12Symmetry Invariant Signals and Systems
12.1 LSI Systems Invariant to a Group of Symmetries
A symmetry is a transformation that leaves a certain entity unchanged. For example, the translation of a lattice by an element of that lattice leaves the lattice unchanged (see Section 13.3). However, translations are not the only symmetries of a lattice. More generally, an isometry of 
that leaves a lattice 
invariant is called a symmetry of 
. An isometry of 
is a permutation of 
that leaves the Euclidean distance between any pair of points unchanged. Besides translations, these consist of rotations and reflections. These symmetries have been well studied, especially in two or three dimensions in crystallography (e.g., Miller (1972)), and the four‐dimensional case is completely covered in Brown et al. (1978). We first present the idea of symmetry invariance in general terms, then present some specific examples of particular interest in signal processing. An example of symmetry invariance in one ...
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