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

Abstract

We propose a new volume reconstruction technique based on the six-direction cubic box-spline M6. M6 is C1 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, M6 has the lowest degree among the symmetric box-splines on Z3 with at least C1 continuity. We analyze the polynomial structure induced by the shifts of M6 and propose an efficient analytic evaluation algorithm for splines and their derivatives (gradient and Hessian) based on the high symmetry of M6. 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 2413 dataset on 5122 framebuffer) on a modern graphics hardware. Moreover, we analyze M6 as a reconstruction filter and state that it is comparable to the tri-cubic B-spline, which possesses a higher approximation order.