Directional Frames for Image Recovery: Multi-scale Discrete Gabor Frames

Gao, Hui; Shen, Zuowei; Zhao, Yufei

doi:10.1007/s00041-016-9487-5

Cited by 20 publications

(18 citation statements)

References 39 publications

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“…Since B-splines are nonnegative, one may also take their square root as window function for a tight frame. This idea has been used in [75] to construct complex-valued discrete tight Gabor frames, which exhibit good orientation selectivity and are useful in various image processing problems.…”

Section: Piecewise Polynomialsmentioning

confidence: 99%

“…For example, the filters used for image restoration in [1,13,91] are learned from the image, resulting in filters that capture certain features of the image and lead to a transform that gives a better sparse representation. In [75] Gabor frame filter banks are designed to achieve high orientation selectivity that adapts to the geometry of image edges for sparse image approximation. Wavelet filters can be considered as discrete approximations of certain differential operators and thus the tight wavelet frame based approach for image processing has close ties with the PDE based approaches.…”

Section: Introductionmentioning

confidence: 99%

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Duality for Frames

Fan

Heinecke

Shen

2015

J Fourier Anal Appl

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The subject of this article is the duality principle, which, well beyond its stand at the heart of Gabor analysis, is a universal principle in frame theory that gives insight into many phenomena. Its fiber matrix formulation for Gabor systems is the driving principle behind seemingly different results. We show how the classical duality identities, operator representations and constructions for dual Gabor frames are in fact aspects of the dual Gramian matrix fiberization and its sole duality principle, giving a unified view to all of them. We show that the same duality principle, via dual Gramian matrix analysis, holds for dual (or bi-) systems in abstract Hilbert spaces. The essence of the duality principle is the unitary equivalence of the frame operator and the Gramian of certain adjoint systems. An immediate consequence is, for example, that, even on this level of generality, dual frames are characterized in terms of biorthogonality relations of adjoint systems. We formulate the duality principle for irregular Gabor systems which have no structure whatsoever to the sampling of the shifts and modulations of the generating window. In case the shifts and modulations are sampled from lattices we show how the abstract matrices can be reduced to the simple structured fiber matrices of shift-invariant systems, thus arriving back in the well understood territory. Moreover, in the arena of multiresolution analysis J Fourier Anal Appl (MRA)-wavelet frames, the mixed unitary extension principle can be viewed as the duality principle in a sequence space. This perspective leads to a construction scheme for dual wavelet frames which is strikingly simple in the sense that it only needs the completion of an invertible constant matrix. Under minimal conditions on the MRA, our construction guarantees the existence and easy constructability of non-separable multivariate dual MRA-wavelet frames. The wavelets have compact support and we show examples for multivariate interpolatory refinable functions. Finally, we generalize the duality principle to the case of transforms that are no longer defined by discrete systems, but may have discrete adjoint systems.

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Section: Piecewise Polynomialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Duality for Frames

Fan

Heinecke

Shen

2015

J Fourier Anal Appl

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“…Additionally, Gabor transformation is an effective tool for characterizing textures and subtle structures. In MR images, structures such as the brain sulcus can be considered as textures . With the combination of two redundant systems, the characterization ability can be guaranteed.…”

Section: Methodsmentioning

confidence: 99%

“…In MR images, structures such as the brain sulcus can be considered as textures. 23,25 With the combination of two redundant systems, the characterization ability can be guaranteed. In addition, without the training of dictionary atoms and the sparse coding procedures, the efficiency is acceptable.…”

Section: A Motivationmentioning

confidence: 99%

A cosparse analysis model with combined redundant systems for MRI reconstruction

Luo¹,

Ling²,

Gong³

et al. 2019

Medical Physics

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Purpose Magnetic resonance imaging (MRI) is widely used due to its noninvasive and nonionizing properties. However, MRI requires a long scanning time. In this paper, our goal is to reconstruct a high‐quality MR image from its sampled k‐space data to accelerate the data acquisition in MRI. Methods We propose a cosparse analysis model with combined redundant systems to fully exploit the sparsity of MR images. Two fixed redundant systems are used to characterize different structures, namely, the wavelet tight frame and Gabor frame. An alternating iteration scheme is used for reconstruction with simple implementation and good performance. Results The proposed method is tested on two MR images under three sampling patterns with sampling ratios ranging from 10% to 60%. The results show that the proposed method outperforms other state‐of‐the‐art MRI reconstruction methods in terms of both subjective visual quality and objective quantitative measurement. For instance, for brain images under random sampling with a ratio of 10%, compared to the other three methods, the proposed method improves the peak signal‐to‐noise ratio (PSNR) by more than 9 dB. Conclusions To better characterize different sparsities of different structures of MRI, a cosparse analysis model combining the wavelet tight frame and Gabor frame is proposed. A partial ℓ2 norm regularization is leveraged to obtain the optimal solution in a lower dimension. Compared to other state‐of‐the‐art MRI reconstruction methods, the proposed method improves the reconstruction quality of MRI, especially highly undersampled MRI.

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“…Hence, in order to obtain a high quality recovery from the ill-posed linear inverse problem (1.1), a proper regularization on the images to be recovered is needed. Successful regularization based methods include the Rudin-Osher-Fatemi model [54] and its nonlocal variants [38,63], the inf-convolution model [17], the total generalized variation (TGV) model [7,8], the combined first and second order total variation model [6,47,52], and the applied harmonic analysis approach such as curvelets [14], Gabor frames [22,40,44,48], shearlets [46], complex tight framelets [41], wavelet frames [4,9,10,13,19,24,30,32,35,36,58,64], etc. The common concept of these methods is to find sparse approximation of images using a properly designed linear transformation together with a sparsity promoting regularization term (such as the widely used 1 norm).…”

Section: Introductionmentioning

confidence: 99%

An edge driven wavelet frame model for image restoration

Choi

Dong

Zhang

2020

Applied and Computational Harmonic Analysis

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Wavelet frame systems are known to be effective in capturing singularities from noisy and degraded images. In this paper, we introduce a new edge driven wavelet frame model for image restoration by approximating images as piecewise smooth functions. With an implicit representation of image singularities sets, the proposed model inflicts different strength of regularization on smooth and singular image regions and edges.The proposed edge driven model is robust to both image approximation and singularity estimation. The implicit formulation also enables an asymptotic analysis of the proposed models and a rigorous connection between the discrete model and a general continuous variational model. Finally, numerical results on image inpainting and deblurring show that the proposed model is compared favorably against several popular image restoration models.

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Directional Frames for Image Recovery: Multi-scale Discrete Gabor Frames

Cited by 20 publications

References 39 publications

Duality for Frames

Duality for Frames

A cosparse analysis model with combined redundant systems for MRI reconstruction

An edge driven wavelet frame model for image restoration

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