Adaptive Directional Subdivision Schemes and Shearlet Multiresolution Analysis.
Submitted (2007).
Abstract: In this paper, we propose a solution for a fundamental problem in computational harmonic analysis, namely, the construction of a multiresolution analysis with directional components. We will do so by constructing subdivision schemes which provide a means to incorporate directionality into the data and thus the limit function. We develop a new type of non-stationary bivariate subdivision schemes, which allow to adapt the subdivision process depending on directionality constraints during its performance, and we derive a complete characterization of those masks for which these adaptive directional subdivision schemes converge. In addition, we present several numerical examples to illustrate how this scheme works. Secondly, we describe a fast decomposition associated with a sparse directional representation system for two dimensional data, where we focus on the recently introduced sparse directional representation system of shearlets. In fact, we show that the introduced adaptive directional subdivision schemes can be used as a framework for deriving a shearlet multiresolution analysis with finitely supported filters, thereby leading to a fast shearlet decomposition.
Shearlet Coorbit Spaces and associated Banach Frames.
Submitted (2007).
Abstract: In this paper, we study the relationships of the newly developed continuous shearlet transform with the coorbit space theory. It turns out that all the conditions that are needed to apply the coorbit space theory can indeed be satisfied for the shearlet group. Consequently, we establish new families of smoothness spaces, the shearlet coorbit spaces. Moreover, our approach yields Banach frames for these spaces in a quite natural way. We also study the approximation power of best n-term approximation schemes and present some first numerical experiments.
Sparse Directional Image Representations using the Discrete Shearlet Transform.
Submitted (2006).
Abstract: It is now widely acknowledged that traditional wavelets are not very effective in dealing with multidimensional signals containing distributed discontinuities. To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. This paper introduces a new discrete multiscale directional representation called the Discrete Shearlet Transform. This approach, which is based on the shearlet transform, combines the power of multiscale methods with a unique ability to capture the geometry of multidimensional data and is optimally efficient in representing images containing edges. We describe two different methods of implementing the shearlet transform. The numerical experiments presented in this paper demonstrate that the Discrete Shearlet Transform is very competitive in denoising applications both in terms of performance and computational efficiency.
Resolution of the Wavefront Set using Continuous Shearlets.
Trans. Amer. Math. Soc., to appear.
Abstract: It is known that the Continuous Wavelet Transform of a function 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 precisely identify the wavefront set of a distribution. In this paper, we employ the framework of affine systems to construct the so-called Continuous Shearlet Transform, which is defined by SH_f(a,s,t) = <f,ψ_{ast}>. The analyzing elements ψ_{ast} are dilated and translated copies of a single generating function \psi, where 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 \in R^2, and oriented along lines of slope s \in R in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution f. Finally, we point out several variations of this approach.
From Wavelets to Shearlets and back again.
Approximation Theory XII (San Antonio, TX, 2007), Nashboro Press, Nashville, TN (2007), to appear.
Abstract: In this paper we will study the Continuous Shearlet Transform from a wavelet point of view, and show how this perspective can be used to derive a new geometric interpretation of this transform providing the possibility for FFT-based fast methods to compute the Continuous Shearlet Transform.
Representation of Fourier Integral Operators using Shearlets.
J. Fourier Anal. Appl., to appear.
Abstract: The traditional methods of time-frequency and multiscale analysis have been successfully employed for representing most classes of pseudodifferential operators. However these methods are not equally effective in dealing with Fourier Integral Operators in general. In this paper, we show that the shearlets, recently introduced by the authors and their collaborators, provide very efficient representations for a large class of Fourier Integral Operators. The shearlets are an a±ne-like system of well-localized waveform at various scales, locations and orientations, which are particularly efficient in representing anisotropic functions. Using this approach, we prove that the matrix representation of a Fourier Integral Operator with respect to a Parseval frame of shearlets is sparse and well-organized. This fact recovers a similar result recently obtained by Candes and Demanet using curvelets. The results presented in this paper show that directional multiscale representations (such as curvelets and shearlets) are a most powerful tool in the study of those functions and operators where more traditional multiscale methods are unable to provide the appropriate geometric analysis in the phase space.
The Uncertainty Principle associated with the Continuous Shearlet Transform.
Int. J. Wavelets Multiresolut. Inf. Process., to appear.
Abstract: In this paper we study the continuous Shearlet transform aiming at deriving mother shearlet functions which ensure optimal accuracy of the parameters of the associated transform. For this, we first show that this transform is associated with a unitary group representation coming from the so-called {\em Shearlet group} and compute the associated admissibility condition. This enables us to employ the general uncertainty principle in order to derive mother shearlet functions that minimize the uncertainty relations derived for the infinitesimal generators of the Shearlet group: scaling, shear and translations. We further discuss methods to ensure square-integrability of the derived minimizers by considering weighted L^2-spaces. Moreover, we study whether the minimizers satisfy the admissibility condition, thereby proposing a method to balance between the minimizing and the admissibility property.
Optimally Sparse Multidimensional Representation using Shearlets.
SIAM J. Math Anal. 39 (2007), 298--318.
Abstract: In this paper we show that the shearlets, an a±ne-like system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2-dimensional functions f that are C2 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 ||f-f_N^S||_2^2 \approx N^{-2} (\log N)^3as N \to \infty, which is essentially optimal, and greatly outperforms the corresponding asymptotic approximation rate N^{-1} associated with wavelet approximations. Unlike the curvelets, that have similar sparsity properties, the 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.
Construction of Regular and Irregular Shearlets.
J. Wavelet Theory and Appl. 1 (2007), 1--10.
Abstract: In this paper, we study the construction of irregular shearlet systems, i.e., systems of the form SH(&psi,&Lambda) = { a^{-3/4} &psi(A_a^{-1}S_s^{-1}(x-t)) : (a,s,t) ∈ &Lambda},$ where &psi ∈ L^2(R^2), &Lambda is an arbitrary sequence in R^+ x R x R^2, A_a is a parabolic scaling matrix and S_s a shear matrix. These systems are obtained by appropriately sampling the Continuous Shearlet Transform. We derive sufficient conditions for such a discrete system to form a frame for L^2(R^2), and provide explicit estimates for the frame bounds. Among the examples of such discrete systems, one is the Parseval frame of shearlets previously introduced by the authors, which is optimal in approximating 2-D smooth functions with discontinuities along C^2-curves. This study provides the framework for the construction of a variety of discrete directional multiscale systems with the ability to detect orientations inherited from the Continuous Shearlet Transform.
Sparse Multidimensional Representations using Anisotropic Dilation and Shear Operators.
Wavelets and Splines (Athens, GA, 2005), Nashboro Press, Nashville, TN (2006), 189-201.
Abstract: Recent advances in applied mathematics and signal processing have shown that, in order to obtain sparse representations of multi-dimensional functions and signals, one has to use representation elements distributed not only at various scales and locations -- as in classical wavelet theory -- but also at various directions. In this paper, we show that we obtain a construction having exactly these properties by using the framework of affine systems. The representation elements that we obtain are generated by translations, dilations, and shear transformations of a single mother function, and are called shearlets. The shearlets provide optimally sparse representations for 2-D functions that are smooth away from discontinuities along curves. Another benefit of this approach is that, thanks to their mathematical structure, these systems provide a Multiresolution analysis similar to the one associated with classical wavelets, which is very useful for the development of fast algorithmic implementations.
Sparse multidimensional representation using shearlets.
Wavelets XI (San Diego, CA, 2005), 254-262, SPIE Proc. 5914, SPIE, Bellingham, WA, 2005.
Abstract: 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.