ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm

Kutyniok, Gitta; Shahram, Morteza; Zhuang, Xiaosheng

doi:10.1137/110854497

Cited by 95 publications

(66 citation statements)

References 22 publications

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“…To date, several distinct numerical implementations of the Discrete Shearlet Transform have been proposed [22,46,58,60,64] and some additional implementations were introduced to address specific applications such as edge detection [74]. Furthermore, several attempts were made to develop a multiresolution analysis similar to the one associated with wavelets, in an effort to develop MRA-based implementations [41,46,58].…”

Section: Algorithmic Implementations Of the Shearlet Transformmentioning

confidence: 99%

“…Finally, we remark that numerical shearlet algorithms are available and downloadable at the webpages www.math.uh.edu/∼dlabate (associated with [22]) and www.ShearLab.org (associated with [60,64]). …”

Section: Algorithmic Implementations Of the Shearlet Transformmentioning

confidence: 99%

“…• A different approach, which was introduced in [59,60], consists of a carefully weighted Pseudo-Polar transform ensuring isometry followed by windowing and inverse FFT. This transform is associated with band-limited tight shearlet frames, thereby allowing the adjoint frame operator for reconstruction.…”

Section: Fourier-based Implementationsmentioning

confidence: 99%

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Introduction to Shearlets

Kutyniok

Labate

2012

Applied and Numerical Harmonic Analysis

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119

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Shearlets emerged in recent years among the most successful frameworks for the efficient representation of multidimensional data. Indeed, after it was recognized that traditional multiscale methods are not very efficient at capturing edges and other anisotropic features which frequently dominate multidimensional phenomena, several methods were introduced to overcome their limitations. The shearlet representation stands out since it offers a unique combinations of some highly desirable properties: it has a single or finite set of generating functions, it provides optimally sparse representations for a large class of multidimensional data, it is possible to use compactly supported analyzing functions, it has fast algorithmic implementations and it allows a unified treatment of the continuum and digital realms. In this chapter, we present a self-contained overview of the main results concerning the theory and applications of shearlets.

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Section: Algorithmic Implementations Of the Shearlet Transformmentioning

confidence: 99%

Section: Algorithmic Implementations Of the Shearlet Transformmentioning

confidence: 99%

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Introduction to Shearlets

Kutyniok

Labate

2012

Applied and Numerical Harmonic Analysis

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119

124

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“…4 provides an overview of the main steps of of the FDST and its inverse. For a more detailed description of FDST, Adjoint FDST, and Inverse FDST in form of pseudo-code, we refer to [20]. For the sake of brevity, we now let P, w, and W denote the Fast PPFT from Subsection 2.1.2, the weighting on the pseudo-polar grid described in Subsection 2.2.3, and windowing operator consisting of the application of the shearlet windows followed by 2D-iFFT to each array as detailed in Subsection 2.3.7, respectively.…”

Section: Algorithmic Realization Of the Fdstmentioning

confidence: 99%

“…We next present the framework for quantifying performance of implementations of directional transforms, which was originally introduced in [20,14]. This set of test measures was designed to analyze particular well-understood properties of a given algorithm, which in this case are the desiderata proposed at the beginning of this chapter.…”

Section: Algorithmic Realizationmentioning

confidence: 99%

Digital Shearlet Transforms

2012

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Over the past years, various representation systems which sparsely approximate functions governed by anisotropic features such as edges in images have been proposed. We exemplarily mention the systems of contourlets, curvelets, and shearlets. Alongside the theoretical development of these systems, algorithmic realizations of the associated transforms were 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 fact, shearlet systems are the only systems so far which satisfy this property, yet still deliver optimally sparse approximations of cartoon-like images. In this chapter, we provide an introduction to digital shearlet theory with a particular focus on a unified treatment of the continuum and digital realm. In our survey we will present the implementations of two shearlet transforms, one based on band-limited shearlets and the other based on compactly supported shearlets. We will moreover discuss various quantitative measures, which allow an objective comparison with other directional transforms and an objective tuning of parameters. The codes for both presented transforms as well as the framework for quantifying performance are provided in the Matlab toolbox ShearLab.

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Microlocal Analysis of the Geometric Separation Problem

Donoho

Kutyniok

2012

Comm Pure Appl Math

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100

190

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Image data are often composed of two or more geometrically distinct constituents; in galaxy catalogs, for instance, one sees a mixture of pointlike structures (galaxy superclusters) and curvelike structures (filaments). It would be ideal to process a single image and extract two geometrically 'pure' images, each one containing features from only one of the two geometric constituents. This seems to be a seriously underdetermined problem, but recent empirical work achieved highly persuasive separations.We present a theoretical analysis showing that accurate geometric separation of point and curve singularities can be achieved by minimizing the ℓ 1 norm of the representing coefficients in two geometrically complementary frames: wavelets and curvelets. Driving our analysis is a specific property of the ideal (but unachievable) representation where each content type is expanded in the frame best adapted to it. This ideal representation has the property that important coefficients are clustered geometrically in phase space, and that at fine scales, there is very little coherence between a cluster of elements in one frame expansion and individual elements in the complementary frame. We formally introduce notions of cluster coherence and clustered sparsity and use this machinery to show that the underdetermined systems of linear equations can be stably solved by ℓ 1 minimization; microlocal phase space helps organize the calculations that cluster coherence requires.

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ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm

Cited by 95 publications

References 22 publications

Introduction to Shearlets

Introduction to Shearlets

Digital Shearlet Transforms

Microlocal Analysis of the Geometric Separation Problem

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