Study of shearlet transform using block matrix dilation

Amiri, Zahra; Bagherzadeh, Hojjat; Harati, Ahad; Gol, Rajab Ali Kamyabi

doi:10.1007/s12190-017-1120-5

Cited by 5 publications

(3 citation statements)

References 19 publications

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“…Shearlet introduced a shear matrix to control the direction, which is a multi-scale wavelet transform with good direction sensitivity and anisotropy. It is an extension of multi-dimensional, multi-scale, and multi-direction wavelet transform [ 25 , 26 ]. However, they also have disadvantages in characterizing and describing boundary curves, due to the lack of bending parameters.…”

Section: Methodsmentioning

confidence: 99%

Brain Tumor Segmentation Based on Bendlet Transform and Improved Chan-Vese Model

Meng

Cattani

Villecco

2022

Entropy

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Automated segmentation of brain tumors is a difficult procedure due to the variability and blurred boundary of the lesions. In this study, we propose an automated model based on Bendlet transform and improved Chan-Vese (CV) model for brain tumor segmentation. Since the Bendlet system is based on the principle of sparse approximation, Bendlet transform is applied to describe the images and map images to the feature space and, thereby, first obtain the feature set. This can help in effectively exploring the mapping relationship between brain lesions and normal tissues, and achieving multi-scale and multi-directional registration. Secondly, the SSIM region detection method is proposed to preliminarily locate the tumor region from three aspects of brightness, structure, and contrast. Finally, the CV model is solved by the Hermite-Shannon-Cosine wavelet homotopy method, and the boundary of the tumor region is more accurately delineated by the wavelet transform coefficient. We randomly selected some cross-sectional images to verify the effectiveness of the proposed algorithm and compared with CV, Ostu, K-FCM, and region growing segmentation methods. The experimental results showed that the proposed algorithm had higher segmentation accuracy and better stability.

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Section: Methodsmentioning

confidence: 99%

Brain Tumor Segmentation Based on Bendlet Transform and Improved Chan-Vese Model

Meng

Cattani

Villecco

2022

Entropy

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“…e left Haar measures on S are given by dv � dadsdt/|a| N+1 . For more details about shearlets and their application, the readers are referred to [10][11][12][13] and the references therein.…”

Section: E Shearlet Transform In Arbitrary Space Dimensionsmentioning

confidence: 99%

A Convolution-Based Shearlet Transform in Free Metaplectic Domains

Garg

Lone

Shah

et al. 2021

Journal of Mathematics

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The free metaplectic transformation (FMT) is a multidimensional integral transform that encompasses a broader range of integral transforms, from the classical Fourier to the more recent linear canonical transforms. The aim of this study is to introduce a novel shearlet transform by employing the free metaplectic convolution structures. Besides obtaining the orthogonality relation, inversion formula, and range theorem, we also study the homogeneous approximation property for the proposed transform. Towards the culmination, we formulate the Heisenberg and logarithmic-type uncertainty principles associated with the free metaplectic shearlet transform.

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“…Unlike the classical wavelets, shearlets are non-isotropic in nature, they offer optimally sparse representations, they allow compactly supported analyzing elements, they are associated with fast decomposition algorithms and they provide a unified treatment of continuum and digital data. However, similar to the wavelets, they are an affine-like system of well-localized waveforms at various scales, locations and orientations; that is, they are generated by dilating and translating one single generating function, where the dilation matrix is the product of a parabolic scaling matrix and a shear matrix and hence, they are a specific type of composite dilation wavelets [5,6,7,8,9]. The importance of shearlet transforms have been widely acknowledged and since their inception, they have emerged as one of the most effective frameworks for representing multidimensional data ranging over the areas of signal and image processing, remote sensing, data compression, and several others, where the detection of directional structure of the analyzed signals play a role [10,11].…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Principles for the Continuous Shearlet Transforms in Arbitrary Space Dimensions

Shah¹,

Tantary²

2019

Preprint

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The aim of this article is to formulate some novel uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions. Firstly, we derive an analogue of the Pitt's inequality for the continuous shearlet transforms, then we formulate the Beckner's uncertainty principle via two approaches: one based on a sharp estimate from Pitt's inequality and the other from the classical Beckner's inequality in the Fourier domain. Secondly, we consider a logarithmic Sobolev inequality for the continuous shearlet transforms which has a dual relation with Beckner's inequality. Thirdly, we derive Nazarov's uncertainty principle for the shearlet transforms which shows that it is impossible for a non-trivial function and its shearlet transform to be both supported on sets of finite measure. Towards the culmination, we formulate local uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions.

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Study of shearlet transform using block matrix dilation

Cited by 5 publications

References 19 publications

Brain Tumor Segmentation Based on Bendlet Transform and Improved Chan-Vese Model

Brain Tumor Segmentation Based on Bendlet Transform and Improved Chan-Vese Model

A Convolution-Based Shearlet Transform in Free Metaplectic Domains

Uncertainty Principles for the Continuous Shearlet Transforms in Arbitrary Space Dimensions

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