Affine Systems inL2(Rd): The Analysis of the Analysis Operator

Abstract

Discrete affine systems are obtained by applying dilations to a given shiftinvariant system. The complicated structure of the affine system is due, first and foremost, to the fact that it is not invariant under shifts. Affine frames carry the additional difficulty that they are``global'' in nature: it is the entire interaction between the various dilation levels that determines whether the system is a frame, and not the behaviour of the system within one dilation level. We completely unravel the structure of the affine system with the aid of two new notions: the affine product, and a quasi-affine system. This leads to a characterization of affine frames; the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the Fourier domain. Several results, such as a general oversampling theorem follow from these characterizations. Most importantly, the affine product can be factored during a multiresolution analysis construction, and this leads to a complete characterization of all tight frames that can be constructed by such methods. Moreover, this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution. Of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions, and that the freedom offered by redundancy can be fully exploited in these constructions.

Blind motion deblurring from a single image using sparse approximation

“…translationinvariant wavelet, Gabor transform, Local cosine transform ( [20]), framelets ( [24]) and curvelets ( [8]). The sparsity of nature images under these tight frames has been successfully used to solve many image restoration tasks including image denoising, non-blind image deblurring, image inpainting, etc (e.g.…”

Representations, Wavelets, and Frames

“…Wavelet frames in such spaces are very useful for signal and image processing. The OEP is a generalization of the unitary extension principle (UEP) for wavelet frames first developed by A. Ron and Z. Shen [52,53]. The article by Veronika Furst, Larry's twelfth Ph.D. student, "Characteristic wavelet equations and generalizations of the spectral function," further develops work done in her Ph.D. thesis completed under the direction of Larry Baggett in 2006.…”

Operators commuting with a discrete subgroup of translations

“…The original source for the equivalence of affine and quasi-affine systems is [32] and it is extended in [3,8,16,20,31]. By H we denote the quasi-affine system corresponding to H .…”