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On Near Optimal Lattice Quantization of Multi-Dimensional Data Points

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Abstract

One of the most elementary application of a lattice is the quantization of real-valued s-dimensional vectors into finite bit precision to make them representable by a digital computer. Most often, the simple s-dimensional regular grid is used for this task where each component of the vector is quantized individually. However, it is known that other lattices perform better regarding the average quantization error. A rank-1 lattices is a special type of lattice, where the lattice points can be described by a single s-dimensional generator vector. Further, the number of points inside the unit cube [0, 1)s is arbitrary and can be directly enumerated by a single one-dimensional integer value. By choosing a suitable generator vector the minimum distance between the lattice points can be maximized which, as we show, leads to a nearly optimal mean quantization error. We present methods for finding parameters for s-dimensional maximized minimum distance rank-1 lattices and further show their practical use in computer graphics applications.

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