Transforms associated to square integrable group representations. I. General results

Abstract

Let G be a locally compact group, which need not be unimodular. Let x→U(x) (x∈G) be an irreducible unitary representation of G in a Hilbert space ℋ(U). Assume that U is square integrable, i.e., that there exists in ℋ(U) at least one nonzero vector g such that ∫‖(U(x)g,g)‖2 dx<∞. We give here a reasonably self-contained analysis of the correspondence associating to every vector f∈ℋ(U) the function (U(x)g,f) on G, discussing its isometry, characterization of the range, inversion, and simplest interpolation properties. This correspondence underlies many properties of generalized coherent states.

Preserving properties of object shape by computations in primary visual cortex

“…First, suppose that f( , ) is a wavelet transform (3, 5) of f(x) so that f( , ) ϵ (G , (x), f(x)); the daughter wavelets are generated from the mother by translations and scalings (3,5). The representation of a scaled (by s) and translated (by t) image f s,t ( , ) is given by the following:…”

“…Set ϭ 1 and ϭ 0, and define 1,0 (x) ϵ (x). Then, we have the following: s,t ͑x͒ ϭ ͱs ͑s͑x Ϫ t͒͒ ϭ G s,t ͑x͒, which means that f( , ) is the result of a wavelet transform if (x) has the properties of a mother wavelet [the integral of (x) must vanish, and the square of the function must posses certain convergence properties (3,5)]. Any function (x) is satisfactory as long as the function can be a mother wavelet so that the transformation carried out in the cortex is invertible (and, thus, does not destroy information about the object shape).…”

“…Because objects are not extracted in the primary visual cortex (V1) but rather in later areas (2), the representation of an image in V1 should not eliminate the essential properties of objects needed for computations by other areas. I show that this requirement for preserving abstract shape properties (described below) in the cortical representation of images restricts the computations that V1 can carry out to a particular class known as wavelet transforms (3)(4)(5). Indeed, analysis of experimental data reveals that V1 can be said to perform a wavelet transform of the image presented to the retina because the firing rate of each simple cell can be calculated from the image by a particular wavelet transform, and the image can, in principle, be reconstructed from the cell firing in V1 by an inverse wavelet transform.…”

Informationally complete measurements and group representation

“…2 does not directly apply. However, the displacement representation is square integrable [24], and the main identity (9) still holds in the form…”

Wavelet-based integral representation for solutions of the wave equation

“…The main ideas of wavelet analysis take their origin in group representation theory and in the theory of coherent states (see [5,6] and references therein). The first papers on continuous wavelet analysis theory were motivated by applications to seismic wave propagation [7,8]. These papers stimulated interest in wavelet analysis.…”