masterarbeit/dokumentation/B-Spline-Volumes.md

B-Spline Volumes

B-Spline Volumes are a simple extension of B-Splines to 3 Dimensions. This is a straightforward adaption of the 2-Dimensional version.

Nomenclature

x,y,z denote space-coordinates,
u,v,w denote spline-coordinates (Between 0-1),
P_{ijk} denote the control-Points on the control-Polygon,
N_{i,d,\tau}(u) denote the value of the underlying Basis-Functions at value u using the $i$-th Basis-Function of degree d in range \tau.

For our case we only care about degree-3 splines, so we omit the d furtheron. \tau is defined statically (in each direction) with each P as Position on the whole surface/volume and within [0,1]. For a regular Control-Grid this defaults to \tau_i = i/n

Given n,m,o control points in $x,y,z$-direction each Point on the curve is defined by

C(u,v,w) = \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} P_{ijk} N_{i}(u) N_j(v) N_k(w)

Calculate u, v, w

Given a target-point \textbf{p}^* and an initial guess \textbf{p}=C(u,v,w) we define the error-function as:

Err(u,v,w,\textbf{p}^{*}) = \| \textbf{p}^{*} - C(u,v,w) \|_2^2

As the error is just the sum of the components

Err(u,v,w,\textbf{p}^{*}) = Err_x(u,v,w,\textbf{p}^{*}) + Err_y(u,v,w,\textbf{p}^{*}) + Err_z(u,v,w,\textbf{p}^{*})

we just take one axis into account, as the others are nearly identical. So we yield

Err_x(u,v,w,\textbf{p}^{*}) = \left( p^{*}_x - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {P_{ijk}}_x N_{i}(u) N_j(v) N_k(w) \right)^2 

To solve this we derive:

\begin{array}{rl}
        \displaystyle \frac{\partial}{\partial u} & \left( p^{*}_x - \displaystyle \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {P_{ijk}}_x N_{i}(u) N_j(v) N_k(w) \right)^2 \\
                                  = & \displaystyle - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {P_{ijk}}_x N'_{i}(u) N_j(v) N_k(w) \\
                                    & \cdot \left( p^{*}_x - \displaystyle \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {P_{ijk}}_x N_{i}(u) N_j(v) N_k(w) \right) \\
                                    & \cdot 2
\end{array}

The other partial derivatives follow the same pattern.

Basis-Splines and Derivatives

The previously mentioned N_{i,d,\tau} are defined recursively:

N_{i,0,\tau}(u) = \begin{cases} 1, & u \in [\tau_i, \tau_{i+1}[ \\ 0, & \mbox{otherwise} \end{cases} 

and

N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+d+1} - u}{\tau_{i+d+1}-\tau_{i+1}} N_{i+1,d-1,\tau}(u) 

This fact can be exploited to get the derivative for an arbitrary N:

\frac{d}{du} 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)

Warning: in the case of d=1 the recursion-formula yields a 0 denominator, but N is also 0. The right solution for this case is a derivative of 0