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.

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.

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.

This paper shows that the continuous shearlet transform, a novel directional multiscale transform recently introduced by the authors and their collaborators, provides a precise geometrical characterization for the boundary curves of very general planar regions. This study is motivated by imaging applications, where such boundary curves represent edges of images. The shearlet approach is able to characterize both locations and orientations of the edge points, including corner points and junctions, where the edge curves exhibit abrupt changes in tangent or curvature. Our results encompass and greatly extend previous results based on the shearlet and curvelet transforms which were limited to very special cases such as polygons and smooth boundary curves with nonvanishing curvature.

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