In many inverse problems, the unknown is composed of multiple components with different regularities, for example, in imaging problems, where the unknown can have both rough and smooth features. We investigate linear Bayesian inverse problems, where the unknown consists of two components: one smooth and one piecewise constant. We model the unknown as a sum of two components and assign individual priors on each component to impose the assumed behavior. We propose and compare two prior models: (i) a combination of a Haar wavelet-based Besov prior and a smoothing Besov prior, and (ii) a hierarchical Gaussian prior on the gradient coupled with a smoothing Besov prior. To achieve a balanced reconstruction, we place hyperpriors on the prior parameters and jointly infer both the components and the hyperparameters. We propose Gibbs sampling schemes for posterior inference in both prior models. We demonstrate the capabilities of our approach on 1D and 2D deconvolution problems, where the unknown consists of smooth parts with jumps. The numerical results indicate that our methods improve the reconstruction quality compared to single-prior approaches and that the prior parameters can be successfully estimated to yield a balanced decomposition.

翻译：在许多反问题中，未知量由具有不同正则性的多个分量组成，例如在成像问题中，未知量可能同时包含粗糙和平滑的特征。我们研究线性贝叶斯反问题，其中未知量包含两个分量：一个平滑分量和一个分段常数分量。我们将未知量建模为两个分量的和，并对每个分量分别指定先验以施加假设的特性。我们提出并比较两种先验模型：(i) 基于Haar小波的Besov先验与平滑Besov先验的组合；(ii) 梯度上的分层高斯先验与平滑Besov先验的耦合。为实现平衡的重建，我们在先验参数上设置超先验，并联合推断分量与超参数。针对两种先验模型，我们提出了用于后验推断的Gibbs采样方案。我们在未知量包含跳跃的平滑分量的1维和2维反卷积问题上展示了本方法的性能。数值结果表明，相较于单先验方法，我们的方法提升了重建质量，且先验参数能够被成功估计以产生平衡的分解。