Compactly Supported Shearlets.
Submitted (2010).
Abstract: Shearlet theory has become a central tool in analyzing and representing 2D data with anisotropic features. Shearlet systems are systems of functions generated by one single generator with parabolic scaling, shearing, and translation operators applied to it, in much the same way wavelet systems are dyadic scalings and translations of a single function, but including a precise control of directionality. Of the many directional representation systems proposed in the last decade, shearlets are among the most versatile and successful systems. The reason for this being an extensive list of desirable properties: shearlet systems can be generated by one function, they provide precise resolution of wavefront sets, they allow compactly supported analyzing elements, they are associated with fast decomposition algorithms, and they provide a unified treatment of the continuum and the digital realm.\\ \indent The aim of this paper is to introduce some key concepts in directional representation systems and to shed some light on the success of shearlet systems as directional representation systems. In particular, we will give an overview of the different paths taken in shearlet theory with focus on separable and compactly supported shearlets in 2D and 3D. We will present constructions of compactly supported shearlet frames in those dimensions as well as discuss recent results on the ability of compactly supported shearlet frames satisfying weak decay, smoothness, and directional moment conditions to provide optimally sparse approximations of cartoon-like images in 2D as well as in 3D. Finally, we will show that these compactly supported shearlet systems provide optimally sparse approximations of an even generalized model of cartoon-like images comprising of $C^2$ functions that are smooth apart from piecewise $C^2$ discontinuity edges.
Shearlets on Bounded Domains.
Submitted (2010).
Abstract: Shearlet systems have so far been only considered as a means to analyze $L^2$-functions defined on $\RR^2$, which exhibit curvilinear singularities. However, in applications such as image processing or numerical solvers of partial differential equations the function to be analyzed or efficiently encoded is typically defined on a non-rectangular shaped bounded domain. Motivated by these applications, in this paper, we first introduce a novel model for cartoon-like images defined on a bounded domain. We then prove that compactly supported shearlet frames satisfying some weak decay and smoothness conditions, when orthogonally projected onto the bounded domain, do provide (almost) optimally sparse approximations of elements belonging to this model class.
Construction of Compactly Supported Shearlet Frames.
Submitted (2010).
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.
Compactly Supported Shearlets are Optimally Sparse
Submitted (2010).
Abstract: 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 approximation 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.
Irregular Shearlet Frames: Geometry and Approximation Properties.
Submitted (2010).
Abstract: Recently, shearlet systems were introduced as a means to derive efficient encoding methodologies for anisotropic features in 2-dimensional data with a unified treatment of the continuum and digital setting. However, only very few construction strategies for discrete shearlet systems are known so far. In this paper, we take a geometric approach to this problem. Utilizing the close connection with group representations, we first introduce and analyze an upper and lower weighted shearlet density based on the shearlet group. We then apply this geometric measure to provide necessary conditions on the geometry of the sets of parameters for the associated shearlet systems to form a frame for L^2(\R^2), either when using all possible generators or a large class exhibiting some decay conditions. While introducing such a feasible class of shearlet generators, we analyze approximation properties of the associated shearlet systems, which themselves lead to interesting insights into homogeneous approximation abilities of shearlet frames. We also present examples, such a oversampled shearlet systems and co-shearlet systems, to illustrate the usefulness of our geometric approach to the construction of shearlet frames.
A Unitary Extension Principle for Shearlet Systems.
Submitted (2009).
Abstract: In this paper, we first introduce the concept of an adaptive MRA (AMRA) structure which is a variant of the classical MRA structure suited to the main goal of a fast flexible decomposition strategy adapted to the data at each decomposition level. We then study this novel methodology for the general case of affine-like systems, and derive a Unitary Extension Principle (UEP) for filter design. Finally, we apply our results to the directional representation system of shearlets. This leads to a comprehensive theory for fast decomposition algorithms associated with shearlet systems which encompasses tight shearlet frames with spatially compactly supported generators within such an AMRA structure. Also shearlet-like systems associated with parabolic scaling and unimodular matrices optimally close to rotation as well as 3D shearlet systems are studied within this framework.
Geometric Separation using a Wavelet-Shearlet Dictionary.
Submitted (2009).
Abstract: Astronomical images of galaxies can be modeled as a superposition of pointlike and curvelike structures. Astronomers typically face the problem of extracting those components as accurate as possible. Although this problem seems unsolvable -- as there are two unknowns for every datum -- suggestive empirical results have been achieved by employing a dictionary consisting of wavelets and curvelets combined with l_1 minimization techniques. In this paper we present a theoretical analysis in a model problem showing that accurate geometric separation can be achieved by l_1 minimization. We introduce the notions of cluster coherence and clustered sparse objects as a machinery to show that the underdetermined system of equations can be stably solved by l_1 minimization. We prove that not only a radial wavelet-curvelet dictionary achieves nearly-perfect separation at all sufficiently fine scales, but, in particular, also an orthonormal wavelet-shearlet dictionary, thereby proposing this dictionary as an interesting alternative for geometric separation of pointlike and curvelike structures. To derive this final result we show that curvelets and shearlets are sparsity equivalent in the sense of a finite p-norm (0 < p <= 1) of the cross-Grammian matrix.
Shearlet Coorbit Spaces and associated Banach Frames.
Appl. Comput. Harmon. Anal. 27 (2009), 195-214.
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.
Adaptive Directional Subdivision Schemes and Shearlet Multiresolution Analysis.
SIAM J. Math. Anal. 41 (2009), 1436-1471.
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.
Radon Transform Inversion using the Shearlet Representation.
preprint, pp.31, (2009)
Abstract: The inversion of the Radon transform is a classical ill-posed problem where some method of regularization must be applied in order to accurately recover the objects of interest from the observable data. A well-known consequence of the traditional regularization methods is that some important features to be recovered are lost, as evident in imaging applications where the regularized reconstructions are blurred versions of the original. In this paper, we show that the affine-like system of functions known as the shearlet system can be applied to obtain a highly effective reconstruction algorithm which provides near-optimal rate of convergence in estimating a large class of images from noisy Radon date. This is achieved by introducing a shearlet-based decomposition of the Radon operator and applying a thresholding scheme on the noisy shearlet transform coefficients. For a given noise level, the proposed shearlet shrinkage method can be tuned so that the estimator will attain the essentialy optimal mean square error. Several numerical demonstarations show that its performance improves upon similar competitive strategies based on wavelets and curvelets.
Characterization and analysis of edges using the Continuous Shearlet Transform.
preprint, pp. 24, (2009)
Abstract: 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 the edges of 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 transform which were limited to very special cases such as polygons and smooth boundary curves with nonvanishing curvature.
A Shearlet Approach to Edge Analysis and Detection.
IEEE Trans. Image Process. 18 (5) pp. 929-941 (2009)
Abstract: It is well known that the wavelet transform provides a very effective framework for analysis of multiscale edges. In this paper, we propose a novel approach based on the shearlet transform: a multiscale directional transform with a greater ability to localize distributed discontinuities such as edges. Indeed, unlike traditional wavelets, shearlets are theoretically optimal in representing images with edges and, in particular, have the ability to fully capture directional and other geometrical features. Numerical examples demonstrate that the shearlet approach is highly effective at detecting both the location and orientation of edges, and outperforms methods based on wavelets as well as other standard methods. Furthermore, the shearlet approach is useful to design simple and effective algorithms for the detection of coners and junctions.
Edge Analysis and identification using the Continuous Shearlet Transform.
Appl. Comput. Harmon. Anal., pp. 31, (2009)
Abstract: It is well known that the continuous wavelet transform has the ability to identify the set of singularities of a function or distribution. It was shown that certain multidimensional generalization of the wavelet transform are useful to capture additional information about the geometry of the singularities of an underlying function. In this paper, we show that the continuous shearlet transform, a novel directional multiscale transform recently introduced by the authors and their collaborators, allows one to exactly identify the location and orientation of the edges of planar objects. In particular, we show that one can use the asymptotic decay of the shearlet coefficients to exactly characterize the location and orientation of the smooth singularities of a piecewise smooth function. 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.
Shearlet Based Total Variation for Denoising.
IEEE Trans. Image Process. 18 (2) pp. 260-268 (2009)
Abstract: We propose a shearlet formulation of the total variation (TV) method for denoising images. Shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization. Common approaches in combining wavelet-like representations such as curvelets with TV or diffusion methods aim at reducing Gibbs-type artifacts after obtaining a nearly optimal estimate. We show that it is possible to obtain much better estimates from a shearlets representation by constraining the residual coefficinets using a projected adaptive total variation scheme in the shearlet domain. We also analyze the performance of a shearlets-based diffusion method. Numerical examples demonstrate that these schemes are highly effective at denoising complex images and outperform a related method based on the use of the curvelet transform. Furthermore the shearlet-TV scheme requires far fewer iterations than similar competitors.
Sparse Directional Image Representations using the Discrete Shearlet Transform.
Appl. Comput. Harmon. Anal. 25 pp. 25-46, (2008).
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.