II.5 - Parallelohedra and Uniform Quantization

Abstract

This chapter focuses on parallelohedra and uniform quantization. It describes a method of quantizing values (locating the nearest neighbor) in 3-space. The method was originally intended as an optimal means of color coding, using a non-Cartesian partitioning of space. The solution, based upon the geometry of the truncated octahedron, has general applications, as in heuristics for intersection testing. Implementing a body-centered cubic lattice quantizer is a straightforward task. The chapter shows that a body-centered cubic lattice is equivalent to two interlaced simple cubic lattices, A and B, whose lattice points are represented by filled and open circles. The given color is quantized first on lattice A in the usual way by independently quantizing the L*, a*, and b* coordinates. The color is then quantized on lattice B in the same way. If the lattice point on A that is closest to the color is closer to it than the closest lattice point on B, then the closest lattice point on A is returned. Otherwise, the closest lattice point on B is returned. So a body-centered cubic lattice quantizer is equivalent to a program that compares the quantization errors of two simple cubic lattice quantizers.