Pub Date : 2021-07-20DOI: 10.1142/s0219691322500400
P. Casazza, F. Akrami, A. Rahimi
In this manuscript, we present several new results in finite and countable dimensional real Hilbert space phase retrieval and norm retrieval by vectors and projections. We make a detailed study of when hyperplanes do norm retrieval. Also, we show that the families of norm retrievable frames ${f_{i}}_{i=1}^{m}$ in $mathbb{R}^n$ are not dense in the family of $mleq (2n-2)$-element sets of vectors in $mathbb{R}^n$ for every finite $n$ and the families of vectors which do norm retrieval in $ell^2$ are not dense in the infinite families of vectors in $ell^2$. We also show that if a Riesz basis does norm retrieval in $ell^2$, then it is an orthogonal sequence. We provide numerous examples to show that our results are best possible.
{"title":"A note on phase (norm) retrievable real Hilbert space fusion frames","authors":"P. Casazza, F. Akrami, A. Rahimi","doi":"10.1142/s0219691322500400","DOIUrl":"https://doi.org/10.1142/s0219691322500400","url":null,"abstract":"In this manuscript, we present several new results in finite and countable dimensional real Hilbert space phase retrieval and norm retrieval by vectors and projections. We make a detailed study of when hyperplanes do norm retrieval. Also, we show that the families of norm retrievable frames ${f_{i}}_{i=1}^{m}$ in $mathbb{R}^n$ are not dense in the family of $mleq (2n-2)$-element sets of vectors in $mathbb{R}^n$ for every finite $n$ and the families of vectors which do norm retrieval in $ell^2$ are not dense in the infinite families of vectors in $ell^2$. We also show that if a Riesz basis does norm retrieval in $ell^2$, then it is an orthogonal sequence. We provide numerous examples to show that our results are best possible.","PeriodicalId":158567,"journal":{"name":"Int. J. Wavelets Multiresolution Inf. Process.","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126655423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-10DOI: 10.1142/s0219691321500326
Siva Prasad Murugan, G. P. Youvaraj
The Franklin wavelet is constructed using the multiresolution analysis (MRA) generated from a scaling function [Formula: see text] that is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text] for every [Formula: see text]. For [Formula: see text] and [Formula: see text], it is shown that if a function [Formula: see text] is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text], for [Formula: see text], and generates MRA with dilation factor [Formula: see text], then [Formula: see text]. Conversely, for [Formula: see text], it is shown that there exists a [Formula: see text], as satisfying the above conditions, that generates MRA with dilation factor [Formula: see text]. The frame MRA (FMRA) is useful in signal processing, since the perfect reconstruction filter banks associated with FMRA can be narrow-band. So it is natural to ask, whether the above results can be extended for the case of FMRA. In this paper, for [Formula: see text], we prove that if [Formula: see text] generates FMRA with dilation factor [Formula: see text], then [Formula: see text]. For [Formula: see text], we prove similar results when [Formula: see text]. In addition, for [Formula: see text] we prove that there exists a function [Formula: see text] as satisfying the above conditions, that generates FMRA. Also, we construct tight wavelet frame and wavelet frame for such scaling functions.
{"title":"Frame multiresolution analysis of continuous piecewise linear functions","authors":"Siva Prasad Murugan, G. P. Youvaraj","doi":"10.1142/s0219691321500326","DOIUrl":"https://doi.org/10.1142/s0219691321500326","url":null,"abstract":"The Franklin wavelet is constructed using the multiresolution analysis (MRA) generated from a scaling function [Formula: see text] that is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text] for every [Formula: see text]. For [Formula: see text] and [Formula: see text], it is shown that if a function [Formula: see text] is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text], for [Formula: see text], and generates MRA with dilation factor [Formula: see text], then [Formula: see text]. Conversely, for [Formula: see text], it is shown that there exists a [Formula: see text], as satisfying the above conditions, that generates MRA with dilation factor [Formula: see text]. The frame MRA (FMRA) is useful in signal processing, since the perfect reconstruction filter banks associated with FMRA can be narrow-band. So it is natural to ask, whether the above results can be extended for the case of FMRA. In this paper, for [Formula: see text], we prove that if [Formula: see text] generates FMRA with dilation factor [Formula: see text], then [Formula: see text]. For [Formula: see text], we prove similar results when [Formula: see text]. In addition, for [Formula: see text] we prove that there exists a function [Formula: see text] as satisfying the above conditions, that generates FMRA. Also, we construct tight wavelet frame and wavelet frame for such scaling functions.","PeriodicalId":158567,"journal":{"name":"Int. J. Wavelets Multiresolution Inf. Process.","volume":"205 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116380706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-08DOI: 10.1142/S0219691321500338
Duván Humberto Cataño, C. Rodríguez-Caballero, D. Peña, Chang Chiann
We introduce a high-dimensional factor model with time-varying loadings. We cover both stationary and nonstationary factors to increase the possibilities of applications. We propose an estimation procedure based on two stages. First, we estimate common factors by principal components. In the second step, considering the estimated factors as observed, the time-varying loadings are estimated by an iterative generalized least squares procedure using wavelet functions. We investigate the finite sample features by some Monte Carlo simulations. Finally, we apply the model to study the Nord Pool power market’s electricity prices and loads.
{"title":"Wavelet estimation for factor models with time-varying loadings","authors":"Duván Humberto Cataño, C. Rodríguez-Caballero, D. Peña, Chang Chiann","doi":"10.1142/S0219691321500338","DOIUrl":"https://doi.org/10.1142/S0219691321500338","url":null,"abstract":"We introduce a high-dimensional factor model with time-varying loadings. We cover both stationary and nonstationary factors to increase the possibilities of applications. We propose an estimation procedure based on two stages. First, we estimate common factors by principal components. In the second step, considering the estimated factors as observed, the time-varying loadings are estimated by an iterative generalized least squares procedure using wavelet functions. We investigate the finite sample features by some Monte Carlo simulations. Finally, we apply the model to study the Nord Pool power market’s electricity prices and loads.","PeriodicalId":158567,"journal":{"name":"Int. J. Wavelets Multiresolution Inf. Process.","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130418131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-05DOI: 10.1142/S021969132150034X
Manxia Cao, Wei Huang
In this paper, the [Formula: see text]-analysis model for the phase retrieval problem of sparse unknown signals in the redundant dictionary is extended to the [Formula: see text]-analysis model, where [Formula: see text]. It’s shown that if the measurement matrix [Formula: see text] satisfies the strong restricted isometry property adapted to D (S-DRIP) condition, the unknown signal [Formula: see text] can be stably recovered by analyzing the [Formula: see text] [Formula: see text] minimization model.
{"title":"Sparse phase retrieval via ℓp (0 p ≤ 1) minimization","authors":"Manxia Cao, Wei Huang","doi":"10.1142/S021969132150034X","DOIUrl":"https://doi.org/10.1142/S021969132150034X","url":null,"abstract":"In this paper, the [Formula: see text]-analysis model for the phase retrieval problem of sparse unknown signals in the redundant dictionary is extended to the [Formula: see text]-analysis model, where [Formula: see text]. It’s shown that if the measurement matrix [Formula: see text] satisfies the strong restricted isometry property adapted to D (S-DRIP) condition, the unknown signal [Formula: see text] can be stably recovered by analyzing the [Formula: see text] [Formula: see text] minimization model.","PeriodicalId":158567,"journal":{"name":"Int. J. Wavelets Multiresolution Inf. Process.","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124744074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-05DOI: 10.1142/S0219691321500351
Yu Tian, Hui-Fang Jia, Guo-Liang He
The theory of Gabor frames has been extensively investigated. This paper addresses partial Gabor systems. We introduce the concepts of partial Gabor system, frame and dual frame. We present some conditions for a partial Gabor system to be a partial Gabor frame, and using these conditions, we characterize partial dual frames. We also give some examples. It is noteworthy that the density theorem does not hold for general partial Gabor systems.
{"title":"Partial Gabor frames and dual frames","authors":"Yu Tian, Hui-Fang Jia, Guo-Liang He","doi":"10.1142/S0219691321500351","DOIUrl":"https://doi.org/10.1142/S0219691321500351","url":null,"abstract":"The theory of Gabor frames has been extensively investigated. This paper addresses partial Gabor systems. We introduce the concepts of partial Gabor system, frame and dual frame. We present some conditions for a partial Gabor system to be a partial Gabor frame, and using these conditions, we characterize partial dual frames. We also give some examples. It is noteworthy that the density theorem does not hold for general partial Gabor systems.","PeriodicalId":158567,"journal":{"name":"Int. J. Wavelets Multiresolution Inf. Process.","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134497279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-01DOI: 10.1142/S0219691321500302
M. Y. Bhat, Aamir Hamid Dar
The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.
{"title":"Wavelet packets associated with linear canonical transform on spectrum","authors":"M. Y. Bhat, Aamir Hamid Dar","doi":"10.1142/S0219691321500302","DOIUrl":"https://doi.org/10.1142/S0219691321500302","url":null,"abstract":"The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.","PeriodicalId":158567,"journal":{"name":"Int. J. Wavelets Multiresolution Inf. Process.","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123908246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-26DOI: 10.1142/s0219691321500314
Di Wang, Can Liu, Suixiang Gao, Fu-cheng Liao
In this paper, a design method of optimal tracking control based on finite-time stability for quadratic performance index is proposed. Finite-time stability of tracking control involves dynamical systems whose actual output can track desired output in finite time while satisfying Lyapunov stability. A nonlinear control law which guaranteed finite-time stability is designed depending on the core idea of dynamic programming. By using Hamilton–Jacobi–Bellman (HJB) equation and finite-time stability theory, sufficient conditions involving V-function are provided, and design steps for nonlinear finite-time tracking control law are derived by constructing augmented systems. In addition, the V-function is constructed to obtain corresponding law for given systems, which verified that the design method is feasible. Simulation examples validate the efficiency of the results.
{"title":"Design of nonlinear optimal tracking control law based on finite-time stability for quadratic performance index","authors":"Di Wang, Can Liu, Suixiang Gao, Fu-cheng Liao","doi":"10.1142/s0219691321500314","DOIUrl":"https://doi.org/10.1142/s0219691321500314","url":null,"abstract":"In this paper, a design method of optimal tracking control based on finite-time stability for quadratic performance index is proposed. Finite-time stability of tracking control involves dynamical systems whose actual output can track desired output in finite time while satisfying Lyapunov stability. A nonlinear control law which guaranteed finite-time stability is designed depending on the core idea of dynamic programming. By using Hamilton–Jacobi–Bellman (HJB) equation and finite-time stability theory, sufficient conditions involving V-function are provided, and design steps for nonlinear finite-time tracking control law are derived by constructing augmented systems. In addition, the V-function is constructed to obtain corresponding law for given systems, which verified that the design method is feasible. Simulation examples validate the efficiency of the results.","PeriodicalId":158567,"journal":{"name":"Int. J. Wavelets Multiresolution Inf. Process.","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125453123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-19DOI: 10.1142/S0219691321500296
Majid Jabbar Saadan, M. Janfada, R. A. Kamyabi-Gol
Let [Formula: see text] be a locally compact group. This work aims to study some important properties of the frame of translates for the space [Formula: see text]. Also, we showed that the frame of translates can be a frame for a proper subspace of [Formula: see text], in the case that [Formula: see text] has a contractive automorphism. Furthermore, we characterize the frame properties for such systems and find an expression for the canonical dual frame.
设[公式:见文本]是一个局部紧群。本研究旨在研究空间翻译框架的一些重要性质[公式:见文]。同时，我们证明了在[Formula: see text]具有收缩自同构的情况下，翻译的框架可以是[Formula: see text]的适当子空间的框架。进一步，我们刻画了这类系统的框架性质，并找到了典型对偶框架的表达式。
{"title":"Frame of translates and its canonical dual in L2(G)","authors":"Majid Jabbar Saadan, M. Janfada, R. A. Kamyabi-Gol","doi":"10.1142/S0219691321500296","DOIUrl":"https://doi.org/10.1142/S0219691321500296","url":null,"abstract":"Let [Formula: see text] be a locally compact group. This work aims to study some important properties of the frame of translates for the space [Formula: see text]. Also, we showed that the frame of translates can be a frame for a proper subspace of [Formula: see text], in the case that [Formula: see text] has a contractive automorphism. Furthermore, we characterize the frame properties for such systems and find an expression for the canonical dual frame.","PeriodicalId":158567,"journal":{"name":"Int. J. Wavelets Multiresolution Inf. Process.","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116412206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-19DOI: 10.1142/s0219691321500284
Guangyi Chen, A. Krzyżak
In this paper, we revisit the effects of principal component analysis (PCA) on hyperspectral imagery denoising. Our previous work combined PCA with wavelet shrinkage and particularly good denoising results has been achieved. We debate that any denoising methods can be used to replace wavelet shrinkage in our PCA+wavelet shrinkage algorithm. The major difference between this work and our previous PCA-based denoising method is that we consider a mixture of Gaussian and shot noise in this work whereas our previous methods studied Gaussian white noise alone. In addition, we retain [Formula: see text] [Formula: see text] PCA output components in our forward PCA transform in this paper whereas we keep all PCA output components [Formula: see text] in our previous works. The [Formula: see text] above is the number of spectral bands in the original hyperspectral imagery data cube. In addition, PCA is much better than nonlinear PCA for hyperspectral imagery denoising when Gaussian white noise and shot noise are introduced as demonstrated in this paper. Extensive experiments demonstrate that the method proposed in this paper outperforms the existing methods significantly in terms of signal-to-noise ratio for two testing hyperspectral imagery data cubes.
{"title":"Noise reduction of shot-noise-dominated hyperspectral imagery by combining PCA with existing denoising methods","authors":"Guangyi Chen, A. Krzyżak","doi":"10.1142/s0219691321500284","DOIUrl":"https://doi.org/10.1142/s0219691321500284","url":null,"abstract":"In this paper, we revisit the effects of principal component analysis (PCA) on hyperspectral imagery denoising. Our previous work combined PCA with wavelet shrinkage and particularly good denoising results has been achieved. We debate that any denoising methods can be used to replace wavelet shrinkage in our PCA+wavelet shrinkage algorithm. The major difference between this work and our previous PCA-based denoising method is that we consider a mixture of Gaussian and shot noise in this work whereas our previous methods studied Gaussian white noise alone. In addition, we retain [Formula: see text] [Formula: see text] PCA output components in our forward PCA transform in this paper whereas we keep all PCA output components [Formula: see text] in our previous works. The [Formula: see text] above is the number of spectral bands in the original hyperspectral imagery data cube. In addition, PCA is much better than nonlinear PCA for hyperspectral imagery denoising when Gaussian white noise and shot noise are introduced as demonstrated in this paper. Extensive experiments demonstrate that the method proposed in this paper outperforms the existing methods significantly in terms of signal-to-noise ratio for two testing hyperspectral imagery data cubes.","PeriodicalId":158567,"journal":{"name":"Int. J. Wavelets Multiresolution Inf. Process.","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129609711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-17DOI: 10.1142/S021969132150020X
Haoliang Yuan, Sio-Long Lo, Ming Yin, Yong Liang
In this paper, we propose a sparse tensor regression model for multi-view feature selection. Apart from the most of existing methods, our model adopts a tensor structure to represent multi-view data, which aims to explore their underlying high-order correlations. Based on this tensor structure, our model can effectively select the meaningful feature set for each view. We also develop an iterative optimization algorithm to solve our model, together with analysis about the convergence and computational complexity. Experimental results on several popular multi-view data sets confirm the effectiveness of our model.
{"title":"Multi-view feature selection via sparse tensor regression","authors":"Haoliang Yuan, Sio-Long Lo, Ming Yin, Yong Liang","doi":"10.1142/S021969132150020X","DOIUrl":"https://doi.org/10.1142/S021969132150020X","url":null,"abstract":"In this paper, we propose a sparse tensor regression model for multi-view feature selection. Apart from the most of existing methods, our model adopts a tensor structure to represent multi-view data, which aims to explore their underlying high-order correlations. Based on this tensor structure, our model can effectively select the meaningful feature set for each view. We also develop an iterative optimization algorithm to solve our model, together with analysis about the convergence and computational complexity. Experimental results on several popular multi-view data sets confirm the effectiveness of our model.","PeriodicalId":158567,"journal":{"name":"Int. J. Wavelets Multiresolution Inf. Process.","volume":"666 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116100800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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