Sparsity in Bayesian Signal Estimation

Wickramasingha, Ishan; Sobhy, Michael; Sherif, Sherif S.

doi:10.5772/intechopen.70529

Cited by 3 publications

(3 citation statements)

References 33 publications

(45 reference statements)

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“…A sparse multilinear representation of equation 2.3 could be obtained by rewriting it as an L p minimization problem (Wickramasingha et al, 2017) x = arg min…”

Section: Sparse Multilinear Least-squares Problemmentioning

confidence: 99%

“…Sparse signal representations have gained much interest recently in both signal processing and statistics communities. A sparse signal representation usually results in simpler and faster processing, in addition to lower memory storage requirement for fewer coefficients (Mallat, 2009;Wickramasingha, Sobhy, & Sherif, 2017). However, finding optimal sparse representations for different signals is not a trivial task (Mallat, 2009).…”

Section: Introductionmentioning

confidence: 99%

“…However, finding optimal sparse representations for different signals is not a trivial task (Mallat, 2009). Therefore, redundant signal representations using overcomplete dictionaries have been introduced to facilitate finding more sparse representations for different signals (Mallat & Zhang, 1993;Mallat, 2009;Pati, Rezaiifar, & Krishnaprasad, 1993;Wickramasingha et al, 2017). Undercomplete dictionaries could also be used to obtain approximate signal representations (Kreutz-Delgado et al, 2003;Malioutov, Çetin, & Willsky, 2005;Tosic & Frossard, 2011).…”

Section: Introductionmentioning

confidence: 99%

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Tensor Least Angle Regression for Sparse Representations of Multidimensional Signals

et al. 2020

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Sparse signal representations have gained much interest recently in both signal processing and statistical communities. Compared to orthogonal matching pursuit (OMP) and basis pursuit, which solve the [Formula: see text] and [Formula: see text] constrained sparse least-squares problems, respectively, least angle regression (LARS) is a computationally efficient method to solve both problems for all critical values of the regularization parameter [Formula: see text]. However, all of these methods are not suitable for solving large multidimensional sparse least-squares problems, as they would require extensive computational power and memory. An earlier generalization of OMP, known as Kronecker-OMP, was developed to solve the [Formula: see text] problem for large multidimensional sparse least-squares problems. However, its memory usage and computation time increase quickly with the number of problem dimensions and iterations. In this letter, we develop a generalization of LARS, tensor least angle regression (T-LARS) that could efficiently solve either large [Formula: see text] or large [Formula: see text] constrained multidimensional, sparse, least-squares problems (underdetermined or overdetermined) for all critical values of the regularization parameter [Formula: see text] and with lower computational complexity and memory usage than Kronecker-OMP. To demonstrate the validity and performance of our T-LARS algorithm, we used it to successfully obtain different sparse representations of two relatively large 3D brain images, using fixed and learned separable overcomplete dictionaries, by solving both [Formula: see text] and [Formula: see text] constrained sparse least-squares problems. Our numerical experiments demonstrate that our T-LARS algorithm is significantly faster (46 to 70 times) than Kronecker-OMP in obtaining [Formula: see text]-sparse solutions for multilinear leastsquares problems. However, the [Formula: see text]-sparse solutions obtained using Kronecker-OMP always have a slightly lower residual error (1.55% to 2.25%) than ones obtained by T-LARS. Therefore, T-LARS could be an important tool for numerous multidimensional biomedical signal processing applications.

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“…A sparse multilinear representation of equation 2.3 could be obtained by rewriting it as an L p minimization problem (Wickramasingha et al, 2017) x = arg min…”

Section: Sparse Multilinear Least-squares Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Tensor Least Angle Regression for Sparse Representations of Multidimensional Signals

et al. 2020

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Weighted Tensor Least Angle Regression for Solving Sparse Weighted Multilinear Least Squares Problems

Wickramasingha,

Mezgebo,

Sherif

2024

New Approaches for Multidimensional Signal Processing

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Image Denoising Using Sparse Representation and Principal Component Analysis

Abedini

Haddad²,

Masouleh³

et al. 2021

Int. J. Image Grap.

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This study proposes an image denoising algorithm based on sparse representation and Principal Component Analysis (PCA). The proposed algorithm includes the following steps. First, the noisy image is divided into overlapped [Formula: see text] blocks. Second, the discrete cosine transform is applied as a dictionary for the sparse representation of the vectors created by the overlapped blocks. To calculate the sparse vector, the orthogonal matching pursuit algorithm is used. Then, the dictionary is updated by means of the PCA algorithm to achieve the sparsest representation of vectors. Since the signal energy, unlike the noise energy, is concentrated on a small dataset by transforming into the PCA domain, the signal and noise can be well distinguished. The proposed algorithm was implemented in a MATLAB environment and its performance was evaluated on some standard grayscale images under different levels of standard deviations of white Gaussian noise by means of peak signal-to-noise ratio, structural similarity indexes, and visual effects. The experimental results demonstrate that the proposed denoising algorithm achieves significant improvement compared to dual-tree complex discrete wavelet transform and K-singular value decomposition image denoising methods. It also obtains competitive results with the block-matching and 3D filtering method, which is the current state-of-the-art for image denoising.

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Sparsity in Bayesian Signal Estimation

Cited by 3 publications

References 33 publications

Tensor Least Angle Regression for Sparse Representations of Multidimensional Signals

Tensor Least Angle Regression for Sparse Representations of Multidimensional Signals

Weighted Tensor Least Angle Regression for Solving Sparse Weighted Multilinear Least Squares Problems

Image Denoising Using Sparse Representation and Principal Component Analysis

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