Analytic evaluation of the FCC Voronoi-splines

Abstract

We propose an analytic evaluation algorithm for splines and their derivatives (gradient and Hessian) of FCC volume datasets based on the Voronoi-spline of order two and three, $V_{1}(\varvec{x})$ and $V_{2}(\varvec{x})$, respectively. Based on the ideas by Van De Ville et al. (IEEE Trans. Image Process 13(6), 758–772, 2004) and Mirzargar and Entezari (IEEE Trans. Signal Process 58(9), 4572–4582, 2010), we obtain the analytic formulas by merging those of the box-splines into which the Voronoi-spline basis is decomposed. The polynomial formulas of the high order box-splines are computed using a modified version of the recursive evaluation package by de Boor (Numer. Algorithms 5(1), 5–23, 1993). We also analyze the symmetries of the polynomial structure and the Voronoi-spline to minimize the number of formulas, which hugely improves the performance especially on GPUs. Our GPU isosurface raycaster, which is publicly available, shows that $V_{1}$ and $V_{2}$ are appropriate for high-speed and high-quality interactive FCC volume renderings, respectively. Specifically, $V_{1}$ and $V_{2}$ shows about 240 and 41.5 fps (frames per second), respectively, for the $160^3\times 4$ FCC dataset rendered on the $512^2$ framebuffer. Compared to the approximate technique based on texture look-up, ours shows $\approx 5.7$ times better performance with superior image quality. Moreover, compared to the analytic evaluation algorithm recently proposed (Horacsek and Alim 2022), ours shows $> 2$ times better computational performance in high-quality real-world application.