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Sparse Sampling in Microscopy
Published in Jeffrey P. Simmons, Lawrence F. Drummy, Charles A. Bouman, Marc De Graef, Statistical Methods for Materials Science, 2019
Kurt Larson, Hyrum Anderson, Jason Wheeler
where ‖x‖1=∑i=1N|xi|, and the constraint is replaced by ‖Ax−y‖22≤ε2 for noisy measurements. Myriad efficient methods exist to solve basis pursuit in polynomial time.
Communication Optimization and Edge Analytics for Smart Water Grids
Published in Panagiotis Tsakalides, Athanasia Panousopoulou, Grigorios Tsagkatakis, Luis Montestruque, Smart Water Grids, 2018
Sokratis Kartakis, Julie A. McCann
In the literature, two main approaches have been used to solve the (3.3) optimization problem. In the first approach, the so-called Basis Pursuit (BP) techniques [13,14,16] transform the problem to Linear Program (LP) [9]. In the second approach, the Matching Pursuit (MP) greedy methods [34,52] use an iterative way to solve the optimization problem.In this study, the NESTA algorithm is used, which was shown to achieve a very good trade-off between reconstruction accuracy and computation time [7]. We emphasize that the scope of this section is to illustrate the efficiency of the CS framework in achieving highly compact, yet very accurate, representations of real high-resolution sensor data recorded in water distribution networks. As such, an exhaustive comparison with the various reconstruction algorithms for finding the optimal solution is left as a separate study.
Compressed Sensing: From Theory to Praxis
Published in C.H. Chen, Compressive Sensing of Earth Observations, 2017
Axel Flinth, Ali Hashemi, Gitta Kutyniok
The algorithm, or rather strategy, of solving Equation 1.2 for recovering sparse solutions of Ax= b is known as Basis Pursuit. Researchers have known for several decades that the ℓ1-norm promotes sparsity. But it was as late as 2001 that the Basis Pursuit algorithm was formulated [21]. The Basis Pursuit algorithm can be formulated as a linear program, which in particular proves that it can be solved in polynomial time. It is however by no means clear that the solution of the problem defined by Equation 1.2 is equal to the sparse signal of interest x0. Much of the theory in fact revolves around determining for which type of matrices this is the case. We will discuss this in detail in Section 1.3.
Joint Dual-Structural Constrained and Non-negative Analysis Representation Learning for Pattern Classification
Published in Applied Artificial Intelligence, 2023
Kun Jiang, Lei Zhu, Qindong Sun
The solution of SDL model includes two basic tasks, i.e., sparse approximation and dictionary learning. On one hand, some algorithms, such as matching pursuit (M-P) (Davis, Mallat, and Avellaneda 1997; Mallat and Zhang 1993), basis pursuit (BP) (Chen, Donoho, and Saunders 2001) and shrinkage method (Hyvärinen 1999), have been well developed to find a sparse solution. On the other hand, dictionary learning is dedicated to search an optimal signal space to support the attribution of sparse vector under a certain measure. There exist a variety of numerical algorithms presented to achieve this objective, e.g., method of optimal directions (MOD) (Engan, Aase, and Hakon Husoy 1999) and K-singular value decomposition (K-SVD) (Aharon, Elad, and Bruckstein 2006). K-SVD method learns an overcomplete dictionary from training samples by updating K dictionary atoms and representation coefficients iteratively with the SVD algorithm under a predefined sparse threshold for non-zero elements in each coefficient (Aharon, Elad, and Bruckstein 2006).
Research on compressive sensing of strong earthquake signals for earthquake early warning
Published in Geomatics, Natural Hazards and Risk, 2021
Jiening Xia, Yuanxiang Li, Yuxiu Cheng, Juan Li, Shasha Tian
The high-resolution original signal x can be recovered from the observed value y in formula (2) according to compressive sensing theory, that is, signal reconstruction. The reconstruction algorithm is of great significance for the accurate reconstruction of the compressed signal and the accuracy verification during the sampling process. In 1999, Donoho et al. has proposed the basis pursuit (BP) algorithm (Chen et al. 2001), the BP algorithm uses the 1-norm of the representation coefficient to measure its sparseness, and describe the sparse representation signal problem as a type of constrained optimization problem, i.e., formula (4) by minimizing the 1-norm of the representation coefficient. And finally solve the problem by solving the linear programming problem (Do et al. 2008; Efron et al. 2004; Rosasco et al. 2009).
Power Quality Disturbances Classification Using Compressive Sensing and Maximum Likelihood
Published in IETE Technical Review, 2018
Hui Liu, Fida Hussain, Yue Shen
CS [31–33] is a novel signal and image processing theory for acquiring and reconstructing a sparse signal from a small set of non-adaptive, linear measurements. Assume be a signal, given an orthonormal basis matrix whose basis are columns, can be represented as , where is an column sparse vector with only non-zero elements, that is . The measurement process can be calculated as where is a column vector of the compressive measurement, is an () measurement matrix and incoherent with , and sensing matrix should satisfy the Restricted Isometry Property. Many algorithms have been used for the recovery of signals such as basis pursuit, matching pursuit, and orthogonal matching pursuit.