acf{RBF} as a representation to manipulate geometric objects to
|
using \acf{RBF} as a representation to manipulate geometric objects to
|
||||||
the use of \acf{FFD} we will use the same definition for evolvability
|
the use of \acf{FFD} we will use the same definition for
|
||||||
the original author used, namely \emph{regularity}, \emph{variability},
|
\emph{evolvability} the original author used, namely \emph{regularity},
|
||||||
and \emph{improvement potential}. We introduce these term in detail in
|
\emph{variability}, and \emph{improvement potential}. We introduce these
|
||||||
Chapter \ref{sec:intro:rvi}. In the original publication the author
|
term in detail in Chapter \ref{sec:intro:rvi}. In the original
|
||||||
could show a correlation between these evolvability--criteria with the
|
publication the author could show a correlation between these
|
||||||
quality and convergence speed of such optimization.
|
evolvability--criteria with the quality and convergence speed of such
|
||||||
|
optimization.
|
||||||
|
|
||||||
We will replicate the same setup on the same objects but use \acf{FFD}
|
We will replicate the same setup on the same objects but use \acf{FFD}
|
||||||
instead of \acf{RBF} to create a local deformation near the control
|
instead of \acf{RBF} to create a local deformation near the
|
||||||
points and evaluate if the evolution--criteria still work as a predictor
|
control--points and evaluate if the evolution--criteria still work as a
|
||||||
for \emph{evolvability} of the representation given the different
|
predictor for \emph{evolvability} of the representation given the
|
||||||
deformation scheme, as suspected in \cite{anrichterEvol}.
|
different deformation scheme, as suspected in \cite{anrichterEvol}.
|
||||||
|
|
||||||
First we introduce different topics in isolation in Chapter
|
First we introduce different topics in isolation in Chapter
|
||||||
\ref{sec:back}. We take an abstract look at the definition of \ac{FFD}
|
\ref{sec:back}. We take an abstract look at the definition of \ac{FFD}
|
||||||
@ -258,7 +259,7 @@ is a sensible deformation function (in \ref{sec:back:ffdgood}). Then we
|
|||||||
establish some background--knowledge of evolutionary algorithms (in
|
establish some background--knowledge of evolutionary algorithms (in
|
||||||
\ref{sec:back:evo}) and why this is useful in our domain (in
|
\ref{sec:back:evo}) and why this is useful in our domain (in
|
||||||
\ref{sec:back:evogood}) followed by the definition of the different
|
\ref{sec:back:evogood}) followed by the definition of the different
|
||||||
evolvability criteria established in \cite{anrichterEvol} (in
|
evolvability--criteria established in \cite{anrichterEvol} (in
|
||||||
\ref {sec:intro:rvi}).
|
\ref {sec:intro:rvi}).
|
||||||
|
|
||||||
In Chapter \ref{sec:impl} we take a look at our implementation of
|
In Chapter \ref{sec:impl} we take a look at our implementation of
|
||||||
@ -285,18 +286,19 @@ from \cite{spitzmuller1996bezier} here and go into the extension to the
|
|||||||
|
|
||||||
The main idea of \ac{FFD} is to create a function
|
The main idea of \ac{FFD} is to create a function
|
||||||
\(s : [0,1[^d \mapsto \mathbb{R}^d\) that spans a certain part of a
|
\(s : [0,1[^d \mapsto \mathbb{R}^d\) that spans a certain part of a
|
||||||
vector--space and is only linearly parametrized by some special control
|
vector--space and is only linearly parametrized by some special
|
||||||
points \(p_i\) and an constant attribution--function \(a_i(u)\), so \[
|
control--points \(p_i\) and an constant attribution--function
|
||||||
|
\(a_i(u)\), so \[
|
||||||
s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}
|
s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}
|
||||||
\] can be thought of a representation of the inside of the convex hull
|
\] can be thought of a representation of the inside of the convex hull
|
||||||
generated by the control points where each point can be accessed by the
|
generated by the control--points where each position inside can be
|
||||||
right \(u \in [0,1[^d\).
|
accessed by the right \(u \in [0,1[^d\).
|
||||||
|
|
||||||
\begin{figure}[!ht]
|
\begin{figure}[!ht]
|
||||||
\begin{center}
|
\begin{center}
|
||||||
\includegraphics[width=0.7\textwidth]{img/B-Splines.png}
|
\includegraphics[width=0.7\textwidth]{img/B-Splines.png}
|
||||||
\end{center}
|
\end{center}
|
||||||
\caption[Example of B-Splines]{Example of a parametrization of a line with
|
\caption[Example of B--Splines]{Example of a parametrization of a line with
|
||||||
corresponding deformation to generate a deformed objet}
|
corresponding deformation to generate a deformed objet}
|
||||||
\label{fig:bspline}
|
\label{fig:bspline}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
@ -338,7 +340,8 @@ We can even derive this equation straightforward for an arbitrary
|
|||||||
\[\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)\]
|
\[\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)\]
|
||||||
|
|
||||||
For a B--Spline \[s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i\] these
|
For a B--Spline \[s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i\] these
|
||||||
derivations yield \(\frac{\partial^d}{\partial u} s(u) = 0\).
|
derivations yield
|
||||||
|
\(\left(\frac{\partial}{\partial u}\right)^d s(u) = 0\).
|
||||||
|
|
||||||
Another interesting property of these recursive polynomials is that they
|
Another interesting property of these recursive polynomials is that they
|
||||||
are continuous (given \(d \ge 1\)) as every \(p_i\) gets blended in
|
are continuous (given \(d \ge 1\)) as every \(p_i\) gets blended in
|
||||||
@ -348,17 +351,17 @@ step of the recursion.
|
|||||||
|
|
||||||
This means that all changes are only a local linear combination between
|
This means that all changes are only a local linear combination between
|
||||||
the control--point \(p_i\) to \(p_{i+d+1}\) and consequently this yields
|
the control--point \(p_i\) to \(p_{i+d+1}\) and consequently this yields
|
||||||
to the convex--hull--property of B-Splines --- meaning, that no matter
|
to the convex--hull--property of B--Splines --- meaning, that no matter
|
||||||
how we choose our coefficients, the resulting points all have to lie
|
how we choose our coefficients, the resulting points all have to lie
|
||||||
inside convex--hull of the control--points.
|
inside convex--hull of the control--points.
|
||||||
|
|
||||||
For a given point \(v_i\) we can then calculate the contributions
|
For a given point \(s_i\) we can then calculate the contributions
|
||||||
\(n_{i,j}~:=~N_{j,d,\tau}\) of each control point \(p_j\) to get the
|
\(u_{i,j}~:=~N_{j,d,\tau}\) of each control point \(p_j\) to get the
|
||||||
projection from the control--point--space into the object--space: \[
|
projection from the control--point--space into the object--space: \[
|
||||||
v_i = \sum_j n_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p}
|
s_i = \sum_j u_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p}
|
||||||
\] or written for all points at the same time: \[
|
\] or written for all points at the same time: \[
|
||||||
\vec{v} = \vec{N} \vec{p}
|
\vec{s} = \vec{U} \vec{p}
|
||||||
\] where \(\vec{N}\) is the \(n \times m\) transformation--matrix (later
|
\] where \(\vec{U}\) is the \(n \times m\) transformation--matrix (later
|
||||||
on called \textbf{deformation matrix}) for \(n\) object--space--points
|
on called \textbf{deformation matrix}) for \(n\) object--space--points
|
||||||
and \(m\) control--points.
|
and \(m\) control--points.
|
||||||
|
|
||||||
@ -372,7 +375,7 @@ of the B--spline ($[k_0,k_4]$ on this figure), the B--Spline basis functions sum
|
|||||||
up to one (partition of unity). In this example, we use B--Splines of degree 2.
|
up to one (partition of unity). In this example, we use B--Splines of degree 2.
|
||||||
The horizontal segment below the abscissa axis represents the domain of
|
The horizontal segment below the abscissa axis represents the domain of
|
||||||
influence of the B--splines basis function, i.e. the interval on which they are
|
influence of the B--splines basis function, i.e. the interval on which they are
|
||||||
not null. At a given point, there are at most $ d+1$ non-zero B--Spline basis
|
not null. At a given point, there are at most $ d+1$ non--zero B--Spline basis
|
||||||
functions (compact support).\grqq \newline
|
functions (compact support).\grqq \newline
|
||||||
Note, that Brunet starts his index at $-d$ opposed to our definition, where we
|
Note, that Brunet starts his index at $-d$ opposed to our definition, where we
|
||||||
start at $0$.}
|
start at $0$.}
|
||||||
@ -381,8 +384,8 @@ start at $0$.}
|
|||||||
|
|
||||||
Furthermore B--Spline--basis--functions form a partition of unity for
|
Furthermore B--Spline--basis--functions form a partition of unity for
|
||||||
all, but the first and last \(d\)
|
all, but the first and last \(d\)
|
||||||
control-points\cite{brunet2010contributions}. Therefore we later on use
|
control--points\cite{brunet2010contributions}. Therefore we later on use
|
||||||
the border-points \(d+1\) times, such that \(\sum_j n_{i,j} p_j = p_i\)
|
the border--points \(d+1\) times, such that \(\sum_j u_{i,j} p_j = p_i\)
|
||||||
for these points.
|
for these points.
|
||||||
|
|
||||||
The locality of the influence of each control--point and the partition
|
The locality of the influence of each control--point and the partition
|
||||||
@ -395,13 +398,13 @@ function?}{Why is a good deformation function?}}\label{why-is-a-good-deformatio
|
|||||||
\label{sec:back:ffdgood}
|
\label{sec:back:ffdgood}
|
||||||
|
|
||||||
The usage of \ac{FFD} as a tool for manipulating follows directly from
|
The usage of \ac{FFD} as a tool for manipulating follows directly from
|
||||||
the properties of the polynomials and the correspondence to the control
|
the properties of the polynomials and the correspondence to the
|
||||||
points. Having only a few control points gives the user a nicer
|
control--points. Having only a few control--points gives the user a
|
||||||
high--level--interface, as she only needs to move these points and the
|
nicer high--level--interface, as she only needs to move these points and
|
||||||
model follows in an intuitive manner. The deformation is smooth as the
|
the model follows in an intuitive manner. The deformation is smooth as
|
||||||
underlying polygon is smooth as well and affects as many vertices of the
|
the underlying polygon is smooth as well and affects as many vertices of
|
||||||
model as needed. Moreover the changes are always local so one risks not
|
the model as needed. Moreover the changes are always local so one risks
|
||||||
any change that a user cannot immediately see.
|
not any change that a user cannot immediately see.
|
||||||
|
|
||||||
But there are also disadvantages of this approach. The user loses the
|
But there are also disadvantages of this approach. The user loses the
|
||||||
ability to directly influence vertices and even seemingly simple tasks
|
ability to directly influence vertices and even seemingly simple tasks
|
||||||
@ -479,7 +482,7 @@ through our \emph{fitness--function}, biologically by the ability to
|
|||||||
survive and produce offspring). Any individual in our algorithm thus
|
survive and produce offspring). Any individual in our algorithm thus
|
||||||
experience a biologically motivated life cycle of inheriting genes from
|
experience a biologically motivated life cycle of inheriting genes from
|
||||||
the parents, modified by mutations occurring, performing according to a
|
the parents, modified by mutations occurring, performing according to a
|
||||||
fitness--metric and generating offspring based on this. Therefore each
|
fitness--metric, and generating offspring based on this. Therefore each
|
||||||
iteration in the while--loop above is also often named generation.
|
iteration in the while--loop above is also often named generation.
|
||||||
|
|
||||||
One should note that there is a subtle difference between
|
One should note that there is a subtle difference between
|
||||||
@ -517,8 +520,8 @@ The main algorithm just repeats the following steps:
|
|||||||
\(s : (I^\lambda \cup I^{\mu + \lambda},\Phi) \mapsto I^\mu\) that
|
\(s : (I^\lambda \cup I^{\mu + \lambda},\Phi) \mapsto I^\mu\) that
|
||||||
selects from the previously generated \(I^\lambda\) children and
|
selects from the previously generated \(I^\lambda\) children and
|
||||||
optionally also the parents (denoted by the set \(Q\) in the
|
optionally also the parents (denoted by the set \(Q\) in the
|
||||||
algorithm) using the fitness--function \(\Phi\). The result of this
|
algorithm) using the \emph{fitness--function} \(\Phi\). The result of
|
||||||
operation is the next Population of \(\mu\) individuals.
|
this operation is the next Population of \(\mu\) individuals.
|
||||||
\end{itemize}
|
\end{itemize}
|
||||||
|
|
||||||
All these functions can (and mostly do) have a lot of hidden parameters
|
All these functions can (and mostly do) have a lot of hidden parameters
|
||||||
@ -547,10 +550,10 @@ algorithms}\label{advantages-of-evolutionary-algorithms}
|
|||||||
|
|
||||||
The main advantage of evolutionary algorithms is the ability to find
|
The main advantage of evolutionary algorithms is the ability to find
|
||||||
optima of general functions just with the help of a given
|
optima of general functions just with the help of a given
|
||||||
fitness--function. Components and techniques for evolutionary algorithms
|
\emph{fitness--function}. Components and techniques for evolutionary
|
||||||
are specifically known to help with different problems arising in the
|
algorithms are specifically known to help with different problems
|
||||||
domain of optimization\cite{weise2012evolutionary}. An overview of the
|
arising in the domain of optimization\cite{weise2012evolutionary}. An
|
||||||
typical problems are shown in figure \ref{fig:probhard}.
|
overview of the typical problems are shown in figure \ref{fig:probhard}.
|
||||||
|
|
||||||
\begin{figure}[!ht]
|
\begin{figure}[!ht]
|
||||||
\includegraphics[width=\textwidth]{img/weise_fig3.png}
|
\includegraphics[width=\textwidth]{img/weise_fig3.png}
|
||||||
@ -559,13 +562,14 @@ typical problems are shown in figure \ref{fig:probhard}.
|
|||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
Most of the advantages stem from the fact that a gradient--based
|
Most of the advantages stem from the fact that a gradient--based
|
||||||
procedure has only one point of observation from where it evaluates the
|
procedure has usually only one point of observation from where it
|
||||||
next steps, whereas an evolutionary strategy starts with a population of
|
evaluates the next steps, whereas an evolutionary strategy starts with a
|
||||||
guessed solutions. Because an evolutionary strategy can be modified
|
population of guessed solutions. Because an evolutionary strategy can be
|
||||||
according to the problem--domain (i.e.~by the ideas given above) it can
|
modified according to the problem--domain (i.e.~by the ideas given
|
||||||
also approximate very difficult problems in an efficient manner and even
|
above) it can also approximate very difficult problems in an efficient
|
||||||
self--tune parameters depending on the ancestry at runtime\footnote{Some
|
manner and even self--tune parameters depending on the ancestry at
|
||||||
examples of this are explained in detail in \cite{eiben1999parameter}}.
|
runtime\footnote{Some examples of this are explained in detail in
|
||||||
|
\cite{eiben1999parameter}}.
|
||||||
|
|
||||||
If an analytic best solution exists and is easily computable
|
If an analytic best solution exists and is easily computable
|
||||||
(i.e.~because the error--function is convex) an evolutionary algorithm
|
(i.e.~because the error--function is convex) an evolutionary algorithm
|
||||||
@ -599,8 +603,8 @@ deformation.
|
|||||||
We can also think of the deformation in terms of differences from the
|
We can also think of the deformation in terms of differences from the
|
||||||
original coordinates \[
|
original coordinates \[
|
||||||
\Delta \vec{S} = \vec{U} \cdot \Delta \vec{P}
|
\Delta \vec{S} = \vec{U} \cdot \Delta \vec{P}
|
||||||
\] which is isomorphic to the former due to the linear correlation in
|
\] which is isomorphic to the former due to the linearity of the
|
||||||
the deformation. One can see in this way, that the way the deformation
|
deformation. One can see in this way, that the way the deformation
|
||||||
behaves lies solely in the entries of \(\vec{U}\), which is why the
|
behaves lies solely in the entries of \(\vec{U}\), which is why the
|
||||||
three criteria focus on this.
|
three criteria focus on this.
|
||||||
|
|
||||||
@ -609,14 +613,14 @@ three criteria focus on this.
|
|||||||
In \cite{anrichterEvol} \emph{variability} is defined as
|
In \cite{anrichterEvol} \emph{variability} is defined as
|
||||||
\[\mathrm{variability}(\vec{U}) := \frac{\mathrm{rank}(\vec{U})}{n},\]
|
\[\mathrm{variability}(\vec{U}) := \frac{\mathrm{rank}(\vec{U})}{n},\]
|
||||||
whereby \(\vec{U}\) is the \(n \times m\) deformation--Matrix used to
|
whereby \(\vec{U}\) is the \(n \times m\) deformation--Matrix used to
|
||||||
map the \(m\) control points onto the \(n\) vertices.
|
map the \(m\) control--points onto the \(n\) vertices.
|
||||||
|
|
||||||
Given \(n = m\), an identical number of control--points and vertices,
|
Given \(n = m\), an identical number of control--points and vertices,
|
||||||
this quotient will be \(=1\) if all control points are independent of
|
this quotient will be \(=1\) if all control--points are independent of
|
||||||
each other and the solution is to trivially move every control--point
|
each other and the solution is to trivially move every control--point
|
||||||
onto a target--point.
|
onto a target--point.
|
||||||
|
|
||||||
In praxis the value of \(V(\vec{U})\) is typically \(\ll 1\), because as
|
In praxis the value of \(V(\vec{U})\) is typically \(\ll 1\), because
|
||||||
there are only few control--points for many vertices, so \(m \ll n\).
|
there are only few control--points for many vertices, so \(m \ll n\).
|
||||||
|
|
||||||
This criterion should correlate to the degrees of freedom the given
|
This criterion should correlate to the degrees of freedom the given
|
||||||
@ -641,7 +645,7 @@ As we deform the given Object only based on the parameters as
|
|||||||
\(\vec{p} \mapsto f(\vec{x} + \vec{U}\vec{p})\) this makes sure that
|
\(\vec{p} \mapsto f(\vec{x} + \vec{U}\vec{p})\) this makes sure that
|
||||||
\(\|\vec{Up}\| \propto \|\vec{p}\|\) when \(\kappa(\vec{U}) \approx 1\).
|
\(\|\vec{Up}\| \propto \|\vec{p}\|\) when \(\kappa(\vec{U}) \approx 1\).
|
||||||
The inversion of \(\kappa(\vec{U})\) is only performed to map the
|
The inversion of \(\kappa(\vec{U})\) is only performed to map the
|
||||||
criterion--range to \([0..1]\), whereas \(1\) is the optimal value and
|
criterion--range to \([0..1]\), where \(1\) is the optimal value and
|
||||||
\(0\) is the worst value.
|
\(0\) is the worst value.
|
||||||
|
|
||||||
On the one hand this criterion should be characteristic for numeric
|
On the one hand this criterion should be characteristic for numeric
|
||||||
@ -653,8 +657,8 @@ locality\cite{weise2012evolutionary,thorhauer2014locality}.
|
|||||||
\subsection{Improvement Potential}\label{improvement-potential}
|
\subsection{Improvement Potential}\label{improvement-potential}
|
||||||
|
|
||||||
In contrast to the general nature of \emph{variability} and
|
In contrast to the general nature of \emph{variability} and
|
||||||
\emph{regularity}, which are agnostic of the fitness--function at hand,
|
\emph{regularity}, which are agnostic of the \emph{fitness--function} at
|
||||||
the third criterion should reflect a notion of the potential for
|
hand, the third criterion should reflect a notion of the potential for
|
||||||
optimization, taking a guess into account.
|
optimization, taking a guess into account.
|
||||||
|
|
||||||
Most of the times some kind of gradient \(g\) is available to suggest a
|
Most of the times some kind of gradient \(g\) is available to suggest a
|
||||||
@ -698,9 +702,9 @@ As we have established in Chapter \ref{sec:back:ffd} we can define an
|
|||||||
\Delta_x(u) = \sum_i N_{i,d,\tau_i}(u) \Delta_x c_i
|
\Delta_x(u) = \sum_i N_{i,d,\tau_i}(u) \Delta_x c_i
|
||||||
\end{equation}
|
\end{equation}
|
||||||
|
|
||||||
Note that we only sum up the \(\Delta\)--displacements in the control
|
Note that we only sum up the \(\Delta\)--displacements in the
|
||||||
points \(c_i\) to get the change in position of the point we are
|
control--points \(c_i\) to get the change in position of the point we
|
||||||
interested in.
|
are interested in.
|
||||||
|
|
||||||
In this way every deformed vertex is defined by \[
|
In this way every deformed vertex is defined by \[
|
||||||
\textrm{Deform}(v_x) = v_x + \Delta_x(u)
|
\textrm{Deform}(v_x) = v_x + \Delta_x(u)
|
||||||
@ -722,8 +726,8 @@ v_x \overset{!}{=} \sum_i N_{i,d,\tau_i}(u) c_i
|
|||||||
|
|
||||||
For this we employ the Gauss--Newton algorithm\cite{gaussNewton}, which
|
For this we employ the Gauss--Newton algorithm\cite{gaussNewton}, which
|
||||||
converges into the least--squares solution. An exact solution of this
|
converges into the least--squares solution. An exact solution of this
|
||||||
problem is impossible most of the times, because we usually have way
|
problem is impossible most of the time, because we usually have way more
|
||||||
more vertices than control points (\(\#v~\gg~\#c\)).
|
vertices than control--points (\(\#v~\gg~\#c\)).
|
||||||
|
|
||||||
\section{\texorpdfstring{Adaption of \ac{FFD} for a
|
\section{\texorpdfstring{Adaption of \ac{FFD} for a
|
||||||
3D--Mesh}{Adaption of for a 3D--Mesh}}\label{adaption-of-for-a-3dmesh}
|
3D--Mesh}{Adaption of for a 3D--Mesh}}\label{adaption-of-for-a-3dmesh}
|
||||||
@ -735,8 +739,8 @@ last chapter. But this time things get a bit more complicated. As we
|
|||||||
have a 3--dimensional grid we may have a different amount of
|
have a 3--dimensional grid we may have a different amount of
|
||||||
control--points in each direction.
|
control--points in each direction.
|
||||||
|
|
||||||
Given \(n,m,o\) control points in \(x,y,z\)--direction each Point on the
|
Given \(n,m,o\) control--points in \(x,y,z\)--direction each Point on
|
||||||
curve is defined by
|
the curve is defined by
|
||||||
\[V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.\]
|
\[V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.\]
|
||||||
|
|
||||||
In this case we have three different B--Splines (one for each dimension)
|
In this case we have three different B--Splines (one for each dimension)
|
||||||
@ -814,14 +818,14 @@ behaviour of the evolutionary algorithm.
|
|||||||
|
|
||||||
As mentioned in chapter \ref{sec:back:evo}, the way of choosing the
|
As mentioned in chapter \ref{sec:back:evo}, the way of choosing the
|
||||||
representation to map the general problem (mesh--fitting/optimization in
|
representation to map the general problem (mesh--fitting/optimization in
|
||||||
our case) into a parameter-space is very important for the quality and
|
our case) into a parameter--space is very important for the quality and
|
||||||
runtime of evolutionary algorithms\cite{Rothlauf2006}.
|
runtime of evolutionary algorithms\cite{Rothlauf2006}.
|
||||||
|
|
||||||
Because our control--points are arranged in a grid, we can accurately
|
Because our control--points are arranged in a grid, we can accurately
|
||||||
represent each vertex--point inside the grids volume with proper
|
represent each vertex--point inside the grids volume with proper
|
||||||
B--Spline--coefficients between \([0,1[\) and --- as a consequence ---
|
B--Spline--coefficients between \([0,1[\) and --- as a consequence ---
|
||||||
we have to embed our object into it (or create constant ``dummy''-points
|
we have to embed our object into it (or create constant
|
||||||
outside).
|
``dummy''--points outside).
|
||||||
|
|
||||||
The great advantage of B--Splines is the local, direct impact of each
|
The great advantage of B--Splines is the local, direct impact of each
|
||||||
control point without having a \(1:1\)--correlation, and a smooth
|
control point without having a \(1:1\)--correlation, and a smooth
|
||||||
@ -844,18 +848,18 @@ One would normally think, that the more control--points you add, the
|
|||||||
better the result will be, but this is not the case for our B--Splines.
|
better the result will be, but this is not the case for our B--Splines.
|
||||||
Given any point \(\vec{p}\) only the \(2 \cdot (d-1)\) control--points
|
Given any point \(\vec{p}\) only the \(2 \cdot (d-1)\) control--points
|
||||||
contribute to the parametrization of that point\footnote{Normally these
|
contribute to the parametrization of that point\footnote{Normally these
|
||||||
are \(d-1\) to each side, but at the boundaries the number gets
|
are \(d-1\) to each side, but at the boundaries border points get used
|
||||||
increased to the inside to meet the required smoothness}. This means,
|
multiple times to meet the number of points required}. This means,
|
||||||
that a high resolution can have many control-points that are not
|
that a high resolution can have many control--points that are not
|
||||||
contributing to any point on the surface and are thus completely
|
contributing to any point on the surface and are thus completely
|
||||||
irrelevant to the solution.
|
irrelevant to the solution.
|
||||||
|
|
||||||
We illustrate this phenomenon in figure \ref{fig:enoughCP}, where the
|
We illustrate this phenomenon in figure \ref{fig:enoughCP}, where the
|
||||||
four red central points are not relevant for the parametrization of the
|
red central points are not relevant for the parametrization of the
|
||||||
circle. This leads to artefacts in the deformation--matrix \(\vec{U}\),
|
circle. This leads to artefacts in the deformation--matrix \(\vec{U}\),
|
||||||
as the columns corresponding to those control--points are \(0\).
|
as the columns corresponding to those control--points are \(0\).
|
||||||
|
|
||||||
This leads to useless increased complexity, as the parameters
|
This also leads to useless increased complexity, as the parameters
|
||||||
corresponding to those points will never have any effect, but a naive
|
corresponding to those points will never have any effect, but a naive
|
||||||
algorithm will still try to optimize them yielding numeric artefacts in
|
algorithm will still try to optimize them yielding numeric artefacts in
|
||||||
the best and non--terminating or ill--defined solutions\footnote{One
|
the best and non--terminating or ill--defined solutions\footnote{One
|
||||||
@ -869,22 +873,23 @@ in the first place. We will address this in a special scenario in
|
|||||||
\ref{sec:res:3d:var}.
|
\ref{sec:res:3d:var}.
|
||||||
|
|
||||||
For our tests we chose different uniformly sized grids and added noise
|
For our tests we chose different uniformly sized grids and added noise
|
||||||
onto each control-point\footnote{For the special case of the outer layer
|
onto each control--point\footnote{For the special case of the outer
|
||||||
we only applied noise away from the object, so the object is still
|
layer we only applied noise away from the object, so the object is
|
||||||
confined in the convex hull of the control--points.} to simulate
|
still confined in the convex hull of the control--points.} to simulate
|
||||||
different starting-conditions.
|
different starting--conditions.
|
||||||
|
|
||||||
\chapter{\texorpdfstring{Scenarios for testing evolvability criteria
|
\chapter{\texorpdfstring{Scenarios for testing evolvability--criteria
|
||||||
using
|
using
|
||||||
\ac{FFD}}{Scenarios for testing evolvability criteria using }}\label{scenarios-for-testing-evolvability-criteria-using}
|
\ac{FFD}}{Scenarios for testing evolvability--criteria using }}\label{scenarios-for-testing-evolvabilitycriteria-using}
|
||||||
|
|
||||||
\label{sec:eval}
|
\label{sec:eval}
|
||||||
|
|
||||||
In our experiments we use the same two testing--scenarios, that were
|
In our experiments we use the same two testing--scenarios, that were
|
||||||
also used by \cite{anrichterEvol}. The first scenario deforms a plane
|
also used by Richter et al.\cite{anrichterEvol} The first scenario
|
||||||
into a shape originally defined in \cite{giannelli2012thb}, where we
|
deforms a plane into a shape originally defined by Giannelli et
|
||||||
setup control-points in a 2--dimensional manner and merely deform in the
|
al.\cite{giannelli2012thb}, where we setup control--points in a
|
||||||
height--coordinate to get the resulting shape.
|
2--dimensional manner and merely deform in the height--coordinate to get
|
||||||
|
the resulting shape.
|
||||||
|
|
||||||
In the second scenario we increase the degrees of freedom significantly
|
In the second scenario we increase the degrees of freedom significantly
|
||||||
by using a 3--dimensional control--grid to deform a sphere into a face,
|
by using a 3--dimensional control--grid to deform a sphere into a face,
|
||||||
@ -922,7 +927,7 @@ including a wireframe--overlay of the vertices.}
|
|||||||
\label{fig:1dtarget}
|
\label{fig:1dtarget}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
As the starting-plane we used the same shape, but set all
|
As the starting--plane we used the same shape, but set all
|
||||||
\(z\)--coordinates to \(0\), yielding a flat plane, which is partially
|
\(z\)--coordinates to \(0\), yielding a flat plane, which is partially
|
||||||
already correct.
|
already correct.
|
||||||
|
|
||||||
@ -936,11 +941,11 @@ corresponding vertex
|
|||||||
|
|
||||||
where \(t_i\) are the respective target--vertices to the parametrized
|
where \(t_i\) are the respective target--vertices to the parametrized
|
||||||
source--vertices\footnote{The parametrization is encoded in \(\vec{U}\)
|
source--vertices\footnote{The parametrization is encoded in \(\vec{U}\)
|
||||||
and the initial position of the control points. See
|
and the initial position of the control--points. See
|
||||||
\ref{sec:ffd:adapt}} with the current deformation--parameters
|
\ref{sec:ffd:adapt}} with the current deformation--parameters
|
||||||
\(\vec{p} = (p_1,\dots, p_m)\). We can do this
|
\(\vec{p} = (p_1,\dots, p_m)\). We can do this
|
||||||
one--to--one--correspondence because we have exactly the same number of
|
one--to--one--correspondence because we have exactly the same number of
|
||||||
source and target-vertices do to our setup of just flattening the
|
source and target--vertices do to our setup of just flattening the
|
||||||
object.
|
object.
|
||||||
|
|
||||||
This formula is also the least--squares approximation error for which we
|
This formula is also the least--squares approximation error for which we
|
||||||
@ -975,16 +980,16 @@ Both of these Models can be seen in figure \ref{fig:3dtarget}.
|
|||||||
Opposed to the 1D--case we cannot map the source and target--vertices in
|
Opposed to the 1D--case we cannot map the source and target--vertices in
|
||||||
a one--to--one--correspondence, which we especially need for the
|
a one--to--one--correspondence, which we especially need for the
|
||||||
approximation of the fitting--error. Hence we state that the error of
|
approximation of the fitting--error. Hence we state that the error of
|
||||||
one vertex is the distance to the closest vertex of the other model and
|
one vertex is the distance to the closest vertex of the respective other
|
||||||
sum up the error from the respective source and target.
|
model and sum up the error from the source and target.
|
||||||
|
|
||||||
We therefore define the \emph{fitness--function} to be:
|
We therefore define the \emph{fitness--function} to be:
|
||||||
|
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\mathrm{f}(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} -
|
\mathrm{f}(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} -
|
||||||
\vec{s_i}\|_2^2}_{\textrm{source-to-target--distance}}
|
\vec{s_i}\|_2^2}_{\textrm{source--to--target--distance}}
|
||||||
+ \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} -
|
+ \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} -
|
||||||
\vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}}
|
\vec{t_i}\|_2^2}_{\textrm{target--to--source--distance}}
|
||||||
+ \lambda \cdot \textrm{regularization}(\vec{P})
|
+ \lambda \cdot \textrm{regularization}(\vec{P})
|
||||||
\label{eq:fit3d}
|
\label{eq:fit3d}
|
||||||
\end{equation}
|
\end{equation}
|
||||||
@ -1002,7 +1007,7 @@ calculated coefficients for the \ac{FFD} --- analog to the 1D case ---
|
|||||||
and finally \(\vec{P}\) being the \(m \times 3\)--matrix of the
|
and finally \(\vec{P}\) being the \(m \times 3\)--matrix of the
|
||||||
control--grid defining the whole deformation.
|
control--grid defining the whole deformation.
|
||||||
|
|
||||||
As regularization-term we add a weighted Laplacian of the deformation
|
As regularization--term we add a weighted Laplacian of the deformation
|
||||||
that has been used before by Aschenbach et
|
that has been used before by Aschenbach et
|
||||||
al.\cite[Section 3.2]{aschenbach2015} on similar models and was shown to
|
al.\cite[Section 3.2]{aschenbach2015} on similar models and was shown to
|
||||||
lead to a more precise fit. The Laplacian
|
lead to a more precise fit. The Laplacian
|
||||||
@ -1034,7 +1039,7 @@ To compare our results to the ones given by Richter et
|
|||||||
al.\cite{anrichterEvol}, we also use Spearman's rank correlation
|
al.\cite{anrichterEvol}, we also use Spearman's rank correlation
|
||||||
coefficient. Opposed to other popular coefficients, like the Pearson
|
coefficient. Opposed to other popular coefficients, like the Pearson
|
||||||
correlation coefficient, which measures a linear relationship between
|
correlation coefficient, which measures a linear relationship between
|
||||||
variables, the Spearmans's coefficient assesses \glqq how well an
|
variables, the Spearman's coefficient assesses \glqq how well an
|
||||||
arbitrary monotonic function can describe the relationship between two
|
arbitrary monotonic function can describe the relationship between two
|
||||||
variables, without making any assumptions about the frequency
|
variables, without making any assumptions about the frequency
|
||||||
distribution of the variables\grqq\cite{hauke2011comparison}.
|
distribution of the variables\grqq\cite{hauke2011comparison}.
|
||||||
@ -1072,18 +1077,19 @@ Approximation}\label{procedure-1d-function-approximation}
|
|||||||
\label{sec:proc:1d}
|
\label{sec:proc:1d}
|
||||||
|
|
||||||
For our setup we first compute the coefficients of the
|
For our setup we first compute the coefficients of the
|
||||||
deformation--matrix and use then the formulas for \emph{variability} and
|
deformation--matrix and use the formulas for \emph{variability} and
|
||||||
\emph{regularity} to get our predictions. Afterwards we solve the
|
\emph{regularity} to get our predictions. Afterwards we solve the
|
||||||
problem analytically to get the (normalized) correct gradient that we
|
problem analytically to get the (normalized) correct gradient that we
|
||||||
use as guess for the \emph{improvement potential}. To check we also
|
use as guess for the \emph{improvement potential}. To further test the
|
||||||
consider a distorted gradient \(\vec{g}_{\mathrm{d}}\) \[
|
\emph{improvement potential} we also consider a distorted gradient
|
||||||
|
\(\vec{g}_{\mathrm{d}}\): \[
|
||||||
\vec{g}_{\mathrm{d}} = \frac{\mu \vec{g}_{\mathrm{c}} + (1-\mu)\mathbb{1}}{\|\mu \vec{g}_{\mathrm{c}} + (1-\mu) \mathbb{1}\|}
|
\vec{g}_{\mathrm{d}} = \frac{\mu \vec{g}_{\mathrm{c}} + (1-\mu)\mathbb{1}}{\|\mu \vec{g}_{\mathrm{c}} + (1-\mu) \mathbb{1}\|}
|
||||||
\] where \(\mathbb{1}\) is the vector consisting of \(1\) in every
|
\] where \(\mathbb{1}\) is the vector consisting of \(1\) in every
|
||||||
dimension, \(\vec{g}_\mathrm{c} = \vec{p^{*}} - \vec{p}\) is the
|
dimension, \(\vec{g}_\mathrm{c} = \vec{p^{*}} - \vec{p}\) is the
|
||||||
calculated correct gradient, and \(\mu\) is used to blend between
|
calculated correct gradient, and \(\mu\) is used to blend between
|
||||||
\(\vec{g}_\mathrm{c}\) and \(\mathbb{1}\). As we always start with a
|
\(\vec{g}_\mathrm{c}\) and \(\mathbb{1}\). As we always start with a
|
||||||
gradient of \(p = \mathbb{0}\) this means shortens
|
gradient of \(p = \mathbb{0}\) this means we can shorten the definition
|
||||||
\(\vec{g}_\mathrm{c} = \vec{p^{*}}\).
|
of \(\vec{g}_\mathrm{c}\) to \(\vec{g}_\mathrm{c} = \vec{p^{*}}\).
|
||||||
|
|
||||||
\begin{figure}[ht]
|
\begin{figure}[ht]
|
||||||
\begin{center}
|
\begin{center}
|
||||||
@ -1096,10 +1102,10 @@ random distortion to generate a testcase.}
|
|||||||
|
|
||||||
We then set up a regular 2--dimensional grid around the object with the
|
We then set up a regular 2--dimensional grid around the object with the
|
||||||
desired grid resolutions. To generate a testcase we then move the
|
desired grid resolutions. To generate a testcase we then move the
|
||||||
grid--vertices randomly inside the x--y--plane. As self-intersecting
|
grid--vertices randomly inside the x--y--plane. As self--intersecting
|
||||||
grids get tricky to solve with our implemented newtons--method we avoid
|
grids get tricky to solve with our implemented newtons--method (see
|
||||||
the generation of such self--intersecting grids for our testcases (see
|
section \ref{3dffd}) we avoid the generation of such self--intersecting
|
||||||
section \ref{3dffd}).
|
grids for our testcases.
|
||||||
|
|
||||||
To achieve that we generated a gaussian distributed number with
|
To achieve that we generated a gaussian distributed number with
|
||||||
\(\mu = 0, \sigma=0.25\) and clamped it to the range \([-0.25,0.25]\).
|
\(\mu = 0, \sigma=0.25\) and clamped it to the range \([-0.25,0.25]\).
|
||||||
@ -1130,12 +1136,12 @@ In the case of our 1D--Optimization--problem, we have the luxury of
|
|||||||
knowing the analytical solution to the given problem--set. We use this
|
knowing the analytical solution to the given problem--set. We use this
|
||||||
to experimentally evaluate the quality criteria we introduced before. As
|
to experimentally evaluate the quality criteria we introduced before. As
|
||||||
an evolutional optimization is partially a random process, we use the
|
an evolutional optimization is partially a random process, we use the
|
||||||
analytical solution as a stopping-criteria. We measure the convergence
|
analytical solution as a stopping--criteria. We measure the convergence
|
||||||
speed as number of iterations the evolutional algorithm needed to get
|
speed as number of iterations the evolutional algorithm needed to get
|
||||||
within \(1.05 \times\) of the optimal solution.
|
within \(1.05 \times\) of the optimal solution.
|
||||||
|
|
||||||
We used different regular grids that we manipulated as explained in
|
We used different regular grids that we manipulated as explained in
|
||||||
Section \ref{sec:proc:1d} with a different number of control points. As
|
Section \ref{sec:proc:1d} with a different number of control--points. As
|
||||||
our grids have to be the product of two integers, we compared a
|
our grids have to be the product of two integers, we compared a
|
||||||
\(5 \times 5\)--grid with \(25\) control--points to a \(4 \times 7\) and
|
\(5 \times 5\)--grid with \(25\) control--points to a \(4 \times 7\) and
|
||||||
\(7 \times 4\)--grid with \(28\) control--points. This was done to
|
\(7 \times 4\)--grid with \(28\) control--points. This was done to
|
||||||
@ -1157,7 +1163,7 @@ Note that $7 \times 4$ and $4 \times 7$ have the same number of control--points.
|
|||||||
\label{fig:1dvar}
|
\label{fig:1dvar}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
Variability should characterize the potential for design space
|
\emph{Variability} should characterize the potential for design space
|
||||||
exploration and is defined in terms of the normalized rank of the
|
exploration and is defined in terms of the normalized rank of the
|
||||||
deformation matrix \(\vec{U}\):
|
deformation matrix \(\vec{U}\):
|
||||||
\(V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n}\), whereby \(n\) is the
|
\(V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n}\), whereby \(n\) is the
|
||||||
@ -1166,29 +1172,30 @@ number of vertices. As all our tested matrices had a constant rank
|
|||||||
plotted the errors in the box plot in figure \ref{fig:1dvar}
|
plotted the errors in the box plot in figure \ref{fig:1dvar}
|
||||||
|
|
||||||
It is also noticeable, that although the \(7 \times 4\) and
|
It is also noticeable, that although the \(7 \times 4\) and
|
||||||
\(4 \times 7\) grids have a higher variability, they perform not better
|
\(4 \times 7\) grids have a higher \emph{variability}, they perform not
|
||||||
than the \(5 \times 5\) grid. Also the \(7 \times 4\) and \(4 \times 7\)
|
better than the \(5 \times 5\) grid. Also the \(7 \times 4\) and
|
||||||
grids differ distinctly from each other with a mean\(\pm\)sigma of
|
\(4 \times 7\) grids differ distinctly from each other with a
|
||||||
\(233.09 \pm 12.32\) for the former and \(286.32 \pm 22.36\) for the
|
mean\(\pm\)sigma of \(233.09 \pm 12.32\) for the former and
|
||||||
latter, although they have the same number of control--points. This is
|
\(286.32 \pm 22.36\) for the latter, although they have the same number
|
||||||
an indication of an impact a proper or improper grid--setup can have. We
|
of control--points. This is an indication of an impact a proper or
|
||||||
do not draw scientific conclusions from these findings, as more research
|
improper grid--setup can have. We do not draw scientific conclusions
|
||||||
on non-squared grids seem necessary.
|
from these findings, as more research on non--squared grids seem
|
||||||
|
necessary.
|
||||||
|
|
||||||
Leaving the issue of the grid--layout aside we focused on grids having
|
Leaving the issue of the grid--layout aside we focused on grids having
|
||||||
the same number of prototypes in every dimension. For the
|
the same number of prototypes in every dimension. For the
|
||||||
\(5 \times 5\), \(7 \times 7\) and \(10 \times 10\) grids we found a
|
\(5 \times 5\), \(7 \times 7\) and \(10 \times 10\) grids we found a
|
||||||
\emph{very strong} correlation (\(-r_S = 0.94, p = 0\)) between the
|
\emph{very strong} correlation (\(-r_S = 0.94, p = 0\)) between the
|
||||||
variability and the evolutionary error.
|
\emph{variability} and the evolutionary error.
|
||||||
|
|
||||||
\subsection{Regularity}\label{regularity-1}
|
\subsection{Regularity}\label{regularity-1}
|
||||||
|
|
||||||
\begin{figure}[tbh]
|
\begin{figure}[tbh]
|
||||||
\centering
|
\centering
|
||||||
\includegraphics[width=\textwidth]{img/evolution1d/55_to_1010_steps.png}
|
\includegraphics[width=\textwidth]{img/evolution1d/55_to_1010_steps.png}
|
||||||
\caption[Improvement potential and regularity vs. steps]{\newline
|
\caption[Improvement potential and regularity against iterations]{\newline
|
||||||
Left: Improvement potential against steps until convergence\newline
|
Left: *Improvement potential* against number of iterations until convergence\newline
|
||||||
Right: Regularity against steps until convergence\newline
|
Right: *Regularity* against number of iterations until convergence\newline
|
||||||
Coloured by their grid--resolution, both with a linear fit over the whole
|
Coloured by their grid--resolution, both with a linear fit over the whole
|
||||||
dataset.}
|
dataset.}
|
||||||
\label{fig:1dreg}
|
\label{fig:1dreg}
|
||||||
@ -1201,16 +1208,16 @@ $5 \times 5$ & $7 \times 4$ & $4 \times 7$ & $7 \times 7$ & $10 \times 10$\\
|
|||||||
\hline
|
\hline
|
||||||
$0.28$ ($0.0045$) & \textcolor{red}{$0.21$} ($0.0396$) & \textcolor{red}{$0.1$} ($0.3019$) & \textcolor{red}{$0.01$} ($0.9216$) & \textcolor{red}{$0.01$} ($0.9185$)
|
$0.28$ ($0.0045$) & \textcolor{red}{$0.21$} ($0.0396$) & \textcolor{red}{$0.1$} ($0.3019$) & \textcolor{red}{$0.01$} ($0.9216$) & \textcolor{red}{$0.01$} ($0.9185$)
|
||||||
\end{tabular}
|
\end{tabular}
|
||||||
\caption[Correlation 1D Regularity/Steps]{Spearman's correlation (and p-values)
|
\caption[Correlation 1D *regularity* against iterations]{Inverted Spearman's correlation (and p--values)
|
||||||
between regularity and convergence speed for the 1D function approximation
|
between *regularity* and number of iterations for the 1D function approximation
|
||||||
problem.
|
problem.
|
||||||
\newline Note: Not significant results are marked in \textcolor{red}{red}.
|
\newline Note: Not significant results are marked in \textcolor{red}{red}.
|
||||||
}
|
}
|
||||||
\label{tab:1dreg}
|
\label{tab:1dreg}
|
||||||
\end{table}
|
\end{table}
|
||||||
|
|
||||||
Regularity should correspond to the convergence speed (measured in
|
\emph{Regularity} should correspond to the convergence speed (measured
|
||||||
iteration--steps of the evolutionary algorithm), and is computed as
|
in iteration--steps of the evolutionary algorithm), and is computed as
|
||||||
inverse condition number \(\kappa(\vec{U})\) of the deformation--matrix.
|
inverse condition number \(\kappa(\vec{U})\) of the deformation--matrix.
|
||||||
|
|
||||||
As can be seen from table \ref{tab:1dreg}, we could only show a
|
As can be seen from table \ref{tab:1dreg}, we could only show a
|
||||||
@ -1222,27 +1229,27 @@ datasets into account we even get a \emph{strong} correlation of
|
|||||||
|
|
||||||
To explain this discrepancy we took a closer look at what caused these
|
To explain this discrepancy we took a closer look at what caused these
|
||||||
high number of iterations. In figure \ref{fig:1dreg} we also plotted the
|
high number of iterations. In figure \ref{fig:1dreg} we also plotted the
|
||||||
improvement-potential against the steps next to the regularity--plot.
|
\emph{improvement potential} against the steps next to the
|
||||||
Our theory is that the \emph{very strong} correlation
|
\emph{regularity}--plot. Our theory is that the \emph{very strong}
|
||||||
(\(-r_S = -0.82, p=0\)) between improvement--potential and number of
|
correlation (\(-r_S = -0.82, p=0\)) between \emph{improvement potential}
|
||||||
iterations hints that the employed algorithm simply takes longer to
|
and number of iterations hints that the employed algorithm simply takes
|
||||||
converge on a better solution (as seen in figure \ref{fig:1dvar} and
|
longer to converge on a better solution (as seen in figure
|
||||||
\ref{fig:1dimp}) offsetting any gain the regularity--measurement could
|
\ref{fig:1dvar} and \ref{fig:1dimp}) offsetting any gain the
|
||||||
achieve.
|
regularity--measurement could achieve.
|
||||||
|
|
||||||
\subsection{Improvement Potential}\label{improvement-potential-1}
|
\subsection{Improvement Potential}\label{improvement-potential-1}
|
||||||
|
|
||||||
\begin{figure}[ht]
|
\begin{figure}[ht]
|
||||||
\centering
|
\centering
|
||||||
\includegraphics[width=0.8\textwidth]{img/evolution1d/55_to_1010_improvement-vs-evo-error.png}
|
\includegraphics[width=0.8\textwidth]{img/evolution1d/55_to_1010_improvement-vs-evo-error.png}
|
||||||
\caption[Correlation 1D Improvement vs. Error]{Improvement potential plotted
|
\caption[Correlation 1D Improvement vs. Error]{*Improvement potential* plotted
|
||||||
against the error yielded by the evolutionary optimization for different
|
against the error yielded by the evolutionary optimization for different
|
||||||
grid--resolutions}
|
grid--resolutions}
|
||||||
\label{fig:1dimp}
|
\label{fig:1dimp}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
The improvement potential should correlate to the quality of the
|
The \emph{improvement potential} should correlate to the quality of the
|
||||||
fitting--result. We plotted the results for the tested grid-sizes
|
fitting--result. We plotted the results for the tested grid--sizes
|
||||||
\(5 \times 5\), \(7 \times 7\) and \(10 \times 10\) in figure
|
\(5 \times 5\), \(7 \times 7\) and \(10 \times 10\) in figure
|
||||||
\ref{fig:1dimp}. We tested the \(4 \times 7\) and \(7 \times 4\) grids
|
\ref{fig:1dimp}. We tested the \(4 \times 7\) and \(7 \times 4\) grids
|
||||||
as well, but omitted them from the plot.
|
as well, but omitted them from the plot.
|
||||||
@ -1280,7 +1287,7 @@ Initially we set up the correspondences \(\vec{c_T(\dots)}\) and
|
|||||||
other model. We then calculate the analytical solution given these
|
other model. We then calculate the analytical solution given these
|
||||||
correspondences via \(\vec{P^{*}} = \vec{U^+}\vec{T}\), and also use the
|
correspondences via \(\vec{P^{*}} = \vec{U^+}\vec{T}\), and also use the
|
||||||
first solution as guessed gradient for the calculation of the
|
first solution as guessed gradient for the calculation of the
|
||||||
\emph{improvement--potential}, as the optimal solution is not known. We
|
\emph{improvement potential}, as the optimal solution is not known. We
|
||||||
then let the evolutionary algorithm run up within \(1.05\) times the
|
then let the evolutionary algorithm run up within \(1.05\) times the
|
||||||
error of this solution and afterwards recalculate the correspondences
|
error of this solution and afterwards recalculate the correspondences
|
||||||
\(\vec{c_T(\dots)}\) and \(\vec{c_S(\dots)}\).
|
\(\vec{c_T(\dots)}\) and \(\vec{c_S(\dots)}\).
|
||||||
@ -1310,12 +1317,12 @@ regularization--effect wears off.
|
|||||||
|
|
||||||
The grid we use for our experiments is just very coarse due to
|
The grid we use for our experiments is just very coarse due to
|
||||||
computational limitations. We are not interested in a good
|
computational limitations. We are not interested in a good
|
||||||
reconstruction, but an estimate if the mentioned evolvability criteria
|
reconstruction, but an estimate if the mentioned evolvability--criteria
|
||||||
are good.
|
are good.
|
||||||
|
|
||||||
In figure \ref{fig:setup3d} we show an example setup of the scene with a
|
In figure \ref{fig:setup3d} we show an example setup of the scene with a
|
||||||
\(4\times 4\times 4\)--grid. Identical to the 1--dimensional scenario
|
\(4\times 4\times 4\)--grid. Identical to the 1--dimensional scenario
|
||||||
before, we create a regular grid and move the control-points in the
|
before, we create a regular grid and move the control--points in the
|
||||||
exact same random manner between their neighbours as described in
|
exact same random manner between their neighbours as described in
|
||||||
section \ref{sec:proc:1d}, but in three instead of two
|
section \ref{sec:proc:1d}, but in three instead of two
|
||||||
dimensions\footnote{Again, we flip the signs for the edges, if necessary
|
dimensions\footnote{Again, we flip the signs for the edges, if necessary
|
||||||
@ -1332,9 +1339,9 @@ Right: A $4 \times 4 \times 7$ grid that we expect to perform worse.}
|
|||||||
|
|
||||||
As is clearly visible from figure \ref{fig:3dgridres}, the target--model
|
As is clearly visible from figure \ref{fig:3dgridres}, the target--model
|
||||||
has many vertices in the facial area, at the ears and in the
|
has many vertices in the facial area, at the ears and in the
|
||||||
neck--region. Therefore we chose to increase the grid-resolutions for
|
neck--region. Therefore we chose to increase the grid--resolutions for
|
||||||
our tests in two different dimensions and see how well the criteria
|
our tests in two different dimensions and see how well the criteria
|
||||||
predict a suboptimal placement of these control-points.
|
predict a suboptimal placement of these control--points.
|
||||||
|
|
||||||
\section{Results of 3D Function
|
\section{Results of 3D Function
|
||||||
Approximation}\label{results-of-3d-function-approximation}
|
Approximation}\label{results-of-3d-function-approximation}
|
||||||
@ -1342,8 +1349,8 @@ Approximation}\label{results-of-3d-function-approximation}
|
|||||||
In the 3D--Approximation we tried to evaluate further on the impact of
|
In the 3D--Approximation we tried to evaluate further on the impact of
|
||||||
the grid--layout to the overall criteria. As the target--model has many
|
the grid--layout to the overall criteria. As the target--model has many
|
||||||
vertices in concentrated in the facial area we start from a
|
vertices in concentrated in the facial area we start from a
|
||||||
\(4 \times 4 \times 4\) grid and only increase the number of control
|
\(4 \times 4 \times 4\) grid and only increase the number of
|
||||||
points in one dimension, yielding a resolution of
|
control--points in one dimension, yielding a resolution of
|
||||||
\(7 \times 4 \times 4\) and \(4 \times 4 \times 7\) respectively. We
|
\(7 \times 4 \times 4\) and \(4 \times 4 \times 7\) respectively. We
|
||||||
visualized those two grids in figure \ref{fig:3dgridres}.
|
visualized those two grids in figure \ref{fig:3dgridres}.
|
||||||
|
|
||||||
@ -1374,10 +1381,10 @@ $4 \times 4 \times \mathrm{X}$ & $\mathrm{X} \times 4 \times 4$ & $\mathrm{Y} \t
|
|||||||
\hline
|
\hline
|
||||||
0.89 (0) & 0.9 (0) & 0.91 (0) & 0.94 (0)
|
0.89 (0) & 0.9 (0) & 0.91 (0) & 0.94 (0)
|
||||||
\end{tabular}
|
\end{tabular}
|
||||||
\caption[Correlation between variability and fitting error for 3D]{Correlation
|
\caption[Correlation between *variability* and fitting error for 3D]{Correlation
|
||||||
between variability and fitting error for the 3D fitting scenario.\newline
|
between *variability* and fitting error for the 3D fitting scenario.\newline
|
||||||
Displayed are the negated Spearman coefficients with the corresponding p-values
|
Displayed are the negated Spearman coefficients with the corresponding p--values
|
||||||
in brackets for three cases of increasing variability ($\mathrm{X} \in [4,5,7],
|
in brackets for three cases of increasing *variability* ($\mathrm{X} \in [4,5,7],
|
||||||
\mathrm{Y} \in [4,5,6]$).
|
\mathrm{Y} \in [4,5,6]$).
|
||||||
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
|
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
|
||||||
\label{tab:3dvar}
|
\label{tab:3dvar}
|
||||||
@ -1399,40 +1406,42 @@ we anticipated, which shows that the evolutionary algorithm we employed
|
|||||||
is capable of correcting a purposefully created \glqq bad\grqq ~grid.
|
is capable of correcting a purposefully created \glqq bad\grqq ~grid.
|
||||||
Also this confirms, that in our cases the number of control--points is
|
Also this confirms, that in our cases the number of control--points is
|
||||||
more important for quality than their placement, which is captured by
|
more important for quality than their placement, which is captured by
|
||||||
the variability via the rank of the deformation--matrix.
|
the \emph{variability} via the rank of the deformation--matrix.
|
||||||
|
|
||||||
|
Overall the correlation between \emph{variability} and fitness--error
|
||||||
|
were \emph{significant} and showed a \emph{very strong} correlation in
|
||||||
|
all our tests. The detailed correlation--coefficients are given in table
|
||||||
|
\ref{tab:3dvar} alongside their p--values.
|
||||||
|
|
||||||
|
As introduces in section \ref{sec:impl:grid} and visualized in figure
|
||||||
|
\ref{fig:enoughCP}, we know, that not all control--points have to
|
||||||
|
necessarily contribute to the parametrization of our 3D--model. Because
|
||||||
|
we are starting from a sphere, some control--points are too far away
|
||||||
|
from the surface to contribute to the deformation at all.
|
||||||
|
|
||||||
|
One can already see in 2D in figure \ref{fig:enoughCP}, that this effect
|
||||||
|
starts with a regular \(9 \times 9\) grid on a perfect circle. To make
|
||||||
|
sure we observe this, we evaluated the \emph{variability} for 100
|
||||||
|
randomly moved \(10 \times 10 \times 10\) grids on the sphere we start
|
||||||
|
out with.
|
||||||
|
|
||||||
\begin{figure}[hbt]
|
\begin{figure}[hbt]
|
||||||
\centering
|
\centering
|
||||||
\includegraphics[width=0.8\textwidth]{img/evolution3d/variability2_boxplot.png}
|
\includegraphics[width=0.8\textwidth]{img/evolution3d/variability2_boxplot.png}
|
||||||
\caption[Histogram of ranks of high--resolution deformation--matrices]{
|
\caption[Histogram of ranks of high--resolution deformation--matrices]{
|
||||||
Histogram of ranks of various $10 \times 10 \times 10$ grids with $1000$
|
Histogram of ranks of various $10 \times 10 \times 10$ grids with $1000$
|
||||||
control--points each showing in this case how many control points are actually
|
control--points each showing in this case how many control--points are actually
|
||||||
used in the calculations.
|
used in the calculations.
|
||||||
}
|
}
|
||||||
\label{fig:histrank3d}
|
\label{fig:histrank3d}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
Overall the correlation between variability and fitness--error were
|
As the \emph{variability} is defined by
|
||||||
\emph{significant} and showed a \emph{very strong} correlation in all
|
\(\frac{\mathrm{rank}(\vec{U})}{n}\) we can easily recover the rank of
|
||||||
our tests. The detailed correlation--coefficients are given in table
|
the deformation--matrix \(\vec{U}\). The results are shown in the
|
||||||
\ref{tab:3dvar} alongside their p--values.
|
histogram in figure \ref{fig:histrank3d}. Especially in the centre of
|
||||||
|
the sphere and in the corners of our grid we effectively loose
|
||||||
As introduces in section \ref{sec:impl:grid} and visualized in figure
|
control--points for our parametrization.
|
||||||
\ref{fig:enoughCP}, we know, that not all control points have to
|
|
||||||
necessarily contribute to the parametrization of our 3D--model. Because
|
|
||||||
we are starting from a sphere, some control-points are too far away from
|
|
||||||
the surface to contribute to the deformation at all.
|
|
||||||
|
|
||||||
One can already see in 2D in figure \ref{fig:enoughCP}, that this effect
|
|
||||||
starts with a regular \(9 \times 9\) grid on a perfect circle. To make
|
|
||||||
sure we observe this, we evaluated the variability for 100 randomly
|
|
||||||
moved \(10 \times 10 \times 10\) grids on the sphere we start out with.
|
|
||||||
|
|
||||||
As the variability is defined by \(\frac{\mathrm{rank}(\vec{U})}{n}\) we
|
|
||||||
can easily recover the rank of the deformation--matrix \(\vec{U}\). The
|
|
||||||
results are shown in the histogram in figure \ref{fig:histrank3d}.
|
|
||||||
Especially in the centre of the sphere and in the corners of our grid we
|
|
||||||
effectively loose control--points for our parametrization.
|
|
||||||
|
|
||||||
This of course yields a worse error as when those control--points would
|
This of course yields a worse error as when those control--points would
|
||||||
be put to use and one should expect a loss in quality evident by a
|
be put to use and one should expect a loss in quality evident by a
|
||||||
@ -1440,7 +1449,7 @@ higher reconstruction--error opposed to a grid where they are used.
|
|||||||
Sadly we could not run a in--depth test on this due to computational
|
Sadly we could not run a in--depth test on this due to computational
|
||||||
limitations.
|
limitations.
|
||||||
|
|
||||||
Nevertheless this hints at the notion, that variability is a good
|
Nevertheless this hints at the notion, that \emph{variability} is a good
|
||||||
measure for the overall quality of a fit.
|
measure for the overall quality of a fit.
|
||||||
|
|
||||||
\subsection{Regularity}\label{regularity-2}
|
\subsection{Regularity}\label{regularity-2}
|
||||||
@ -1468,21 +1477,21 @@ $4 \times 4 \times 4$ & $4 \times 4 \times 5$ & $4 \times 4 \times 7$ & $4 \time
|
|||||||
\cline{2-4}
|
\cline{2-4}
|
||||||
\multicolumn{3}{c}{} & all: 0.15 (0) \T
|
\multicolumn{3}{c}{} & all: 0.15 (0) \T
|
||||||
\end{tabular}
|
\end{tabular}
|
||||||
\caption[Correlation between regularity and iterations for 3D]{Correlation
|
\caption[Correlation between *regularity* and iterations for 3D]{Correlation
|
||||||
between regularity and number of iterations for the 3D fitting scenario.
|
between *regularity* and number of iterations for the 3D fitting scenario.
|
||||||
Displayed are the negated Spearman coefficients with the corresponding p--values
|
Displayed are the negated Spearman coefficients with the corresponding p--values
|
||||||
in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
|
in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
|
||||||
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
|
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
|
||||||
\label{tab:3dreg}
|
\label{tab:3dreg}
|
||||||
\end{table}
|
\end{table}
|
||||||
|
|
||||||
Opposed to the predictions of variability our test on regularity gave a
|
Opposed to the predictions of \emph{variability} our test on
|
||||||
mixed result --- similar to the 1D--case.
|
\emph{regularity} gave a mixed result --- similar to the 1D--case.
|
||||||
|
|
||||||
In roughly half of the scenarios we have a \emph{significant}, but
|
In roughly half of the scenarios we have a \emph{significant}, but
|
||||||
\emph{weak} to \emph{moderate} correlation between regularity and number
|
\emph{weak} to \emph{moderate} correlation between \emph{regularity} and
|
||||||
of iterations. On the other hand in the scenarios where we increased the
|
number of iterations. On the other hand in the scenarios where we
|
||||||
number of control--points, namely \(125\) for the
|
increased the number of control--points, namely \(125\) for the
|
||||||
\(5 \times 5 \times 5\) grid and \(216\) for the \(6 \times 6 \times 6\)
|
\(5 \times 5 \times 5\) grid and \(216\) for the \(6 \times 6 \times 6\)
|
||||||
grid we found a \emph{significant}, but \emph{weak}
|
grid we found a \emph{significant}, but \emph{weak}
|
||||||
\textbf{anti}--correlation when taking all three tests into
|
\textbf{anti}--correlation when taking all three tests into
|
||||||
@ -1491,14 +1500,14 @@ contradict the findings/trends for the sets with \(64\), \(80\), and
|
|||||||
\(112\) control--points (first two rows of table \ref{tab:3dreg}).
|
\(112\) control--points (first two rows of table \ref{tab:3dreg}).
|
||||||
|
|
||||||
Taking all results together we only find a \emph{very weak}, but
|
Taking all results together we only find a \emph{very weak}, but
|
||||||
\emph{significant} link between regularity and the number of iterations
|
\emph{significant} link between \emph{regularity} and the number of
|
||||||
needed for the algorithm to converge.
|
iterations needed for the algorithm to converge.
|
||||||
|
|
||||||
\begin{figure}[!htb]
|
\begin{figure}[!htb]
|
||||||
\centering
|
\centering
|
||||||
\includegraphics[width=\textwidth]{img/evolution3d/regularity_montage.png}
|
\includegraphics[width=\textwidth]{img/evolution3d/regularity_montage.png}
|
||||||
\caption[Regularity for different 3D--grids]{
|
\caption[Regularity for different 3D--grids]{
|
||||||
Plots of regularity against number of iterations for various scenarios together
|
Plots of *regularity* against number of iterations for various scenarios together
|
||||||
with a linear fit to indicate trends.}
|
with a linear fit to indicate trends.}
|
||||||
\label{fig:resreg3d}
|
\label{fig:resreg3d}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
@ -1509,10 +1518,10 @@ The regularity--criterion first behaves as we would like to, but then
|
|||||||
switches to behave exactly opposite to our expectations, as can be seen
|
switches to behave exactly opposite to our expectations, as can be seen
|
||||||
in the first three plots. While the number of control--points increases
|
in the first three plots. While the number of control--points increases
|
||||||
from red to green to blue and the number of iterations decreases, the
|
from red to green to blue and the number of iterations decreases, the
|
||||||
regularity seems to increase at first, but then decreases again on
|
\emph{regularity} seems to increase at first, but then decreases again
|
||||||
higher grid--resolutions.
|
on higher grid--resolutions.
|
||||||
|
|
||||||
This can be an artefact of the definition of regularity, as it is
|
This can be an artefact of the definition of \emph{regularity}, as it is
|
||||||
defined by the inverse condition--number of the deformation--matrix
|
defined by the inverse condition--number of the deformation--matrix
|
||||||
\(\vec{U}\), being the fraction
|
\(\vec{U}\), being the fraction
|
||||||
\(\frac{\sigma_{\mathrm{min}}}{\sigma_{\mathrm{max}}}\) between the
|
\(\frac{\sigma_{\mathrm{min}}}{\sigma_{\mathrm{max}}}\) between the
|
||||||
@ -1524,12 +1533,12 @@ small minimal right singular value occurring on higher grid--resolutions
|
|||||||
seems likely the problem.
|
seems likely the problem.
|
||||||
|
|
||||||
Adding to this we also noted, that in the case of the
|
Adding to this we also noted, that in the case of the
|
||||||
\(10 \times 10 \times 10\)--grid the regularity was always \(0\), as a
|
\(10 \times 10 \times 10\)--grid the \emph{regularity} was always \(0\),
|
||||||
non--contributing control-point yields a \(0\)--column in the
|
as a non--contributing control--point yields a \(0\)--column in the
|
||||||
deformation--matrix, thus letting \(\sigma_\mathrm{min} = 0\). A better
|
deformation--matrix, thus letting \(\sigma_\mathrm{min} = 0\). A better
|
||||||
definition for regularity (i.e.~using the smallest non--zero right
|
definition for \emph{regularity} (i.e.~using the smallest non--zero
|
||||||
singular value) could solve this particular issue, but not fix the trend
|
right singular value) could solve this particular issue, but not fix the
|
||||||
we noticed above.
|
trend we noticed above.
|
||||||
|
|
||||||
\subsection{Improvement Potential}\label{improvement-potential-2}
|
\subsection{Improvement Potential}\label{improvement-potential-2}
|
||||||
|
|
||||||
@ -1556,8 +1565,8 @@ $4 \times 4 \times 4$ & $4 \times 4 \times 5$ & $4 \times 4 \times 7$ & $4 \time
|
|||||||
\cline{2-4}
|
\cline{2-4}
|
||||||
\multicolumn{3}{c}{} & all: 0.95 (0) \T
|
\multicolumn{3}{c}{} & all: 0.95 (0) \T
|
||||||
\end{tabular}
|
\end{tabular}
|
||||||
\caption[Correlation between improvement--potential and fitting--error for 3D]{Correlation
|
\caption[Correlation between *improvement potential* and fitting--error for 3D]{Correlation
|
||||||
between improvement--potential and fitting--error for the 3D fitting scenario.
|
between *improvement potential* and fitting--error for the 3D fitting scenario.
|
||||||
Displayed are the negated Spearman coefficients with the corresponding p--values
|
Displayed are the negated Spearman coefficients with the corresponding p--values
|
||||||
in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
|
in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
|
||||||
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
|
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
|
||||||
@ -1576,21 +1585,22 @@ gradients anyway.
|
|||||||
\centering
|
\centering
|
||||||
\includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png}
|
\includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png}
|
||||||
\caption[Improvement potential for different 3D--grids]{
|
\caption[Improvement potential for different 3D--grids]{
|
||||||
Plots of improvement potential against error given by our fitness--function
|
Plots of *improvement potential* against error given by our *fitness--function*
|
||||||
after convergence together with a linear fit of each of the plotted data to
|
after convergence together with a linear fit of each of the plotted data to
|
||||||
indicate trends.}
|
indicate trends.}
|
||||||
\label{fig:resimp3d}
|
\label{fig:resimp3d}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
We plotted our findings on the improvement potential in a similar way as
|
We plotted our findings on the \emph{improvement potential} in a similar
|
||||||
we did before with the regularity. In figure \ref{fig:resimp3d} one can
|
way as we did before with the \emph{regularity}. In figure
|
||||||
clearly see the correlation and the spread within each setup and the
|
\ref{fig:resimp3d} one can clearly see the correlation and the spread
|
||||||
behaviour when we increase the number of control--points.
|
within each setup and the behaviour when we increase the number of
|
||||||
|
control--points.
|
||||||
|
|
||||||
Along with this we also give the Spearman--coefficients along with their
|
Along with this we also give the Spearman--coefficients along with their
|
||||||
p--values in table \ref{tab:3dimp}. Within one scenario we only find a
|
p--values in table \ref{tab:3dimp}. Within one scenario we only find a
|
||||||
\emph{weak} to \emph{moderate} correlation between the improvement
|
\emph{weak} to \emph{moderate} correlation between the \emph{improvement
|
||||||
potential and the fitting error, but all findings (except for
|
potential} and the fitting error, but all findings (except for
|
||||||
\(7 \times 4 \times 4\) and \(6 \times 6 \times 6\)) are significant.
|
\(7 \times 4 \times 4\) and \(6 \times 6 \times 6\)) are significant.
|
||||||
|
|
||||||
If we take multiple datasets into account the correlation is \emph{very
|
If we take multiple datasets into account the correlation is \emph{very
|
||||||
@ -1598,30 +1608,30 @@ strong} and \emph{significant}, which is good, as this functions as a
|
|||||||
litmus--test, because the quality is naturally tied to the number of
|
litmus--test, because the quality is naturally tied to the number of
|
||||||
control--points.
|
control--points.
|
||||||
|
|
||||||
All in all the improvement potential seems to be a good and sensible
|
All in all the \emph{improvement potential} seems to be a good and
|
||||||
measure of quality, even given gradients of varying quality.
|
sensible measure of quality, even given gradients of varying quality.
|
||||||
|
|
||||||
Lastly, a small note on the behaviour of improvement potential and
|
Lastly, a small note on the behaviour of \emph{improvement potential}
|
||||||
convergence speed, as we used this in the 1D case to argue, why the
|
and convergence speed, as we used this in the 1D case to argue, why the
|
||||||
\emph{regularity} defied our expectations. As a contrast we wanted to
|
\emph{regularity} defied our expectations. As a contrast we wanted to
|
||||||
show, that improvement potential cannot serve for good predictions of
|
show, that \emph{improvement potential} cannot serve for good
|
||||||
the convergence speed. In figure \ref{fig:imp1d3d} we show improvement
|
predictions of the convergence speed. In figure \ref{fig:imp1d3d} we
|
||||||
potential against number of iterations for both scenarios. As one can
|
show \emph{improvement potential} against number of iterations for both
|
||||||
see, in the 1D scenario we have a \emph{strong} and \emph{significant}
|
scenarios. As one can see, in the 1D scenario we have a \emph{strong}
|
||||||
correlation (with \(-r_S = -0.72\), \(p = 0\)), whereas in the 3D
|
and \emph{significant} correlation (with \(-r_S = -0.72\), \(p = 0\)),
|
||||||
scenario we have the opposite \emph{significant} and \emph{strong}
|
whereas in the 3D scenario we have the opposite \emph{significant} and
|
||||||
effect (with \(-r_S = 0.69\), \(p=0\)), so these correlations clearly
|
\emph{strong} effect (with \(-r_S = 0.69\), \(p=0\)), so these
|
||||||
seem to be dependent on the scenario and are not suited for
|
correlations clearly seem to be dependent on the scenario and are not
|
||||||
generalization.
|
suited for generalization.
|
||||||
|
|
||||||
\begin{figure}[hbt]
|
\begin{figure}[hbt]
|
||||||
\centering
|
\centering
|
||||||
\includegraphics[width=\textwidth]{img/imp1d3d.png}
|
\includegraphics[width=\textwidth]{img/imp1d3d.png}
|
||||||
\caption[Improvement potential and convergence speed for 1D and 3D--scenarios]{
|
\caption[Improvement potential and convergence speed\newline for 1D and 3D--scenarios]{
|
||||||
\newline
|
\newline
|
||||||
Left: Improvement potential against convergence speed for the
|
Left: *Improvement potential* against convergence speed for the
|
||||||
1D--scenario\newline
|
1D--scenario\newline
|
||||||
Right: Improvement potential against convergence speed for the 3D--scnario
|
Right: *Improvement potential* against convergence speed for the 3D--scnario
|
||||||
}
|
}
|
||||||
\label{fig:imp1d3d}
|
\label{fig:imp1d3d}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
@ -1630,36 +1640,36 @@ Right: Improvement potential against convergence speed for the 3D--scnario
|
|||||||
|
|
||||||
\label{sec:dis}
|
\label{sec:dis}
|
||||||
|
|
||||||
In this thesis we took a look at the different criteria for evolvability
|
In this thesis we took a look at the different criteria for
|
||||||
as introduced by Richter et al.\cite{anrichterEvol}, namely
|
\emph{evolvability} as introduced by Richter et al.\cite{anrichterEvol},
|
||||||
\emph{variability}, \emph{regularity} and \emph{improvement potential}
|
namely \emph{variability}, \emph{regularity} and \emph{improvement
|
||||||
under different setup--conditions. Where Richter et al. used \acf{RBF},
|
potential} under different setup--conditions. Where Richter et al. used
|
||||||
we employed \acf{FFD} to set up a low--complexity parametrization of a
|
\acf{RBF}, we employed \acf{FFD} to set up a low--complexity
|
||||||
more complex vertex--mesh.
|
parametrization of a more complex vertex--mesh.
|
||||||
|
|
||||||
In our findings we could show in the 1D--scenario, that there were
|
In our findings we could show in the 1D--scenario, that there were
|
||||||
statistically significant very strong correlations between
|
statistically \emph{significant} \emph{very strong} correlations between
|
||||||
\emph{variability and fitting error} (\(0.94\)) and
|
\emph{variability and fitting error} (\(0.94\)) and \emph{improvement
|
||||||
\emph{improvement--potential and fitting error} (\(1.0\)) with
|
potential and fitting error} (\(1.0\)) with comparable results than
|
||||||
comparable results than Richter et al. (with \(0.31\) to \(0.88\) for
|
Richter et al. (with \(0.31\) to \(0.88\) for the former and \(0.75\) to
|
||||||
the former and \(0.75\) to \(0.99\) for the latter), whereas we found
|
\(0.99\) for the latter), whereas we found only \emph{weak} correlations
|
||||||
only weak correlations for \emph{regularity and convergence--speed}
|
for \emph{regularity and convergence--speed} (\(0.28\)) opposed to
|
||||||
(\(0.28\)) opposed to Richter et al. with \(0.39\) to
|
Richter et al. with \(0.39\) to \(0.91\).\footnote{We only took
|
||||||
\(0.91\).\footnote{We only took statistically \emph{significant} results
|
statistically \emph{significant} results into consideration when
|
||||||
into consideration when compiling these numbers. Details are given in
|
compiling these numbers. Details are given in the respective chapters.}
|
||||||
the respective chapters.}
|
|
||||||
|
|
||||||
For the 3D--scenario our results show a very strong, significant
|
For the 3D--scenario our results show a \emph{very strong},
|
||||||
correlation between \emph{variability and fitting error} with \(0.89\)
|
\emph{significant} correlation between \emph{variability and fitting
|
||||||
to \(0.94\), which are pretty much in line with the findings of Richter
|
error} with \(0.89\) to \(0.94\), which are pretty much in line with the
|
||||||
et al. (\(0.65\) to \(0.95\)). The correlation between \emph{improvement
|
findings of Richter et al. (\(0.65\) to \(0.95\)). The correlation
|
||||||
potential and fitting error} behave similar, with our findings having a
|
between \emph{improvement potential and fitting error} behave similar,
|
||||||
significant coefficient of \(0.3\) to \(0.95\) depending on the
|
with our findings having a significant coefficient of \(0.3\) to
|
||||||
grid--resolution compared to the \(0.61\) to \(0.93\) from Richter et
|
\(0.95\) depending on the grid--resolution compared to the \(0.61\) to
|
||||||
al. In the case of the correlation of \emph{regularity and convergence
|
\(0.93\) from Richter et al. In the case of the correlation of
|
||||||
speed} we found very different (and often not significant) correlations
|
\emph{regularity and convergence speed} we found very different (and
|
||||||
and anti--correlations ranging from \(-0.25\) to \(0.46\), whereas
|
often not significant) correlations and anti--correlations ranging from
|
||||||
Richter et al. reported correlations between \(0.34\) to \(0.87\).
|
\(-0.25\) to \(0.46\), whereas Richter et al. reported correlations
|
||||||
|
between \(0.34\) to \(0.87\).
|
||||||
|
|
||||||
Taking these results into consideration, one can say, that
|
Taking these results into consideration, one can say, that
|
||||||
\emph{variability} and \emph{improvement potential} are very good
|
\emph{variability} and \emph{improvement potential} are very good
|
||||||
@ -1683,14 +1693,11 @@ manipulation in \cite{anrichterEvol}, whereas we merely used an indirect
|
|||||||
indirect manipulations, the usage of \acf{DM--FFD} could also work
|
indirect manipulations, the usage of \acf{DM--FFD} could also work
|
||||||
better with the criteria we examined. This can also solve the problem of
|
better with the criteria we examined. This can also solve the problem of
|
||||||
bad singular values for the \emph{regularity} as the incorporation of
|
bad singular values for the \emph{regularity} as the incorporation of
|
||||||
the parametrization of the points on the surface, which are the
|
the parametrization of the points on the surface --- which are the
|
||||||
essential part of a direct--manipulation, could cancel out a bad
|
essential part of a direct--manipulation --- could cancel out a bad
|
||||||
control--grid as the bad control--points are never or negligibly used to
|
control--grid as the bad control--points are never or negligibly used to
|
||||||
parametrize those surface--points.
|
parametrize those surface--points.
|
||||||
|
|
||||||
\improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt
|
|
||||||
Direktlinks des Autors.}
|
|
||||||
|
|
||||||
% \backmatter
|
% \backmatter
|
||||||
\cleardoublepage
|
\cleardoublepage
|
||||||
|
|
||||||
@ -1725,10 +1732,10 @@ Direktlinks des Autors.}
|
|||||||
% \addtocounter{chapter}{1}
|
% \addtocounter{chapter}{1}
|
||||||
\newpage
|
\newpage
|
||||||
% \listoftables
|
% \listoftables
|
||||||
\listoftodos
|
% \listoftodos
|
||||||
% \addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}TODOs}
|
% \addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}TODOs}
|
||||||
% \addtocounter{chapter}{1}
|
% \addtocounter{chapter}{1}
|
||||||
\newpage
|
% \newpage
|
||||||
% \printindex
|
% \printindex
|
||||||
|
|
||||||
%%%%%%%%%%%%%%% Erklaerung %%%%%%%%%%%%%%%
|
%%%%%%%%%%%%%%% Erklaerung %%%%%%%%%%%%%%%
|
||||||
|
|||||||
@ -175,10 +175,10 @@ $body$
|
|||||||
% \addtocounter{chapter}{1}
|
% \addtocounter{chapter}{1}
|
||||||
\newpage
|
\newpage
|
||||||
% \listoftables
|
% \listoftables
|
||||||
\listoftodos
|
% \listoftodos
|
||||||
% \addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}TODOs}
|
% \addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}TODOs}
|
||||||
% \addtocounter{chapter}{1}
|
% \addtocounter{chapter}{1}
|
||||||
\newpage
|
% \newpage
|
||||||
% \printindex
|
% \printindex
|
||||||
|
|
||||||
%%%%%%%%%%%%%%% Erklaerung %%%%%%%%%%%%%%%
|
%%%%%%%%%%%%%%% Erklaerung %%%%%%%%%%%%%%%
|
||||||
|
|||||||
@ -14,7 +14,7 @@ set xtics norangelimit
|
|||||||
set xtics ()
|
set xtics ()
|
||||||
set ytics border in scale 1,0.5 nomirror norotate autojustify
|
set ytics border in scale 1,0.5 nomirror norotate autojustify
|
||||||
set title "Fitting Errors of 1D Function Approximation for various grids\n"
|
set title "Fitting Errors of 1D Function Approximation for various grids\n"
|
||||||
set ylabel "Squared Error of Vertex-Difference"
|
set ylabel "Fitting-Error according to fitness-function"
|
||||||
|
|
||||||
header ="`head -1 errors.csv | sed -s "s/\"//g" | sed -s "s/,/ /g"`"
|
header ="`head -1 errors.csv | sed -s "s/\"//g" | sed -s "s/,/ /g"`"
|
||||||
set for [i=1:words(header)] xtics (word(header,i) i)
|
set for [i=1:words(header)] xtics (word(header,i) i)
|
||||||
|
|||||||
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|
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Loading…
Reference in New Issue
Block a user