Wang-Q Lim

Sparse multidimensional representation using shearlets

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.

The Discrete Shearlet Transform: A New Directional Transform and Compactly Supported Shearlet Frames

It is now widely acknowledged that analyzing the intrinsic geometrical features of the underlying image is essential in many applications including image processing. In order to achieve this, several directional image representation schemes have been proposed. In this paper, we develop the discrete shearlet transform (DST) which provides efficient multiscale directional representation and show that the implementation of the transform is built in the discrete framework based on a multiresolution analysis (MRA). We assess the performance of the DST in image denoising and approximation applications. In image approximations, our approximation scheme using the DST outperforms the discrete wavelet transform (DWT) while the computational cost of our scheme is comparable to the DWT. Also, in image denoising, the DST compares favorably with other existing transforms in the literature.

Full length article: Compactly supported shearlets are optimally sparse

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 (nonlinear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit sparse approximations within this model, which are optimal up to a log-factor. 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 present the first complete proof of 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 (non-tight) shearlet frames with shearlet generators having compact support and satisfying some weak directional vanishing moment conditions.

Wavelets with composite dilations

A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for L2(Rn) under the action of lattice translations and dilations by products of elements drawn from non-commuting matrix sets A and B. Typically, the members of B are shear matrices (all eigenvalues are one) while the members of A are matrices expanding or contracting on a proper subspace of Rn. These wavelets are of interest in applications because of their tendency to produce “long, narrow” window functions well suited to edge detection. In this paper, we discuss the remarkable extent to which the theory of wavelets with composite dilations parallels the theory of classical wavelets, and present several examples of such systems.

The Theory of Wavelets with Composite Dilations

A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for L 2(ℝn) under the action of lattice translations and dilations by products of elements drawn from non-commuting sets of matrices A and B. Typically, the members of B are matrices whose eigenvalues have magnitude one, while the members of A are matrices expanding on a proper subspace of ℝn. The theory of these systems generalizes the classical theory of wavelets and provides a simple and flexible framework for the construction of orthonormal bases and related systems that exhibit a number of geometric features of great potential in applications. For example, composite wavelets have the ability to produce “long and narrow” window functions, with various orientations, well-suited to applications in image processing.

Image separation using wavelets and shearlets

In this paper, we present an image separation method for separating images into point- and curvelike parts by employing a combined dictionary consisting of wavelets and compactly supported shearlets utilizing the fact that they sparsely represent point and curvilinear singularities, respectively. Our methodology is based on the very recently introduced mathematical theory of geometric separation, which shows that highly precise separation of the morphologically distinct features of points and curves can be achieved by l1 minimization. Finally, we present some experimental results showing the effectiveness of our algorithm, in particular, the ability to accurately separate points from curves even if the curvature is relatively large due to the excellent localization property of compactly supported shearlets.

Nonseparable Shearlet Transform

Over the past few years, various representation systems which sparsely approximate functions governed by anisotropic features, such as edges in images, have been proposed. Alongside the theoretical development of these systems, algorithmic realizations of the associated transforms are provided. However, one of the most common shortcomings of these frameworks is the lack of providing a unified treatment of the continuum and digital world, i.e., allowing a digital theory to be a natural digitization of the continuum theory. In this paper, we introduce a new shearlet transform associated with a nonseparable shearlet generator, which improves the directional selectivity of previous shearlet transforms. Our approach is based on a discrete framework, which allows a faithful digitization of the continuum domain directional transform based on compactly supported shearlets introduced as means to sparsely encode anisotropic singularities of multivariate data. We show numerical experiments demonstrating the potential of our new shearlet transform in 2D and 3D image processing applications.

ShearLab 3D: Faithful Digital Shearlet Transforms Based on Compactly Supported Shearlets

Wavelets and their associated transforms are highly efficient when approximating and analyzing one-dimensional signals. However, multivariate signals such as images or videos typically exhibit curvilinear singularities, which wavelets are provably deficient in sparsely approximating and also in analyzing in the sense of, for instance, detecting their direction. Shearlets are a directional representation system extending the wavelet framework, which overcomes those deficiencies. Similar to wavelets, shearlets allow a faithful implementation and fast associated transforms. In this article, we will introduce a comprehensive carefully documented software package coined ShearLab 3D (www.ShearLab.org) and discuss its algorithmic details. This package provides MATLAB code for a novel faithful algorithmic realization of the 2D and 3D shearlet transform (and their inverses) associated with compactly supported universal shearlet systems incorporating the option of using CUDA. We will present extensive numerical experiments in 2D and 3D concerning denoising, inpainting, and feature extraction, comparing the performance of ShearLab 3D with similar transform-based algorithms such as curvelets, contourlets, or surfacelets. In the spirit of reproducible research, all scripts are accessible on www.ShearLab.org.

Compactly Supported Shearlets

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. 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.

Optimally sparse approximations of 3D functions by compactly supported shearlet frames

We study efficient and reliable methods of capturing and sparsely representing anisotropic structures in 3D data. As a model class for multidimensional data with anisotropic features, we introduce generalized three-dimensional cartoon-like images. This function class will have two smoothness parameters: one parameter \beta controlling classical smoothness and one parameter \alpha controlling anisotropic smoothness. The class then consists of piecewise C^\beta-smooth functions with discontinuities on a piecewise C^\alpha-smooth surface. We introduce a pyramid-adapted, hybrid shearlet system for the three-dimensional setting and construct frames for L^2(R^3) with this particular shearlet structure. For the smoothness range 1<\alpha =< \beta =< 2 we show that pyramid-adapted shearlet systems provide a nearly optimally sparse approximation rate within the generalized cartoon-like image model class measured by means of non-linear N-term approximations.