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Haichao Wang - One of the best experts on this subject based on the ideXlab platform.
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phase retrieval of real valued signals in a shift invariant Space
Applied and Computational Harmonic Analysis, 2018Co-Authors: Yang Chen, Cheng Cheng, Haichao WangAbstract:Abstract In this paper, we consider an infinite-dimensional phase retrieval problem to reconstruct real-valued signals living in a Shift-Invariant Space from their phaseless samples taken either on the whole line or on a discrete set with finite sampling density. We characterize all phase retrievable signals in a real-valued Shift-Invariant Space using their nonseparability. For nonseparable signals generated by some function with support length L, we show that they can be well approximated, up to a sign, from their noisy phaseless samples taken on a discrete set with sampling density 2 L − 1 . In this paper, we also propose an algorithm with linear computational complexity to reconstruct nonseparable signals in a Shift-Invariant Space from their phaseless samples corrupted by bounded noises.
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phase retrieval of real valued signals in a shift invariant Space
arXiv: Information Theory, 2016Co-Authors: Yang Chen, Cheng Cheng, Haichao WangAbstract:Phase retrieval arises in various fields of science and engineering and it is well studied in a finite-dimensional setting. In this paper, we consider an infinite-dimensional phase retrieval problem to reconstruct real-valued signals living in a Shift-Invariant Space from its phaseless samples taken either on the whole line or on a set with finite sampling rate. We find the equivalence between nonseparability of signals in a linear Space and its phase retrievability with phaseless samples taken on the whole line. For a spline signal of order $N$, we show that it can be well approximated, up to a sign, from its noisy phaseless samples taken on a set with sampling rate $2N-1$. We propose an algorithm to reconstruct nonseparable signals in a Shift-Invariant Space generated by a compactly supported continuous function. The proposed algorithm is robust against bounded sampling noise and it could be implemented in a distributed manner.
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Principal Shift-Invariant Spaces with extra invariance nearest to observed data
Collectanea Mathematica, 2011Co-Authors: Akram Aldroubi, Ilya A. Krishtal, Romain Tessera, Haichao WangAbstract:Given a set of functions \({\mathcal{F}=\{f_1, \dots, f_m\}\subset L^2(\mathbb{R}),}\) we construct a principal Shift-Invariant Space V nearest to \({\mathcal{F}}\) in the sense that V minimizes the expression $$\sum_{i=1}^{m}\|f_i-P_{V}f_i\|^2,$$ among all the principal Shift-Invariant Spaces with an orthonormal generator which are also translation invariant, or among all the principal Shift-Invariant Spaces with an orthonormal generator which are also \({\frac{1}{n}\mathbb{Z}}\) -invariant for some fixed \({n\in\mathbb{N}}\) .
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Uncertainty Principles and Balian-Low type Theorems in Principal Shift-Invariant Spaces
Applied and Computational Harmonic Analysis, 2011Co-Authors: Akram Aldroubi, Haichao WangAbstract:Abstract In this paper, we consider the time-frequency localization of the generator of a principal Shift-Invariant Space on the real line which has additional shift-invariance. We prove that if a principal Shift-Invariant Space on the real line is translation-invariant then any of its orthonormal (or Riesz) generators is non-integrable. However, for any n ⩾ 2 , there exist principal Shift-Invariant Spaces on the real line that are also 1 n Z -invariant with an integrable orthonormal (or a Riesz) generator ϕ, but ϕ satisfies ∫ R | ϕ ( x ) | 2 | x | 1 + ϵ d x = ∞ for any ϵ > 0 and its Fourier transform ϕ ˆ cannot decay as fast as ( 1 + | ξ | ) − r for any r > 1 2 . Examples are constructed to demonstrate that the above decay properties for the orthonormal generator in the time domain and in the frequency domain are optimal.
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uncertainty principles and balian low type theorems in principal shift invariant Spaces
arXiv: Functional Analysis, 2010Co-Authors: Akram Aldroubi, Qiyu Sun, Haichao WangAbstract:In this paper, we consider the time-frequency localization of the generator of a principal Shift-Invariant Space on the real line which has additional shift-invariance. We prove that if a principal Shift-Invariant Space on the real line is translation-invariant then any of its orthonormal (or Riesz) generators is non-integrable. However, for any $n\ge2$, there exist principal Shift-Invariant Spaces on the real line that are also $\nZ$-invariant with an integrable orthonormal (or a Riesz) generator $\phi$, but $\phi$ satisfies $\int_{\mathbb R} |\phi(x)|^2 |x|^{1+\epsilon} dx=\infty$ for any $\epsilon>0$ and its Fourier transform $\hat\phi$ cannot decay as fast as $ (1+|\xi|)^{-r}$ for any $r>1/2$. Examples are constructed to demonstrate that the above decay properties for the orthormal generator in the time domain and in the frequency domain are optimal.
Akram Aldroubi - One of the best experts on this subject based on the ideXlab platform.
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Principal Shift-Invariant Spaces with extra invariance nearest to observed data
Collectanea Mathematica, 2011Co-Authors: Akram Aldroubi, Ilya A. Krishtal, Romain Tessera, Haichao WangAbstract:Given a set of functions \({\mathcal{F}=\{f_1, \dots, f_m\}\subset L^2(\mathbb{R}),}\) we construct a principal Shift-Invariant Space V nearest to \({\mathcal{F}}\) in the sense that V minimizes the expression $$\sum_{i=1}^{m}\|f_i-P_{V}f_i\|^2,$$ among all the principal Shift-Invariant Spaces with an orthonormal generator which are also translation invariant, or among all the principal Shift-Invariant Spaces with an orthonormal generator which are also \({\frac{1}{n}\mathbb{Z}}\) -invariant for some fixed \({n\in\mathbb{N}}\) .
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Uncertainty Principles and Balian-Low type Theorems in Principal Shift-Invariant Spaces
Applied and Computational Harmonic Analysis, 2011Co-Authors: Akram Aldroubi, Haichao WangAbstract:Abstract In this paper, we consider the time-frequency localization of the generator of a principal Shift-Invariant Space on the real line which has additional shift-invariance. We prove that if a principal Shift-Invariant Space on the real line is translation-invariant then any of its orthonormal (or Riesz) generators is non-integrable. However, for any n ⩾ 2 , there exist principal Shift-Invariant Spaces on the real line that are also 1 n Z -invariant with an integrable orthonormal (or a Riesz) generator ϕ, but ϕ satisfies ∫ R | ϕ ( x ) | 2 | x | 1 + ϵ d x = ∞ for any ϵ > 0 and its Fourier transform ϕ ˆ cannot decay as fast as ( 1 + | ξ | ) − r for any r > 1 2 . Examples are constructed to demonstrate that the above decay properties for the orthonormal generator in the time domain and in the frequency domain are optimal.
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uncertainty principles and balian low type theorems in principal shift invariant Spaces
arXiv: Functional Analysis, 2010Co-Authors: Akram Aldroubi, Qiyu Sun, Haichao WangAbstract:In this paper, we consider the time-frequency localization of the generator of a principal Shift-Invariant Space on the real line which has additional shift-invariance. We prove that if a principal Shift-Invariant Space on the real line is translation-invariant then any of its orthonormal (or Riesz) generators is non-integrable. However, for any $n\ge2$, there exist principal Shift-Invariant Spaces on the real line that are also $\nZ$-invariant with an integrable orthonormal (or a Riesz) generator $\phi$, but $\phi$ satisfies $\int_{\mathbb R} |\phi(x)|^2 |x|^{1+\epsilon} dx=\infty$ for any $\epsilon>0$ and its Fourier transform $\hat\phi$ cannot decay as fast as $ (1+|\xi|)^{-r}$ for any $r>1/2$. Examples are constructed to demonstrate that the above decay properties for the orthormal generator in the time domain and in the frequency domain are optimal.
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invariance of a shift invariant Space
Journal of Fourier Analysis and Applications, 2010Co-Authors: Akram Aldroubi, Keri Kornelson, Christopher Heil, Carlos Cabrelli, Ursula MolterAbstract:A Shift-Invariant Space is a Space of functions that is invariant under integer translations. Such Spaces are often used as models for Spaces of signals and images in mathematical and engineering applications. This paper characterizes those Shift-Invariant subSpaces S that are also invariant under additional (non-integer) translations. For the case of finitely generated Spaces, these Spaces are characterized in terms of the generators of the Space. As a consequence, it is shown that principal Shift-Invariant Spaces with a compactly supported generator cannot be invariant under any non-integer translations.
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Determining sets of shift invariant Spaces
2003Co-Authors: Akram Aldroubi, Carlos Cabrelli, D. Hardin, U. Molter, A. RodadoAbstract:Abstract. The problem of determining an appropriate signal or image model from experimental data is addressed. Specifically, given a finite set of signals or images belonging to a fixed but unknown shift invariant Space, the problem is whether the known signals at hand are sufficient for determining the unknown shift invariant Space to which they belong. This problem gives rise to the concept of determining sets for shift invariant Spaces, and we obtain necessary and sufficient conditions needed for determining an unknown shift invariant Space V (Φ) from a finite subset F = {f1, f2,..., fm} of V (Φ). 1. introduction In many signal and image processing applications, images and signals are as-sumed to belong to some shift invariant Space of the form: V (Φ): = {f = n∑ i=1 j∈Zd αi(j)φi(·+ j) : αi ∈ l2, i = 1,..., n} (1.1) where Φ = [φ1, φ2... φn]t is a column vector consisting of functions in L2(Rd) called a generator for the Space V = V (Φ) (see e.g., [2]). For example, if n = 1, d = 1 and φ(x) = sinc(x), then the underlying Space is the Space of band limited functions (often used in communications). However, the assumed model (often the band limitedness assumption) is seldom derived from experimental data. Thus, given a class of signals belonging to a certain fixed- but unknown- shift invariant Space V, the problem is whether it is possible to determine the Space V from a set of m experimental data F = {f1, f2,..., fm}, where fi are observed functions (signals) belonging to V (Φ). If a finite set F is sufficient to determine V (Φ), we will call it a determining set for V (Φ). The goal is to see if we can perform operations on the observations F = {f1, f2,..., fm} to deduce whether they are sufficient to determine the unknown shift invariant Space V (Φ), and if so, use them to find some generator Ψ for V (Φ), i.e., find Ψ such that V (Ψ) = V (Φ). If the observations are not sufficient to determine V (Φ), then we need to obtain more observations until a determining set is found. In this paper we give necessary and sufficient conditions for a finite subset F of the Space V (Φ) to be a determining set for V (Φ). In the case F is a determining set, we shall exhibit an orthonormal generator for V (Φ) that is written in terms of the elements of F and thereby reconstruct the whole Space
Yang Chen - One of the best experts on this subject based on the ideXlab platform.
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phase retrieval of real valued signals in a shift invariant Space
Applied and Computational Harmonic Analysis, 2018Co-Authors: Yang Chen, Cheng Cheng, Haichao WangAbstract:Abstract In this paper, we consider an infinite-dimensional phase retrieval problem to reconstruct real-valued signals living in a Shift-Invariant Space from their phaseless samples taken either on the whole line or on a discrete set with finite sampling density. We characterize all phase retrievable signals in a real-valued Shift-Invariant Space using their nonseparability. For nonseparable signals generated by some function with support length L, we show that they can be well approximated, up to a sign, from their noisy phaseless samples taken on a discrete set with sampling density 2 L − 1 . In this paper, we also propose an algorithm with linear computational complexity to reconstruct nonseparable signals in a Shift-Invariant Space from their phaseless samples corrupted by bounded noises.
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phase retrieval of real valued signals in a shift invariant Space
arXiv: Information Theory, 2016Co-Authors: Yang Chen, Cheng Cheng, Haichao WangAbstract:Phase retrieval arises in various fields of science and engineering and it is well studied in a finite-dimensional setting. In this paper, we consider an infinite-dimensional phase retrieval problem to reconstruct real-valued signals living in a Shift-Invariant Space from its phaseless samples taken either on the whole line or on a set with finite sampling rate. We find the equivalence between nonseparability of signals in a linear Space and its phase retrievability with phaseless samples taken on the whole line. For a spline signal of order $N$, we show that it can be well approximated, up to a sign, from its noisy phaseless samples taken on a set with sampling rate $2N-1$. We propose an algorithm to reconstruct nonseparable signals in a Shift-Invariant Space generated by a compactly supported continuous function. The proposed algorithm is robust against bounded sampling noise and it could be implemented in a distributed manner.
Cheng Cheng - One of the best experts on this subject based on the ideXlab platform.
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phase retrieval of real valued signals in a shift invariant Space
Applied and Computational Harmonic Analysis, 2018Co-Authors: Yang Chen, Cheng Cheng, Haichao WangAbstract:Abstract In this paper, we consider an infinite-dimensional phase retrieval problem to reconstruct real-valued signals living in a Shift-Invariant Space from their phaseless samples taken either on the whole line or on a discrete set with finite sampling density. We characterize all phase retrievable signals in a real-valued Shift-Invariant Space using their nonseparability. For nonseparable signals generated by some function with support length L, we show that they can be well approximated, up to a sign, from their noisy phaseless samples taken on a discrete set with sampling density 2 L − 1 . In this paper, we also propose an algorithm with linear computational complexity to reconstruct nonseparable signals in a Shift-Invariant Space from their phaseless samples corrupted by bounded noises.
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Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces
2017Co-Authors: Cheng Cheng, Jiang Junzheng, Sun QiyuAbstract:Sampling in Shift-Invariant Spaces is a realistic model for signals with smooth spectrum. In this paper, we consider phaseless sampling and reconstruction of real-valued signals in a Shift-Invariant Space from their magnitude measurements on the whole Euclidean Space and from their phaseless samples taken on a discrete set with finite sampling density. We introduce an undirected graph to a signal and use connectivity of the graph to characterize whether the signal can be determined, up to a sign, from its magnitude measurements on the whole Euclidean Space. Under the local complement property assumption on a Shift-Invariant Space, we find a discrete set with finite sampling density such that signals in the Shift-Invariant Space, that are determined from their magnitude measurements on the whole Euclidean Space, can be reconstructed in a stable way from their phaseless samples taken on that discrete set. In this paper, we also propose a reconstruction algorithm which provides a suboptimal approximation to the original signal when its noisy phaseless samples are available only. Finally, numerical simulations are performed to demonstrate the robust reconstruction of box spline signals from their noisy phaseless samples
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phase retrieval of real valued signals in a shift invariant Space
arXiv: Information Theory, 2016Co-Authors: Yang Chen, Cheng Cheng, Haichao WangAbstract:Phase retrieval arises in various fields of science and engineering and it is well studied in a finite-dimensional setting. In this paper, we consider an infinite-dimensional phase retrieval problem to reconstruct real-valued signals living in a Shift-Invariant Space from its phaseless samples taken either on the whole line or on a set with finite sampling rate. We find the equivalence between nonseparability of signals in a linear Space and its phase retrievability with phaseless samples taken on the whole line. For a spline signal of order $N$, we show that it can be well approximated, up to a sign, from its noisy phaseless samples taken on a set with sampling rate $2N-1$. We propose an algorithm to reconstruct nonseparable signals in a Shift-Invariant Space generated by a compactly supported continuous function. The proposed algorithm is robust against bounded sampling noise and it could be implemented in a distributed manner.
Wenchang Sun - One of the best experts on this subject based on the ideXlab platform.
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random phaseless sampling for causal signals in shift invariant Spaces a zero distribution perspective
IEEE Transactions on Signal Processing, 2020Co-Authors: Wenchang SunAbstract:We proved that the phaseless sampling (PLS) in the linear-phase modulated Shift-Invariant Space (SIS) $V(e^{{\bf i}\alpha \cdot }\varphi), \alpha \ne 0,$ is impossible even though the real-valued function $\varphi$ enjoys the full spark property (so does $e^{{\bf i}\alpha \cdot }\varphi$ ). Stated another way, the PLS in the complex-generated SISs is essentially different from that in the real-generated ones. Motivated by this, we first establish the condition on the complex-valued generator $\phi$ such that the PLS of nonseparable causal (NC) signals in $V(\phi)$ can be achieved by random sampling. The condition is established from the generalized Haar condition (GHC) perspective. Based on the proposed reconstruction approach, it is proved that if the GHC holds then with probability 1, the random sampling density (SD) $=3$ is sufficient for the PLS of NC signals in the complex-generated SISs. For the real-valued case we also prove that, if the GHC holds then with probability 1, the random SD $=2$ is sufficient for the PLS of real-valued NC signals in the real-generated SISs. For the local reconstruction of highly oscillatory signals such as chirps, a great number of deterministic samples are required. Compared with deterministic sampling, the proposed random approach enjoys not only the greater sampling flexibility but the much smaller number of samples. To verify our results, numerical simulations were conducted to reconstruct highly oscillatory NC signals in the chirp-modulated SISs.