No abstract

In this paper we show that shearlets, an affine-like system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2-dimensional functions f which are C 2 except for discontinuities along C 2 curves. More specifically, if f S N is the N -term reconstruction of f obtained by using the N largest coefficients in the shearlet representation, then the asymptotic approximation error decays as f − f S N 2 2 N −2 (log N ) 3 , N → ∞, which is essentially optimal, and greatly outperforms the corresponding asymptotic approximation rate N −1 associated with wavelet approximations. Unlike curvelets, which have similar sparsity properties, shearlets form an affine-like system and have a simpler mathematical structure. In fact, the elements of this system form a Parseval frame and are generated by applying dilations, shear transformations, and translations to a single well-localized window function.

Abstract. It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f . However, the Continuous Wavelet Transform is unable to describe the geometry of the set of singularities of f and, in particular, identify the wavefront set of a distribution. In this paper, we employ the same framework of affine systems which is at the core of the construction of the wavelet transform to introduce the Continuous Shearlet Transform. This is defined by SH ψ f (a, s, t) = f ψ ast , where the analyzing elements ψ ast are dilated and translated copies of a single generating function ψ. The dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements {ψ ast } form a system of smooth functions at continuous scales a > 0, locations t ∈ R 2 , and oriented along lines of slope s ∈ R in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution f .

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.

By a "reproducing" method for H = L 2 (R n) we mean the use of two countable families {e α : α ∈ A}, {f α : α ∈ A}, in H, so that the first "analyzes" a function h ∈ H by forming the inner products {< h, e α >: α ∈ A}, and the second "reconstructs" h from this information: h = α∈A < h, e α > f α. A variety of such systems have been used successfully in both pure and applied mathematics. They have the following feature in common: they are generated by a single or a finite collection of functions by applying to the generators two countable families of operators that consist of two of the following three actions: dilations, modulations, and translations. The Gabor systems, for example, involve a countable collection of modulations and translations; the affine systems (that produce a variety of wavelets) involve translations and dilations. A considerable amount of research has been conducted in order to characterize those generators of such systems. In this article we establish a result that "unifies" all of these characterizations by means of a relatively simple system of equalities. Such unification has been presented in a work by one of the authors. One of the novelties here is the use of a different approach that provides us with a considerably more general class of such reproducing systems; for example, in the affine case, we need not to restrict the dilation matrices to ones that preserve the integer lattice and are expanding on R n. Another novelty is a detailed analysis, in the case of affine and quasi-affine systems, of the characterizing equations for different kinds of dilation matrices. Math Subject Classifications. 42C15, 42C40.

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.

Shearlets emerged in recent years among the most successful frameworks for the efficient representation of multidimensional data. Indeed, after it was recognized that traditional multiscale methods are not very efficient at capturing edges and other anisotropic features which frequently dominate multidimensional phenomena, several methods were introduced to overcome their limitations. The shearlet representation stands out since it offers a unique combinations of some highly desirable properties: it has a single or finite set of generating functions, it provides optimally sparse representations for a large class of multidimensional data, it is possible to use compactly supported analyzing functions, it has fast algorithmic implementations and it allows a unified treatment of the continuum and digital realms. In this chapter, we present a self-contained overview of the main results concerning the theory and applications of shearlets.

It is well known that the continuous wavelet transform has the ability to identify the set of singularities of a function or distribution f . It was recently shown that certain multidimensional generalizations of the wavelet transform are useful to capture additional information about the geometry of the singularities of f . In this paper, we consider the continuous shearlet transform, which is the mappingf , ψ ast , where the analyzing elements ψ ast form an affine system of well localized functions at continuous scales a > 0, locations t ∈ R 2 , and oriented along lines of slope s ∈ R in the frequency domain. We show that the continuous shearlet transform allows one to exactly identify the location and orientation of the edges of planar objects. In particular, if f = N n=1 f n χ Ωn where the functions f n are smooth and the sets Ω n have smooth boundaries, then one can use the asymptotic decay of SH ψ f (a, s, t), as a → 0 (fine scales), to exactly characterize the location and orientation of the boundaries ∂Ω n . This improves similar results recently obtained in the literature and provides the theoretical background for the development of improved algorithms for edge detection and analysis.