Introduction to Shearlets

Kutyniok, Gitta; Labate, Demetrio

doi:10.1007/978-0-8176-8316-0_1

Cited by 116 publications

(132 citation statements)

References 64 publications

(89 reference statements)

Supporting

Mentioning

129

Contrasting

Order By: Relevance

“…14,15 Thanks to this property, shearlets provide optimally sparse approximations with images containing C 2 -edges, outperforming conventional wavelets. 14 This is particularly relevant in CT-like applications, because point-like structures in the image domain map onto sine-shaped curvilinear structures in the projection domain.…”

Section: Resultsmentioning

confidence: 99%

Shearlet-based regularized ROI reconstruction in fan beam computed tomography

et al. 2015

View full text Add to dashboard Cite

Region-of-interest (ROI) reconstruction in computed tomography (CT) is a problem receiving increasing attention in the medical imaging community, due to its potential to lower exposure to X-ray radiation and to reduce the scanning time. Since the ROI reconstruction problem requires to deal with truncated projection images, classical CT reconstruction algorithms tend to become very unstable and the solution of this problem requires either ad hoc analytic formulas or more sophisticated numerical schemes. In this paper, we introduce a novel approach for ROI CT reconstruction, formulated as a convex optimization problem with a regularized functional based on shearlets or wavelets. Our numerical implementation consists of an iterative algorithm based on the scaled gradient projection method. As illustrated by numerical tests in the context of fan beam CT, our algorithm is insensitive to the location of the ROI and remains very stable also when the ROI size is rather small.

show abstract

Section: Resultsmentioning

confidence: 99%

Shearlet-based regularized ROI reconstruction in fan beam computed tomography

et al. 2015

View full text Add to dashboard Cite

show abstract

“…Most of the background, theory and applications of shearlets, and especially a lot of valuable references can be found in the still quite recent book [8]. There even exist excellent numerical implementations [5,9,11] meanwhile, but they are still implementations of a continuous transform, which thus discretizes parameters of a continuous nature and acts on samples of functions.…”

Section: Some Remarks On Shearletsmentioning

confidence: 99%

An anisotropic directional subdivision and multiresolution scheme

et al. 2014

View full text Add to dashboard Cite

In order to handle directional singularities, standard wavelet approaches have been extended to the concept of discrete shearlets in Kutyniok and Sauer (SIAM J. Math. Anal. 41, 1436-1471, 2009). One disadvantage of this extension, however, is the relatively large determinant of the scaling matrices used there which results in a substantial data complexity. This motivates the question whether some of the features of the discrete shearlets can also be obtained by means of different geometries. In this paper, we give a positive answer by presenting a different approach, based on a matrix with small determinant which therefore offers a larger recursion depth for the same amount of data.

show abstract

“…where X is a matrix whose i-th column X i is the vectorized version of the patches extracted from ,x and A is a rearranged version of α where each column is correlated to DX i (9). The following proposing gives the exact solution to (16), or equivalently to (11 …”

Section: Mri Reconstruction Using Adaptive Wavelet Tight Framementioning

confidence: 99%

Adaptive wavelet tight frame construction for accelerating MRI reconstruction

Zhou¹,

Huang

2017

Stat., optim. inf. comput.

View full text Add to dashboard Cite

The sparsity regularization approach, which assumes that the image of interest is likely to have sparse representation in some transform domain, has been an active research area in image processing and medical image reconstruction. Although various sparsifying transforms have been used in medical image reconstruction such as wavelet, contourlet, and total variation (TV) etc., the efficiency of these transforms typically rely on the special structure of the underlying image. A better way to address this issue is to develop an overcomplete dictionary from the input data in order to get a better sparsifying transform for the underlying image. However, the general overcomplete dictionaries do not satisfy the so-called perfect reconstruction property which ensures that the given signal can be perfectly represented by its canonical coefficients in a manner similar to orthonormal bases, resulting in time consuming in the iterative image reconstruction. This work is to develop an adaptive wavelet tight frame method for magnetic resonance image reconstruction. The proposed scheme incorporates the adaptive wavelet tight frame approach into the magnetic resonance image reconstruction by solving a ℓ 0 -regularized minimization problem. Numerical results show that the proposed approach provides significant time savings as compared to the over-complete dictionary based methods with comparable performance in terms of both peak signal-tonoise ratio and subjective visual quality.

show abstract

Introduction to Shearlets

Cited by 116 publications

References 64 publications

Shearlet-based regularized ROI reconstruction in fan beam computed tomography

Shearlet-based regularized ROI reconstruction in fan beam computed tomography

An anisotropic directional subdivision and multiresolution scheme

Adaptive wavelet tight frame construction for accelerating MRI reconstruction

Contact Info

Product

Resources

About