Volume reconstruction based on the six-direction cubic box-spline

We propose a new volume reconstruction technique based on the six-direction cubic box-spline $M_6$. $M_6$ is $C^1$ continuous and possesses an approximation order of three, the same as that of the tri-quadratic B-spline but with much lower degree. In fact, $M_6$ has the lowest degree among the symmetric box-splines on $Z^3$ with at least $C^1$ continuity. We analyze the polynomial structure induced by the shifts of $M_6$ and propose an efficient analytic evaluation algorithm for splines and their derivatives (gradient and Hessian) based on the high symmetry of $M_6$. To verify the evaluation algorithm, we implement a real-time GPU (graphics processing unit) isosurface raycaster which exhibits interactive performance (54.5 fps (frames per second) with $241^3$ dataset on $512^2$ framebuffer) on a modern graphics hardware. Moreover, we analyze $M_6$ as a reconstruction filter and state that it is comparable to the tri-cubic B-spline, which possesses a higher approximation order.