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Product(α1,α2)-modulation spaces Dedicated to the Memory of Michel Marias

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Science China Mathematics 2022年第8期65卷 1599-1640页

作者： Galatia Cleanthous Athanasios G.Georgiadis Department of Mathematics and Statistics National University of IrelandMaynooth W 23 F 2K8Ireland Faculty of Engineering Mathematics and ScienceTrinity College of DublinDublin D02 PN40Ireland

We study fundamental properties of product(α,α)-modulation spaces built by(α,α)-coverings of R× R.Precisely we prove embedding theorems between these spaces with different parameters and other classical spaces.Furthermore,we specify their duals.The characterization of product modulation spaces via the short time Fourier transform is also obtained.Families of tight frames are constructed and discrete representations in terms of corresponding sequence spaces are derived.Fourier multipliers are studied and as applications we extract lifting properties and the identification of our spaces with(fractional) Sobolev spaces with mixed smoothness.

关键词： product(α1,α2)-modulation space product space mixed smoothness short time Fourier transform embedding dual tight frame discrete decomposition

A Robust Digital Watermarking Algorithm Based on Finite-Set Discrete Radon Transform tight frame

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Journal of Computer and Communications 2020年第12期8卷 123-133页

作者： Jiangui Zhang Huibin Qi Guangdong Huiyu Intelligent Survey Technology Co. Ltd. Zhuhai China Hunan Jiajia Car-Parking Technology Co. Ltd. Changsha China

Digital watermarking technology plays a powerful role in the effective protection of digital media copyright, image authentication, image sharing, image information transmission and other fields. Driven by strong demand, digital image watermarking technology has aroused widespread research interest and has gradually developed into one of the most active research directions in information science. In this paper, we present a novel robust digital watermarking algorithm based on discrete radon transform tight frame in finite-set (FDRT). FDRT of the zero mean image is a tight frame, the frame boundary A = B = 1, the dual of the frame is itself. The decomposition and reconstruction of the FDRT tight frame will not cause the phenomenon of image distortion. The embedding of hidden watermark is to add a weak signal to the strong background of the original image. Watermark extraction is to effectively identify the embedded weak signal. The feasibility of the watermarking algorithm is analyzed from two aspects of information hiding and robustness. We select the independent Gaussian random vector as the watermark series, and the peak signal-to-noise ratio (PSNR) as the visual degradation criterion of the watermark image. Basing the FDRT compact stand dual operator, we derived the relationship among the strength parameter, square sum of watermark series, the PSNR. Using Checkmark system, the simulation results show that the algorithm is robust enough to some very important image processing attacks such as lossy compression, MAP, filtering, segmentation, edge enhancement, jitter, quadratic modulation and general geometric attack (scaling, rotation, shearing), etc.

关键词： Digital Watermarking Data Mining Discrete Radon Transform tight frame Copyright Protection Information Hiding Finite-Set

frameLET BASED DECONVOLUTION

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Journal of Computational Mathematics 2010年第3期28卷 289-308页

作者： Jian-Feng Cai Zuowei Shen Department of Mathematics University of California Los Angeles CA 90095 Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543

In this paper, two framelet based deconvolution algorithms are proposed. The basic idea of framelet based approach is to convert the deconvolution problem to the problem of inpainting in a frame domain by constructing a framelet system with one of the masks being the given （discrete） convolution kernel via the unitary extension principle of [26], as introduced in [6-9] . The first algorithm unifies our previous works in high resolution image reconstruction and infra-red chopped and nodded image restoration, and the second one is a combination of our previous frame-based deconvolution algorithm and the iterative thresholding algorithm given by [14, 16]. The strong convergence of the algorithms in infinite dimensional settings is given by employing proximal forward-backward splitting （PFBS） method. Consequently, it unifies iterative algorithms of infinite and finite dimensional setting and simplifies the proof of the convergence of the aluorithms of [6].

关键词： framelet deconvolution wavelet tight frame soft-thresholding

Expansion of frames to tight frames

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Acta Mathematica Sinica,English Series 2009年第2期25卷 287-298页

作者： Deng Feng LI Wen Chang SUN Institute of Applied Mathematics College of Mathematies and Information Science He'nan University Kaifeng 475001 P. R. China Department of Mathematics and LPMC Nankai University Tianjin 300071 P. R. China

We show that every Bessel sequence （and therefore every frame） in a separable Hilbert space can be expanded to a tight frame by adding some elements. The proof is based on a recent generalization of the frame concept, the g-frame, which illustrates that g-frames could be useful in the study of frame theory. As an application, we prove that any Gabor frame can be expanded to a tight frame by adding one window function.

关键词： frame tight frame g-frame Gabor frame frame expansion

STABLE RECOVERY OF SIGNALS WITH THE HIGH ORDER D-RIP CONDITION

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Acta Mathematica Scientia 2016年第6期36卷 1721-1730页

作者：谌稳固李亚玲 Institute of Applied Physics and Computational Mathematics Graduate School China Academy of Engineering Physics

This paper establishes a high order condition on the restricted isometry property adapted to a frame D （D-RIF） for the signal recovery. It is shown that if the measurementmatrix A satisfies the D-RIP condition δtk 〈t-1/t for t 〉 1, then all signals f which aresparse in terms of a tight frame D can be recovered stably or exactly via the l1-analysis model based on y= Af ＋ z in 12 and Dantzig selector bounded noise setting.

关键词： compressed sensing D-restricted isometry property tight frame

Lifting scheme of symmetric tight wavelets frames

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Science in China(Series F) 2008年第8期51卷 1117-1124页

作者： ZHUANG BoJin YUAN WeiTao PENG LiZhong Laboratory of Mathematics and Applied Mathematics School of Mathematical Sciences Peking UniversityBeijing 100871 China

This paper proposes a method to realize the lifting scheme of tight frame wavelet filters. As for 4-channel tight frame wavelet filter, the tight frame transforms＇ matrix is 2×4, but the lifting scheme transforms＇ matrix must be 4×4. And in the case of 3-channel tight frame wavelet filter, the transforms＇ matrix is 2×3, but the lifting scheme transforms＇ matrix must be 3×3. In order to solve this problem, we introduce two concepts： transferred polyphase matrix for 4-channel filters and transferred unitary matrix for 3-channel filters. The transferred polyphase matrix is symmetric/antisymmetric. Thus, we use this advantage to realize the lifting scheme.

关键词： lifting scheme tight frame wavelet transform

A Robust Digital Watermarking Algorithm Based on Finite-Set Discrete Radon Transform tight frame

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A Robust Digital Watermarking Algorithm Based on Finite-Set ...

Construction of Multivariate tight framelet Packets Associated with Dilation Matrix

作者： Jiangui Zhang Huibin Qi Guangdong Huiyu Intelligent Survey Technology Co.Ltd.

Digital watermarking technology plays a powerful role in the effective protection of digital media copyright, image authentication, image sharing, image information transmission and other fields. Driven by strong demand, digital image watermarking technology has aroused widespread research interest and has gradually developed into one of the most active research directions in information science. In this paper, we present a novel robust digital watermarking algorithm based on discrete radon transform tight frame in finite-set（FDRT）. FDRT of the zero mean image is a tight frame, the frame boundary A = B = 1, the dual of the frame is itself. The decomposition and reconstruction of the FDRT tight frame will not cause the phenomenon of image distortion. The embedding of hidden watermark is to add a weak signal to the strong background of the original image. Watermark extraction is to effectively identify the embedded weak signal. The feasibility of the watermarking algorithm is analyzed from two aspects of information hiding and robustness. We select the independent Gaussian random vector as the watermark series, and the peak signal-to-noise ratio（PSNR） as the visual degradation criterion of the watermark image. Basing the FDRT compact stand dual operator, we derived the relationship among the strength parameter, square sum of watermark series, the PSNR. Using Checkmark system, the simulation results show that the algorithm is robust enough to some very important image processing attacks such as lossy compression, MAP, filtering, segmentation, edge enhancement, jitter, quadratic modulation and general geometric attack（scaling, rotation, shearing）, etc.

关键词： Digital Watermarking Data Mining Discrete Radon Transform tight frame Copyright Protection Information Hiding Finite-Set

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Analysis in Theory and Applications 2015年第2期31卷 109-122页

作者： Firdous A.Shah Abdullah Department of Mathematics University of Kashmir Department of Mathematics ZHDC University of Delhi

In this paper, we present a method for constructing multivariate tight framelet packets associated with an arbitrary dilation matrix using unitary extension principles. We also prove how to construct various tight fra... 详细信息

关键词： Wavelet tight frame framelet packet matrix dilation extension principle Fouriertransform.