In this paper,a Littlewood-Paley function characterization of the spaces L p（R）,1〈p〈∞,is first established by means of the equivalent conditions of tight wavelet frames,wherein the Littlewood-Paley function is as...In this paper,a Littlewood-Paley function characterization of the spaces L p（R）,1〈p〈∞,is first established by means of the equivalent conditions of tight wavelet frames,wherein the Littlewood-Paley function is associated with a tight wavelet frame generated by the so-called extension principles.With the above characterization,another characterization of L p（R）,1〈p〈∞,is also established in terms of the weighted l 2-norm of the wavelet frame coefficients,which can be a useful tool in harmonic analysis,approximation theory,and image processing.展开更多
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 ...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.展开更多
A muitisensor image fusion algorithm is described using 2-dimensional nonseparable wavelet frame （NWF） transform. The source muitisensor images are first decomposed by the NWF transform. Then, the NWF transform coef...A muitisensor image fusion algorithm is described using 2-dimensional nonseparable wavelet frame （NWF） transform. The source muitisensor images are first decomposed by the NWF transform. Then, the NWF transform coefficients of the source images are combined into the composite NWF transform coefficients. Inverse NWF transform is performed on the composite NWF transform coefficients in order to obtain the intermediate fused image. Finally, intensity adjustment is applied to the intermediate fused image in order to maintain the dynamic intensity range. Experiment resuits using real data show that the proposed algorithm works well in muitisensor image fusion.展开更多
Extension Principles play a significant role in the construction of MRA based wavelet frames and have attracted much attention for their potential applications in various scientific fields. A novel and simple procedur...Extension Principles play a significant role in the construction of MRA based wavelet frames and have attracted much attention for their potential applications in various scientific fields. A novel and simple procedure for the construction of tight wavelet frames generated by the Walsh polynomials using Extension Principles was recently considered by Shah in [Tight wavelet frames generated by the Walsh poly- nomials, Int. J. Wavelets, Multiresolut. Inf. Process., 11（6） （2013）, 1350042]. In this paper, we establish a complete characterization of tight wavelet frames generated by the Walsh polynomials in terms of the polyphase matrices formed by the polyphase components of the Walsh polynomials.展开更多
In this article, we introduce a notion of nonuniform wavelet frames on local fields of positive characteristic. Furthermore, we gave a complete characterization of tight nonuniform wavelet frames on local fields of po...In this article, we introduce a notion of nonuniform wavelet frames on local fields of positive characteristic. Furthermore, we gave a complete characterization of tight nonuniform wavelet frames on local fields of positive characteristic via Fourier transform. Our results also hold for the Cantor dyadic group and the Vilenkin groups as they are local fields of positive characteristic.展开更多
The dropping off of data during information transmission and the storage device’s damage etc.often leads the sampled data to be non-uniform.The paper, based on the stability theory of irregular wavelet frame and the ...The dropping off of data during information transmission and the storage device’s damage etc.often leads the sampled data to be non-uniform.The paper, based on the stability theory of irregular wavelet frame and the irregular weighted wavelet frame operator,proposed an irregular weighted wavelet fame conjugate gradient iterative algorithm for the reconstruction of non-uniformly sampling signal. Compared the experiment results with the iterative algorithm of the Ref.[5],the new algorithm has remarkable advantages in approximation error,running time and so on.展开更多
The aim of this paper is to study wavelet frame packets in which there are many frames. It is a generalization of wavelet packets. We derive few results on wavelet frame packets and have obtained the corresponding fra...The aim of this paper is to study wavelet frame packets in which there are many frames. It is a generalization of wavelet packets. We derive few results on wavelet frame packets and have obtained the corresponding frame bounds.展开更多
Pointwise convergence and uniform convergence for wavelet frame series is a new topic. With the help of band-limited dual wavelet frames, this topic is first researched.
This paper addresses the construction of wavelet frame from a frame multiresolution analysis （FMRA） associated with a dilation matrix of determinant ±2. The dilation matrices of determinant ±2 can be class...This paper addresses the construction of wavelet frame from a frame multiresolution analysis （FMRA） associated with a dilation matrix of determinant ±2. The dilation matrices of determinant ±2 can be classified as six classes according to integral similarity. In this paper, for four classes of them, the construction of wavelet frame from an FMRA is obtained, and, as examples, Shannon type wavelet frames are constructed, which have an independent value for their optimality in some sense.展开更多
An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125： 68-186]. We exten...An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125： 68-186]. We extend their results to the dilation matrix cases in two aspects. We first show that the periodization of any wavelet frame constructed by the unitary extension principle formulated by Ron and Shen is still a periodic wavelet frame under weaker conditions than that given by Zhang and Saito, and then prove that the periodization of those generated by the mixed extension principle is also a periodic wavelet frame if the scaling functions have compact supports.展开更多
The homogeneous approximation property （HAP） states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we ...The homogeneous approximation property （HAP） states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.展开更多
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 prese...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.展开更多
In this paper, we present the conditions on dilation parameter {sj}j that ensure a discrete irregular wavelet system to be a frame on L2(Rn), and for the wavelet frame we consider the perturbations of translation para...In this paper, we present the conditions on dilation parameter {sj}j that ensure a discrete irregular wavelet system to be a frame on L2(Rn), and for the wavelet frame we consider the perturbations of translation parameter b and frame function ψ respectively.展开更多
THE inner product 【f, g】 of f and g in L<sub>2</sub> （R） is defined by 【f, g】, and is the norm of f ; moreover, dx is the Fourier transform of f. ForΩ】O, we
Suppose that η1,...,η_n are measurable functions in L2（R）.We call the n-tuple（η1,...,ηn） a Parseval super frame wavelet of length n if {2^（k/2） η1（2~kt-l） ⊕···⊕2^（k/2） ηn（2kt-l）：k,l...Suppose that η1,...,η_n are measurable functions in L2（R）.We call the n-tuple（η1,...,ηn） a Parseval super frame wavelet of length n if {2^（k/2） η1（2~kt-l） ⊕···⊕2^（k/2） ηn（2kt-l）：k,l∈Z} is a Parseval frame for L2（R）⊕n.In high dimensional case,there exists a similar notion of Parseval super frame wavelet with some expansive dilation matrix.In this paper,we will study the Parseval super frame wavelets of length n,and will focus on the path-connectedness of the set of all s-elementary Parseval super frame wavelets in one-dimensional and high dimensional cases.We will prove the corresponding path-connectedness theorems.展开更多
For the non-band-limited functionΨ, a sufficient condition is presented under which $\{ \sqrt {s_j } \psi (s_j \cdot - kb)\} $ is a frame for L2(R). The stability of these frames is studied. For the wavelets frequent...For the non-band-limited functionΨ, a sufficient condition is presented under which $\{ \sqrt {s_j } \psi (s_j \cdot - kb)\} $ is a frame for L2(R). The stability of these frames is studied. For the wavelets frequently used in signal processing, some concrete results are given.展开更多
The construction and properties of interval minimum-energy wavelet frame are systematically studied in this paper. They are as follows： 1） give the definition of interval minimum-energy wavelet frame; 2） give the n...The construction and properties of interval minimum-energy wavelet frame are systematically studied in this paper. They are as follows： 1） give the definition of interval minimum-energy wavelet frame; 2） give the necessary and sufficient conditions for the minimum-energy frames for L^2[0,1]; 3） present the construction algorithm for minimum-energy wavelet frame associated with refinable functions on the interval with any support y; 4） give the decomposition and reconstruction formulas of the minimum-energy frame on the interval [0,1],展开更多
A new method for constructing locally supported radial wavelet frame or basis, which is different from the multiresolution analysis, is proposed. A continuously differentiable radial function with a local, support is ...A new method for constructing locally supported radial wavelet frame or basis, which is different from the multiresolution analysis, is proposed. A continuously differentiable radial function with a local, support is chosen at first. Then a radial wavelet is obtained by the first and second derivatives of the radial function. If the radial function is both locally supported and infinitely differentiable, so is the radial wavelet. It is shown that the radial wavelet is a multidimensional dyadic one. I. Daubechies’ wavelet frame Theorem is extended from one dimension ton dimensions. It is proven that the family generated by dilations and translations from a single radial wavelet can constitute a frame inL 2(? n ). Consequently, it is concluded that the radial wavelet family generated by dilations and translations combined with their linear combination can constitute an orthonormal basis inL 2(? n ). Finally, an example of the radial wavelet, which is inseparable and with a local support and infinitely high regularity, is given based on the framework given here. As an application, a class of wavelet network for image denoising is designed, and an underrelaxation iterative fastlearning algorithm with varied learning rate is given as well.展开更多
基于高斯差分小波(Difference of Gaussians,DOG)向球面上拓展产生的球面DOG小波函数,提出了基于球面DOG小波框架重构骨料形状的新方法。为探讨该方法的优势,采用该方法和传统球谐重构法,对采用三维扫描仪获取机制砂的表面点云数据进行...基于高斯差分小波(Difference of Gaussians,DOG)向球面上拓展产生的球面DOG小波函数,提出了基于球面DOG小波框架重构骨料形状的新方法。为探讨该方法的优势,采用该方法和传统球谐重构法,对采用三维扫描仪获取机制砂的表面点云数据进行重构,分析了球面DOG小波框架方法的参数设置,以及准确性。结果显示,随着球面DOG小波框架qmax的增大,重构骨料的形貌逐渐接近真实形态。且当qmax达到4时,支撑集足够小的球面DOG小波可有效避免高阶次球谐级数重构多棱角骨料时由于振铃效应导致的重构精度下降问题。展开更多
A new framework of region-based dynamic image fusion is proposed. First, the technique of target detection is applied to dynamic images (image sequences) to segment images into different targets and background regions...A new framework of region-based dynamic image fusion is proposed. First, the technique of target detection is applied to dynamic images (image sequences) to segment images into different targets and background regions. Then different fusion rules are employed in different regions so that the target information is preserved as much as possible. In addition, steerable non-separable wavelet frame transform is used in the process of multi-resolution analysis, so the system achieves favorable characters of orientation and invariant shift. Compared with other image fusion methods, experimental results showed that the proposed method has better capabilities of target recognition and preserves clear background information.展开更多
基金Supported by the National High Technology Research and Development Program of China (863 Program) (2009AA12Z203,2008AA 12Z201)
文摘In this paper,a Littlewood-Paley function characterization of the spaces L p（R）,1〈p〈∞,is first established by means of the equivalent conditions of tight wavelet frames,wherein the Littlewood-Paley function is associated with a tight wavelet frame generated by the so-called extension principles.With the above characterization,another characterization of L p（R）,1〈p〈∞,is also established in terms of the weighted l 2-norm of the wavelet frame coefficients,which can be a useful tool in harmonic analysis,approximation theory,and image processing.
基金supported by NBHM, Department of Atomic Energy, Government of India (Grant No. 2/48(8)/2016/NBHM(R.P)/R&D II/13924)
文摘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.
文摘A muitisensor image fusion algorithm is described using 2-dimensional nonseparable wavelet frame （NWF） transform. The source muitisensor images are first decomposed by the NWF transform. Then, the NWF transform coefficients of the source images are combined into the composite NWF transform coefficients. Inverse NWF transform is performed on the composite NWF transform coefficients in order to obtain the intermediate fused image. Finally, intensity adjustment is applied to the intermediate fused image in order to maintain the dynamic intensity range. Experiment resuits using real data show that the proposed algorithm works well in muitisensor image fusion.
文摘Extension Principles play a significant role in the construction of MRA based wavelet frames and have attracted much attention for their potential applications in various scientific fields. A novel and simple procedure for the construction of tight wavelet frames generated by the Walsh polynomials using Extension Principles was recently considered by Shah in [Tight wavelet frames generated by the Walsh poly- nomials, Int. J. Wavelets, Multiresolut. Inf. Process., 11（6） （2013）, 1350042]. In this paper, we establish a complete characterization of tight wavelet frames generated by the Walsh polynomials in terms of the polyphase matrices formed by the polyphase components of the Walsh polynomials.
文摘In this article, we introduce a notion of nonuniform wavelet frames on local fields of positive characteristic. Furthermore, we gave a complete characterization of tight nonuniform wavelet frames on local fields of positive characteristic via Fourier transform. Our results also hold for the Cantor dyadic group and the Vilenkin groups as they are local fields of positive characteristic.
基金supported by Hunan Education Office Foundation under Grant 06C260
文摘The dropping off of data during information transmission and the storage device’s damage etc.often leads the sampled data to be non-uniform.The paper, based on the stability theory of irregular wavelet frame and the irregular weighted wavelet frame operator,proposed an irregular weighted wavelet fame conjugate gradient iterative algorithm for the reconstruction of non-uniformly sampling signal. Compared the experiment results with the iterative algorithm of the Ref.[5],the new algorithm has remarkable advantages in approximation error,running time and so on.
文摘The aim of this paper is to study wavelet frame packets in which there are many frames. It is a generalization of wavelet packets. We derive few results on wavelet frame packets and have obtained the corresponding frame bounds.
文摘Pointwise convergence and uniform convergence for wavelet frame series is a new topic. With the help of band-limited dual wavelet frames, this topic is first researched.
基金Excellent Talent Training Foundation of Beijing(20051D0501522)
文摘This paper addresses the construction of wavelet frame from a frame multiresolution analysis （FMRA） associated with a dilation matrix of determinant ±2. The dilation matrices of determinant ±2 can be classified as six classes according to integral similarity. In this paper, for four classes of them, the construction of wavelet frame from an FMRA is obtained, and, as examples, Shannon type wavelet frames are constructed, which have an independent value for their optimality in some sense.
基金Acknowledgements The authors express their gratitude to the anonymous referees for their kind suggestions and useful comments on the original manuscript, which resulted in this final version. This work was supported by the National Natural Science Foundation of China （No. 61071189）, the Natural Science Foundation for the Education Department of Henan Province of China （No. 13A110072）, and the Natural Science Foundation of Henan University （No. 2011YBZR001）.
文摘An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125： 68-186]. We extend their results to the dilation matrix cases in two aspects. We first show that the periodization of any wavelet frame constructed by the unitary extension principle formulated by Ron and Shen is still a periodic wavelet frame under weaker conditions than that given by Zhang and Saito, and then prove that the periodization of those generated by the mixed extension principle is also a periodic wavelet frame if the scaling functions have compact supports.
基金Supported partially by National Natural Science Foundation of China(Grant Nos.10971105and10990012)Natural Science Foundation of Tianjin(Grant No.09JCYBJC01000)
文摘The homogeneous approximation property （HAP） states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.
基金This work was supported by CONICET and Universidad Nacional de San Luis
文摘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.
基金The authors thank the referees for their valuable suggestions.This work was supported by the National Natural Science Foundation of China(Grant No.60472042)the Research Fund for the Doctoral Program of Higher Education.
文摘In this paper, we present the conditions on dilation parameter {sj}j that ensure a discrete irregular wavelet system to be a frame on L2(Rn), and for the wavelet frame we consider the perturbations of translation parameter b and frame function ψ respectively.
文摘THE inner product 【f, g】 of f and g in L<sub>2</sub> （R） is defined by 【f, g】, and is the norm of f ; moreover, dx is the Fourier transform of f. ForΩ】O, we
基金Supported by the National Natural Science Foundation of China(11071065,11101142,11171306,10671062)the China Postdoctoral Science Foundation(20100480942)+1 种基金the Ph.D.Programs Foundation of the Ministry of Education of China(20094306110004)the Program for Science and Technology Research Team in Higher Educational Institutions of Hunan Province
文摘Suppose that η1,...,η_n are measurable functions in L2（R）.We call the n-tuple（η1,...,ηn） a Parseval super frame wavelet of length n if {2^（k/2） η1（2~kt-l） ⊕···⊕2^（k/2） ηn（2kt-l）：k,l∈Z} is a Parseval frame for L2（R）⊕n.In high dimensional case,there exists a similar notion of Parseval super frame wavelet with some expansive dilation matrix.In this paper,we will study the Parseval super frame wavelets of length n,and will focus on the path-connectedness of the set of all s-elementary Parseval super frame wavelets in one-dimensional and high dimensional cases.We will prove the corresponding path-connectedness theorems.
基金This work was supported by the National Natural Science Foundation of China(Grant No.16971047),the Mathemalical Center of State Education Commission and RFDP.
文摘For the non-band-limited functionΨ, a sufficient condition is presented under which $\{ \sqrt {s_j } \psi (s_j \cdot - kb)\} $ is a frame for L2(R). The stability of these frames is studied. For the wavelets frequently used in signal processing, some concrete results are given.
基金the National Natural Science Foundation of China (Grant No.60375021)the Natural Science Foundation of Hunan Province,China (Grant No.05JJ10011)the Scientific Research Fund of Hunan Provincial Education Department of China (Grant Nos.04A056 and 06C836)
文摘The construction and properties of interval minimum-energy wavelet frame are systematically studied in this paper. They are as follows： 1） give the definition of interval minimum-energy wavelet frame; 2） give the necessary and sufficient conditions for the minimum-energy frames for L^2[0,1]; 3） present the construction algorithm for minimum-energy wavelet frame associated with refinable functions on the interval with any support y; 4） give the decomposition and reconstruction formulas of the minimum-energy frame on the interval [0,1],
基金Project supported by the National Natural Science Foundation of China (Grant No. 69735010) and Doctorate Foundation of Xi' an Jiaotong University.
文摘A new method for constructing locally supported radial wavelet frame or basis, which is different from the multiresolution analysis, is proposed. A continuously differentiable radial function with a local, support is chosen at first. Then a radial wavelet is obtained by the first and second derivatives of the radial function. If the radial function is both locally supported and infinitely differentiable, so is the radial wavelet. It is shown that the radial wavelet is a multidimensional dyadic one. I. Daubechies’ wavelet frame Theorem is extended from one dimension ton dimensions. It is proven that the family generated by dilations and translations from a single radial wavelet can constitute a frame inL 2(? n ). Consequently, it is concluded that the radial wavelet family generated by dilations and translations combined with their linear combination can constitute an orthonormal basis inL 2(? n ). Finally, an example of the radial wavelet, which is inseparable and with a local support and infinitely high regularity, is given based on the framework given here. As an application, a class of wavelet network for image denoising is designed, and an underrelaxation iterative fastlearning algorithm with varied learning rate is given as well.
文摘基于高斯差分小波(Difference of Gaussians,DOG)向球面上拓展产生的球面DOG小波函数,提出了基于球面DOG小波框架重构骨料形状的新方法。为探讨该方法的优势,采用该方法和传统球谐重构法,对采用三维扫描仪获取机制砂的表面点云数据进行重构,分析了球面DOG小波框架方法的参数设置,以及准确性。结果显示,随着球面DOG小波框架qmax的增大,重构骨料的形貌逐渐接近真实形态。且当qmax达到4时,支撑集足够小的球面DOG小波可有效避免高阶次球谐级数重构多棱角骨料时由于振铃效应导致的重构精度下降问题。
基金Project (No. 2004CB719401) supported by the National Basic Research Program (973) of China
文摘A new framework of region-based dynamic image fusion is proposed. First, the technique of target detection is applied to dynamic images (image sequences) to segment images into different targets and background regions. Then different fusion rules are employed in different regions so that the target information is preserved as much as possible. In addition, steerable non-separable wavelet frame transform is used in the process of multi-resolution analysis, so the system achieves favorable characters of orientation and invariant shift. Compared with other image fusion methods, experimental results showed that the proposed method has better capabilities of target recognition and preserves clear background information.
