Multidimensional Signal and Color Image Processing Using Lattices

13Lattices

13.1 Introduction

This chapter gathers in one place the main definitions and results on lattices used throughout this book. Proofs of many results are also provided here. Lattices are used in many fields including the geometry of numbers [Cassels (1997)], crystallography [Miller (1972)], communications [Wübben et al. (2011)], cryptography [Micciancio and Goldwasser (2002)], vector quantization [Gersho and Gray (1992)], and in our domain of interest, the sampling of multidimensional signals. Some of these applications involve a relatively high number of dimensions, whereas in multidimensional sampling we generally deal with two–four dimensions. Thus, the choice of properties and algorithms to present here is guided by our application domain, where complexity issues in a high number of dimensions do not really concern us.

13.2 Basic Definitions

A lattice is a regular, discrete set of points in a ‐dimensional Euclidean space . It is discrete in the sense that it is composed of discrete isolated points in , and it is regular in the sense that the neighborhood of every lattice point looks the same. We assume that is equipped with an orthonormal basis, typically corresponding ...