Radon transform inversion using the shearlet representation

Colonna, Flavia; Easley, Glenn R.; Guo, Kanghui; Labate, Demetrio

doi:10.1016/j.acha.2009.10.005

Cited by 71 publications

(77 citation statements)

References 22 publications

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“…This property is very useful in applications which require the analysis or detection of edge discontinuities. For example, using these observations, a shearlet-based algorithm for edge detection and analysis was developed in [74], and related ideas were exploited to develop algorithms for the regularized inversion of the Radon transform in [6]. A more detailed discussion of these issues, including the extensions to higher dimensions, will be the content of Chapters 2 and 3 of this volume.…”

Section: Proposition 1 ([52]mentioning

confidence: 99%

Introduction to Shearlets

2012

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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: Proposition 1 ([52]mentioning

confidence: 99%

Introduction to Shearlets

2012

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“…For functions f in a certain class (specifically, if f is in a certain family of Besov spaces), then the WVD method converges to f with the optimal rate, provided that the thresholding parameters are properly selected [21,54]. By adapting the WVD approach within the shearlet framework, it is shown in [17,29] that a function f ∈ L 2 (R 2 ) is recovered from the Radon data R f using the expansion …”

Section: Inverting the Radon Transformmentioning

confidence: 99%

“…By taking advantage of the ability of the shearlet representation to represent some important classes of operators, a novel method for the regularized inversion of the Radon transform was introduced in [17,29]. This is a problem of great interest since the Radon transform is the underlying mathematical framework of Computerized Tomography (CT), which has become an essential tool in medical diagnostics and preventive medicine 2 .…”

Section: Inverting the Radon Transformmentioning

confidence: 99%

“…A typical application of the shearlet-based regularized reconstruction algorithm is reported in Figure 5 where the method is compared against the curvelet-based algorithm and the wavelet-based one (corresponding to WVD). We refer to [17,29] for additional numerical tests and details on the algorithm. A related method of using shearlets to control noise amplification when inverting the Radon transform was presented in [1] in the case when the sampled data is compressed.…”

Section: Inverting the Radon Transformmentioning

confidence: 99%

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Image Processing Using Shearlets

Easley

Labate

2012

Shearlets

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Since shearlets provide nearly optimally sparse representations for a large class of functions that are useful to model natural images, many image processing methods benefit from their use. In particular, the error rates of data estimation from noise are highly dependent on the sparsity properties of the representation, so that many successful applications of shearlets center around restoration tasks such as denoising and inverse problems. Other imaging problems, where also the application of the shearlet representation turns out to be very beneficial, include image enhancement, image separation, edge detection, and estimation of the geometric features of an object.

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“…This is another fundamental difference with respect to curvelets, since the shear matrices preserve the integer lattice and this allows the shearlet systems to provide a unified treatment of the continuum and digital setting. These properties and the special flexibility of the shearlet framework have made this approach very successful both as a theoretical and an applicable tool (see, for example, publications [6,14,16,19,28,30] for the main theoretical results, and [5,9,10,11,30,33,34,35] for applications).…”

Section: Introductionmentioning

confidence: 99%

The Construction of Smooth Parseval Frames of Shearlets

Guo

Labate

2013

Math. Model. Nat. Phenom.

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Abstract. The shearlet representation has gained increasing recognition in recent years as a framework for the efficient representation of multidimensional data. This representation consists of a countable collection of functions defined at various locations, scales and orientations, where the orientations are obtained through the use of shearing matrices. While shearing matrices offer the advantage of preserving the integer lattice and being more appropriate than rotations for digital implementations, the drawback is that the action of the shearing matrices is restricted to coneshaped regions in the frequency domain. Hence, in the standard construction, a Parseval frame of shearlets is obtained by combining different systems of cone-based shearlets which are projected onto certain subspaces of L 2 (R D ) with the consequence that the elements of the shearlet system corresponding to the boundary of the cone regions lose their good spatial localization property. In this paper, we present a new construction yielding smooth Parseval frame of shearlets for L 2 (R D ). Specifically, all elements of the shearlet systems obtained from this construction are compactly supported and C ∞ in the frequency domain, hence ensuring that the system has also excellent spatial localization.

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Radon transform inversion using the shearlet representation

Cited by 71 publications

References 22 publications

Introduction to Shearlets

Introduction to Shearlets

Image Processing Using Shearlets

The Construction of Smooth Parseval Frames of Shearlets

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