Sparse Reconstruction Using Hyperbolic Tangent as Smooth l1-Norm Approximation
department or school: Applied Science, Faculty of
department or school: Non UBC
department or school: Electrical and Computer Engineering, Department of
data provider: DSpace
identifier citation: Computation 11 (1): 7 (2023)
creator: Haider, Hassaan
creator: Shah, Jawad Ali
creator: Kadir, Kushsairy
creator: Khan, Najeeb
issued: 2024-12-18T19:29:22Z
issued: 2023-01-04
description: In the Compressed Sensing (CS) framework, the underdetermined system of linear equation (USLE) can have infinitely many possible solutions. However, we intend to find the sparsest possible solution, which is l₀-norm minimization. However, finding an l₀ norm solution out of infinitely many possible solutions is NP-hard problem that becomes non-convex optimization problem. It has been a practically proven fact that l₀ norm penalty can be adequately estimated by l₁ norm, which recasts a non-convex minimization problem to a convex problem. However, l₁ norm non-differentiable and gradient-based minimization algorithms are not applicable, due to this very reason there is a need to approximate l₁ norm by its smooth approximation. Iterative shrinkage algorithms provide an efficient method to numerically minimize l₁-regularized least square optimization problem. These algorithms are required to induce sparsity in their solutions to meet the CS recovery requirement. In this research article, we have developed a novel recovery method that uses hyperbolic tangent function to recover undersampled signal/images in CS framework. In our work, l₁ norm and soft thresholding are both approximated with the hyperbolic tangent functions. We have also proposed the criteria to tune optimization parameters to get optimal results. The error bounds for the proposed l₁ norm approximation are evaluated. To evaluate performance of our proposed method, we have utilized a dataset comprised of 1-D sparse signal, compressively sampled MR image and cardiac cine MRI. The MRI is an important imaging modality for assessing cardiac vascular function. It provides the ejection fraction and cardiac output of the heart. However, this advantage comes at the cost of a slow acquisition process. Hence, it is essential to speed up the acquisition process to take the full benefits of cardiac cine MRI. Numerical results based on performance metrics, such as Structural Similarity (SSIM), Peak Signal to Noise Ratio (…
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/89921
note: Citation: Haider, H.; Shah, J.A.;Kadir, K.; Khan, N. SparseReconstruction Using HyperbolicTangent as Smooth l1-NormApproximation. Computation 2023, 11,7. https://doi.org/10.3390/computation11010007Academic Editors: Anando Sen andDemos T. TsahalisReceived: 11 October 2022Revised: 29 November 2022Accepted: 16 December 2022Published: 4 January 2023Copyright: © 2023 by the authors.Licensee MDPI, Basel, Switzerland.This article is an open access articledistributed under the terms andconditions of the Creative CommonsAttribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).computationArticleSparse Reconstruction Using Hyperbolic Tangent as Smoothl1-Norm ApproximationHassaan Haider 1 , Jawad Ali Shah 1,*, Kushsairy Kadir 2 and Najeeb Khan 31 Department of Electrical and Computer Engineering, Faculty of Engineering and Technology,International Islamic University, Islamabad 44000, Pakistan2 Electrical Section, British Malaysian Institute, Universiti Kuala Lumpur, Gombak 53100, Malaysia3 Department of Electrical and Computer Engineering, University of British Columbia,Vancouver, BC V6T 1Z4, Canada* Correspondence: jawad.shah@iiu.edu.pk; Tel.: +92-333-5274448Abstract: In the Compressed Sensing (CS) framework, the underdetermined system of linear equation(USLE) can have infinitely many possible solutions. However, we intend to find the sparsest possiblesolution, which is l0-norm minimization. However, finding an l0 norm solution out of infinitely manypossible solutions is NP-hard problem that becomes non-convex optimization problem. It has been apractically proven fact that l0 norm penalty can be adequately estimated by l1 norm, which recasts anon-convex minimization problem to a convex problem. However, l1 norm non-differentiable andgradient-based minimization algorithms are not applicable, due to this very reason there is a needto approximate l1 norm by its smooth approximation. Iterative shrinkage algorithms provide anefficient method to numerically minimiz…
has type: Article
is shown at: 10.14288/1.0447544
language: eng
peer review status: Reviewed
provider: Vancouver : University of British Columbia Library
publisher: Multidisciplinary Digital Publishing Institute
publisher DOI: 10.3390/computation11010007
rights: CC BY 4.0
rights URI: https://creativecommons.org/licenses/by/4.0/
scholar level: Faculty
scholar level: Researcher
subject: compressed sensing
subject: holography cardiac cine MRI
subject: l1-norm smooth approximations
subject: hyperbolic tangent function
subject: soft thresholding
type: Text
identifier URI: http://hdl.handle.net/2429/89921