Lattice points enumeration for image coding

The use of lattices as vector quantizers in still image and video compression schemes has grown increasingly in the last few years. In order to obtain a compromise between minimum distortion and bit rate, lattices have to be truncated so that the chosen lattice points, the codebook size, lie within a finite boundary. The determination of this boundary is source dependent. The geometric properties of a memoryless Laplacian source fit properly to model transform coded image statistics. In this case, the l₁ norm is more suitable than the it norm (then the classical lattice Θ series should no longer be used). In this paper we define the contour points, which count how many lattice points are at l₁ distance m from a given lattice point; i.e., they help to establish the lattice boundary or equivalently the codebook size. Explicit combinatorial expressions for the codebook size for lattices Z^d, A_d, D_d, D_d *, D_d⁺, Construction A and Construction B are given. These expressions pave the way for algorithms for labeling lattice points that achieve full efficiency