分段平滑信号恢复的估计和不确定性量化
Estimation and uncertainty quantification for piecewise smooth signal recovery
摘要：
针对高阶全变差（HOTV）稀疏先验下的线性反问题，提出了一种稀疏贝叶斯学习（SBL）算法。对于稀疏信号恢复的问题，SBL通常产生比最大后验估计更准确的估计，包括那些依赖于l1正则化的估计。此外，而不是一个单一的信号估计，SBL产生一个完整的后验密度估计，可用于不确定性量化。然而，SBL仅直接适用于具有直接稀疏性先验的问题，或者可以通过合成形成的问题。本文演示了如何通过综合来制定HOTV稀疏先验问题，然后开发了相应的贝叶斯学习方法。这扩展了贝叶斯学习可用的问题类别，包括，例如，从数据中恢复分段平滑函数或信号的逆问题。数值例子来证明这种新技术是如何有效地采用。
英文摘要：
This paper presents a sparse Bayesian learning (SBL) algorithm for 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 rely on l1 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 develops a corresponding Bayesian learning method. 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.
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