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 - <p style="text-align: justify;">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&nbsp;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&nbsp;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.</p>