Fast and stable evaluation of splines and their derivatives generated by the seven-direction quartic box-spline

In this paper, we propose a fast and stable evaluation method for the analytic derivatives of splines generated by the 7-direction quartic box-spline. We can maintain the spline structure by determining the derivative functions that can be represented as finite differences of box-splines defined by the sub-directions. Thus, the evaluation overhead can be reduced. We demonstrate that the first and second derivative functions are composed of only three cubic and six quadratic polynomial formulas, respectively, owing to their symmetries. Moreover, for each derivative order, all of the required functions possess change-of-variables relation with each other. Therefore, additional formulas are not required. As a result, for a given point, we only need to evaluate one quartic, three cubic, and six quadratic polynomial formulas to evaluate its spline value, gradient, and Hessian, respectively. This reduction in cases is especially advantageous for graphics processing unit (GPU) kernels, where conditional statements significantly degrade performance. To verify our technique, we implemented a real-time curvature-based GPU isosurface raycaster. Compared with other implementations, our method (i) achieves superior accuracy, (ii) is more than four times faster, and (iii) requires less than 15% of memory.