In this paper we describe a new class of multidimensional representation systems, called shearlets. They are obtained by applying the actions of dilation, shear transformation and translation to a fixed function, and exhibit the geometric and mathematical properties, e.g., directionality, elongated shapes, scales, oscillations, recently advocated by many authors for sparse image processing applications. These systems can be studied within the framework of a generalized multiresolution analysis. This approach leads to a recursive algorithm for the implementation of these systems, that generalizes the classical cascade algorithm.
Cartoon-like images, i.e., C 2 functions which are smooth apart from a C 2 discontinuity curve, have by now become a standard model for measuring sparse (non-linear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit (almost) optimally sparse approximations within this model. However, all those results are only applicable to band-limited generators, whereas, in particular, spatially compactly supported generators are of uttermost importance for applications.In this paper, we now present the first complete proof of (almost) optimally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of compactly supported elements. This class will be chosen as a subset of shearlet frames -not necessarily required to be tight -with shearlet generators having compact support and satisfying some weak moment conditions.
Abstract. Shearlet tight frames have been extensively studied during the last years due to their optimal approximation properties of cartoon-like images and their unified treatment of the continuum and digital setting. However, these studies only concerned shearlet tight frames generated by a band-limited shearlet, whereas for practical purposes compact support in spatial domain is crucial.In this paper, we focus on cone-adapted shearlet systems which -accounting for stability questions -are associated with a general irregular set of parameters. We first derive sufficient conditions for such cone-adapted irregular shearlet systems to form a frame and provide explicit estimates for their frame bounds. Secondly, exploring these results and using specifically designed wavelet scaling functions and filters, we construct a family of cone-adapted shearlet frames consisting of compactly supported shearlets. For this family, we derive estimates for the ratio of their frame bounds and prove that they provide optimally sparse approximations of cartoon-like images.
Affine systems are reproducing systems of the form A C = {D c T k ψ : 1 L, k ∈ Z n , c ∈ C}, which arise by applying lattice translation operators T k to one or more generators ψ in L 2 (R n ), followed by the application of dilation operators D c , associated with a countable set C of invertible matrices. In the wavelet literature, C is usually taken to be the group consisting of all integer powers of a fixed expanding matrix. In this paper, we develop the properties of much more general systems, for which C = {c = ab: a ∈ A, b ∈ B} where A and B are not necessarily commuting matrix sets. C need not contain a single expanding matrix. Nonetheless, for many choices of A and B, there are wavelet systems with multiresolution properties very similar to those of classical dyadic wavelets. Typically, A expands or contracts only in certain directions, while B acts by volume-preserving maps in transverse directions. Then the resulting wavelets exhibit the geometric properties, e.g., directionality, elongated shapes, scales, oscillations, recently advocated by many authors for multidimensional signal and image processing applications. Our method is a systematic approach to the theory of affine-like systems yielding these and more general features.
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