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Register a new userGeneralized Alternating Projection Based Total Variation Minimization for Compressive Sensing
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Added by
Xin Yuan
Publication date
2015
fields
Informatics Engineering
and research's language is
English
Authors
Xin Yuan
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We consider the total variation (TV) minimization problem used for compressive sensing and solve it using the generalized alternating projection (GAP) algorithm. Extensive results demonstrate the high performance of proposed algorithm on compressive sensing, including two dimensional images, hyperspectral images and videos. We further derive the Alternating Direction Method of Multipliers (ADMM) framework with TV minimization for video and hyperspectral image compressive sensing under the CACTI and CASSI framework, respectively. Connections between GAP and ADMM are also provided.
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This paper considers solving the unconstrained $ell_q$-norm ($0leq q<1$) regularized least squares ($ell_q$-LS) problem for recovering sparse signals in compressive sensing. We propose two highly efficient first-order algorithms via incorporating the proximity operator for nonconvex $ell_q$-norm functions into the fast iterative shrinkage/thresholding (FISTA) and the alternative direction method of multipliers (ADMM) frameworks, respectively. Furthermore, in solving the nonconvex $ell_q$-LS problem, a sequential minimization strategy is adopted in the new algorithms to gain better global convergence performance. Unlike most existing $ell_q$-minimization algorithms, the new algorithms solve the $ell_q$-minimization problem without smoothing (approximating) the $ell_q$-norm. Meanwhile, the new algorithms scale well for large-scale problems, as often encountered in image processing. We show that the proposed algorithms are the fastest methods in solving the nonconvex $ell_q$-minimization problem, while offering competent performance in recovering sparse signals and compressible images compared with several state-of-the-art algorithms.
Distributed Compressive Sensing (DCS) improves the signal recovery performance of multi signal ensembles by exploiting both intra- and inter-signal correlation and sparsity structure. However, the existing DCS was proposed for a very limited ensemble of signals that has single common information cite{Baron:2009vd}. In this paper, we propose a generalized DCS (GDCS) which can improve sparse signal detection performance given arbitrary types of common information which are classified into not just full common information but also a variety of partial common information. The theoretical bound on the required number of measurements using the GDCS is obtained. Unfortunately, the GDCS may require much a priori-knowledge on various inter common information of ensemble of signals to enhance the performance over the existing DCS. To deal with this problem, we propose a novel algorithm that can search for the correlation structure among the signals, with which the proposed GDCS improves detection performance even without a priori-knowledge on correlation structure for the case of arbitrarily correlated multi signal ensembles.
This work considers the use of Total variation (TV) minimization in the recovery of a given gradient sparse vector from Gaussian linear measurements. It has been shown in recent studies that there exist a sharp phase transition behavior in TV minimization in asymptotic regimes. The phase transition curve specifies the boundary of success and failure of TV minimization for large number of measurements. It is a challenging task to obtain a theoretical bound that reflects this curve. In this work, we present a novel upper-bound that suitably approximates this curve and is asymptotically sharp. Numerical results show that our bound is closer to the empirical TV phase transition curve than the previously known bound obtained by Kabanava.
Quantized compressive sensing (QCS) deals with the problem of representing compressive signal measurements with finite precision representation, i.e., a mandatory process in any practical sensor design. To characterize the signal reconstruction quality in this framework, most of the existing theoretical analyses lie heavily on the quantization of sub-Gaussian random projections (e.g., Gaussian or Bernoulli). We show here that a simple uniform scalar quantizer is compatible with a large class of random sensing matrices known to respect, with high probability, the restricted isometry property (RIP). Critically, this compatibility arises from the addition of a uniform random vector, or dithering, to the linear signal observations before quantization. In this setting, we prove the existence of (at least) one signal reconstruction method, i.e., the projected back projection (PBP), whose reconstruction error decays when the number of quantized measurements increases. This holds with high probability in the estimation of sparse signals and low-rank matrices. We validate numerically the predicted error decay as the number of measurements increases.
Characterizing the phase transitions of convex optimizations in recovering structured signals or data is of central importance in compressed sensing, machine learning and statistics. The phase transitions of many convex optimization signal recovery methods such as $ell_1$ minimization and nuclear norm minimization are well understood through recent years research. However, rigorously characterizing the phase transition of total variation (TV) minimization in recovering sparse-gradient signal is still open. In this paper, we fully characterize the phase transition curve of the TV minimization. Our proof builds on Donoho, Johnstone and Montanaris conjectured phase transition curve for the TV approximate message passing algorithm (AMP), together with the linkage between the minmax Mean Square Error of a denoising problem and the high-dimensional convex geometry for TV minimization.
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