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Some New Inequalities for wavelet frames on Local Fields

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Analysis in Theory and Applications 2017年第2期33卷 134-148页

作者： Firdous A.Shah Owais Ahmad Neyaz A.Sheikh Department of Mathematics University of Kashmir South Campus Department of Mathematics National Institute of Technology

wavelet frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of science and engineering. Finding general and verifiable conditions which imply that the wavelet systems are wavelet frames is among the core problems in time-frequency analysis. In this article, we establish some new inequalities for wavelet frames on local fields of positive characteristic by means of the Fourier transform. As an application, an improved version of the Li-Jiang inequality for wavelet frames on local fields is obtained.

关键词： frame inequalities wavelet frame local field Fourier transform.

PERTURBATION OF wavelet AND GABOR frameS

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Analysis in Theory and Applications 2003年第3期19卷 238-254页

作者： Ivana Carrizo Sergio Favier Instituto de Matemática Aplicada San Luis Universidad Nacional de San Luis Instituto de Matemática Aplicada San Luis Universidad Nacional de San Luis Av.Ejército de Los Andes 950.5700 San LuisArgentinaAv.Ejército de Los Andes 950.5700 San LuisArgentina

In this work two aspects of theory of frames are presented: a side necessary condition on irregular wavelet frames is obtained, another perturbation of wavelet and Gabor frames is considered. Specifically, we present the results obtained on frame stability when one disturbs the mother of wavelet frame, or the parameter of dilatation, and in Gabor frames when the generating function or the parameter of translation are perturbed. In all cases we work without demanding compactness of the support, neither on the generating function, nor on its Fourier transform.

关键词： Riesz basis Gabor frame wavelet frame perturbation

ROBUST frame BASED X-RAY CT RECONSTRUCTION

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Journal of Computational Mathematics 2016年第6期34卷 683-704页

作者： Jia Li Chuang Miao Zuowei Shen Ge Wang Hengyong Yu Department of Mathematical Science Rensselaer Polytechnic Institute Troy NY 12180 USA Department of Biomedical Engineering Wake Forest University Health Sciences Winston-Salem NC 27157 USA and Biomedical Imaging Division VT- WFU School of Biomedical Engineering and Sciences Wake Forest University Health Sciences Winston-Salem NC 27157 USA Department of Mathematics National University of Singapore Singapore 119076 Biomedical Imaging Cluster Department of Biomedical Engineering Rensselaer Polytechnic Institute Troy NY 12180 USA Department of Electrical and Computer Engineering University of Massachusetts Lowell MA 01854 USA

As X-ray computed tomography （CT） is widely used in diagnosis and radiotherapy, it is important to reduce the radiation dose as low as reasonably achievable. For this pur- pose, one may use the TV based methods or wavelet frame based methods to reconstruct high quality images from reduced number of projections. Furthermore, by using the in- terior tomography scheme which only illuminates a region-of-interest （ROI）, one can save more radiation dose. In this paper, a robust wavelet frame regularization based model is proposed for both global reconstruction and interior tomography. The model can help to reduce the errors caused by mismatch of the huge sparse projection matrix. A three-system decomposition scheme is applied to decompose the reconstructed images into three differ- ent parts： cartoon, artifacts and noise. Therefore, by discarding the estimated artifacts and noise parts, the reconstructed images can be obtained with less noise and artifacts. Similar to other frame based image restoration models, the model can be efficiently solved by the split Bregman algorithm. Numerical simulations show that the proposed model outperforms the FBP and SART＋TV methods in terms of preservation of sharp edges, mean structural similarity （SSIM）, contrast-to-noise ratio, relative error and correlation- s. For example, for real sheep lung reconstruction, the proposed method can reach the mean structural similarity as high as 0.75 using only 100 projections while the FBP and the SART^TV methods need more than 200 projections. Additionally, the proposed ro- bust method is applicable for interior and exterior tomography with better performance compared to the FBP and the SART＋TV methods.

关键词： Computed tomography wavelet frame Split Bregman algorithm.

Multi-window dilation-and-modulation frames on the half real line

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Science China Mathematics 2020年第12期63卷 2423-2438页

作者： Yunzhang Li Wei Zhang College of Applied Sciences Beijing University of TechnologyBeijing 100124China School of Mathematics and Information Sciences Henan University of Economics and LawZhengzhou 450046China

wavelet and Gabor systems are based on translation-and-dilation and translation-and-modulation operators,respectively,and have been studied ***,dilation-and-modulation systems cannot be derived from wavelet or Gabor *** study aims to investigate a class of dilation-and-modulation systems in the causal signal space L^2(R+).L^2(R+)can be identified as a subspace of L^2(R),which consists of all L^2(R)-functions supported on R+but not closed under the Fourier ***,the Fourier transform method does not work in L^2(R+).Herein,we introduce the notion ofΘa-transform in L^2(R+)and characterize the dilation-and-modulation frames and dual frames in L^2(R+)using theΘa-transform;and present an explicit expression of all duals with the same structure for a general dilation-and-modulation frame for L^2(R+).Furthermore,it has been proven that an arbitrary frame of this form is always nonredundant whenever the number of the generators is 1 and is always redundant whenever the number is greater than ***,some examples are provided to illustrate the generality of our results.

关键词： frame wavelet frame Gabor frame dilation-and-modulation frame multi-window dilation-and-modulation frame

LIMITED TOMOGRAPHY RECONSTRUCTION VIA TIGHT frame AND SIMULTANEOUS SINOGRAM EXTRAPOLATION

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Journal of Computational Mathematics 2016年第6期34卷 575-589页

作者： Jae Kyu Choi Bin Dong Xiaoqun Zhang Institute of Natural Sciences Shanghai Jiao Tong University Shanghai China Beijing International Center for Mathematical Research Peking University Beijing China Institute of Natural Sciences School of Mathematical Sciences and MOE-LSC Shanghai Jiao Tong University Shanghai China

X-ray computed tomography （CT） is one of widely used diagnostic tools for medical and dental tomographic imaging of the human body. However, the standard filtered back- projection reconstruction method requires the complete knowledge of the projection data. In the case of limited data, the inverse problem of CT becomes more ill-posed, which makes the reconstructed image deteriorated by the artifacts. In this paper, we consider two dimensional CT reconstruction using the projections truncated along the spatial direc- tion in the Radon domain. Over the decades, the numerous results including the sparsity model based approach has enabled the reconstruction of the image inside the region of interest （ROI） from the limited knowledge of the data. However, unlike these existing methods, we try to reconstruct the entire CT image from the limited knowledge of the sinogram via the tight frame regularization and the simultaneous sinogram extrapolation. Our proposed model shows more promising numerical simulation results compared with the existing sparsity model based approach.

关键词： X-ray computed tomography Limited tomography wavelet frame Data driventight frame Bregmanized operator splitting algorithm Sinogram extrapolation.

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一维连续小波变换

高等学校计算数学学报 2007年第1期29卷 1-8页

作者：施云惠阮秋琦北京工业大学计算机学院多媒体与智能软件技术北京市重点实验室北京交通大学信息科学研究所

1引言小波分析是近年来迅速发展起来的一门新兴学科,小波分析最显著的特征是频域和时域具有良好局部化特性,可以观察函数的任意细节,被誉为数学的显微镜．它不仅理论深刻,且理论与应用的发展交织在一起,它成功地应用于信噪分离、图像编... 详细信息

1引言小波分析是近年来迅速发展起来的一门新兴学科,小波分析最显著的特征是频域和时域具有良好局部化特性,可以观察函数的任意细节,被誉为数学的显微镜．它不仅理论深刻,且理论与应用的发展交织在一起,它成功地应用于信噪分离、图像编码、图像的边缘检测、数据压缩、计算机视觉中的多分辨率分析等领域．

关键词： fourier transform continuous wavelet transform wavelet frame