Robust Hierarchical Voxel Models for Representation and Interactive

In this article we present a new method of vox-elizing implicit surfaces in spherical coordinates. The approach has several advantages: (1) The voxeliza-tion algorithm is robust and has low complexity; (2) allowing high resolution voxelizations, including nor-mal calculations, in reasonable time and (3) its vi-sualization using GL points with normals generates Ray-Casting quality images at interactive speeds. The approach is robust because no part of the sur-face will be missed in the voxel model and because no oating point errors will occur. The robustness emerges from the use of interval arithmetic. Voxeliza-tion is done by subdividing the rectangular space in a recursive way until the voxel level is reached. The rectangular intervals are then converted to spherical intervals and interval arithmetic assists in identify-ing whether the surface does not cross a subdivided space at a very early stage. This algorithm has a dual advantage: voxelization is guaranteed to envelop the true surface, and its experimental running time in-creases only by a factor of 4, instead of 8, every time resolution is doubled for every coordinate axis. The resulting voxelization is stored in an octree de ning a new concept called here robust hierarchical voxel model which guarantees that every level of the tree denes a set of voxels which always envelop the continuous surface. Each voxel stores the normal vec-tor, calculated by the gradient of the function during the voxelization. The visualization method used to render the hier-archical voxel model in this article uses GL/OpenGL point primitives and no special hardware is required besides the Z-Buer