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Volume 41, Issue 2
Estimation and Uncertainty Quantification for Piecewise Smooth Signal Recovery

V. Churchill & A. Gelb

DOI: 10.4208/jcm.2110-m2021-0157

J. Comp. Math., 41 (2023), pp. 246-262.

Published online: 2022-11

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  • Abstract

This paper presents an application of the sparse Bayesian learning (SBL) algorithm to linear inverse problems with a high order total variation (HOTV) sparsity prior. For the problem of sparse signal recovery, SBL often produces more accurate estimates than maximum a posteriori estimates, including those that use $\ell_1$ regularization. Moreover, rather than a single signal estimate, SBL yields a full posterior density estimate which can be used for uncertainty quantification. However, SBL is only immediately applicable to problems having a direct sparsity prior, or to those that can be formed via synthesis. This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis, and then utilizes SBL. This expands the class of problems available to Bayesian learning to include, e.g., inverse problems dealing with the recovery of piecewise smooth functions or signals from data. Numerical examples are provided to demonstrate how this new technique is effectively employed.

  • Keywords

High order total variation regularization, Sparse Bayesian learning, Analysis and synthesis, Piecewise smooth function recovery.

  • AMS Subject Headings

62F15, 65C60, 65F22, 94A12

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

churchill.77@osu.edu (V. Churchill)

Anne.E.Gelb@dartmouth.edu (A. Gelb)

  • BibTex
  • RIS
  • TXT
@Article{JCM-41-246, author = {Churchill , V. and Gelb , A.}, title = {Estimation and Uncertainty Quantification for Piecewise Smooth Signal Recovery}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {41}, number = {2}, pages = {246--262}, abstract = {

This paper presents an application of the sparse Bayesian learning (SBL) algorithm to linear inverse problems with a high order total variation (HOTV) sparsity prior. For the problem of sparse signal recovery, SBL often produces more accurate estimates than maximum a posteriori estimates, including those that use $\ell_1$ regularization. Moreover, rather than a single signal estimate, SBL yields a full posterior density estimate which can be used for uncertainty quantification. However, SBL is only immediately applicable to problems having a direct sparsity prior, or to those that can be formed via synthesis. This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis, and then utilizes SBL. This expands the class of problems available to Bayesian learning to include, e.g., inverse problems dealing with the recovery of piecewise smooth functions or signals from data. Numerical examples are provided to demonstrate how this new technique is effectively employed.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2110-m2021-0157}, url = {http://global-sci.org/intro/article_detail/jcm/21179.html} }
TY - JOUR T1 - Estimation and Uncertainty Quantification for Piecewise Smooth Signal Recovery AU - Churchill , V. AU - Gelb , A. JO - Journal of Computational Mathematics VL - 2 SP - 246 EP - 262 PY - 2022 DA - 2022/11 SN - 41 DO - http://doi.org/10.4208/jcm.2110-m2021-0157 UR - https://global-sci.org/intro/article_detail/jcm/21179.html KW - High order total variation regularization, Sparse Bayesian learning, Analysis and synthesis, Piecewise smooth function recovery. AB -

This paper presents an application of the sparse Bayesian learning (SBL) algorithm to linear inverse problems with a high order total variation (HOTV) sparsity prior. For the problem of sparse signal recovery, SBL often produces more accurate estimates than maximum a posteriori estimates, including those that use $\ell_1$ regularization. Moreover, rather than a single signal estimate, SBL yields a full posterior density estimate which can be used for uncertainty quantification. However, SBL is only immediately applicable to problems having a direct sparsity prior, or to those that can be formed via synthesis. This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis, and then utilizes SBL. This expands the class of problems available to Bayesian learning to include, e.g., inverse problems dealing with the recovery of piecewise smooth functions or signals from data. Numerical examples are provided to demonstrate how this new technique is effectively employed.

Churchill , V. and Gelb , A.. (2022). Estimation and Uncertainty Quantification for Piecewise Smooth Signal Recovery. Journal of Computational Mathematics. 41 (2). 246-262. doi:10.4208/jcm.2110-m2021-0157
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