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 $f_N^S$ 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 $\norm{f-f_N^S}_2^2 \asymp N^{-2} (\log N)^3, N \to \infty,$ 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.

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 $\mathcal {SH}_\psi f(a,s,t) = \langle {f}{\psi _{ast}}\rangle$ , where the analyzing elements $\psi _{ast}$ are dilated and translated copies of a single generating function $\psi$ . The dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements $\{\psi _{ast}\}$ form a system of smooth functions at continuous scales $a>0$ , locations $t \in \mathbb {R}^2$ , and oriented along lines of slope $s \in \mathbb {R}$ 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.