基于部分误差支持和信号值知识的正则化修正BPDN算法
Regularized Modified BPDN for Noisy Sparse Reconstruction with Partial Erroneous Support and Signal Value Knowledge
摘要：
我们研究的问题稀疏重建噪声欠采样测量时，以下两件事是可用的。(1)我们得到的是信号支持度的部分知识和部分错误知识，用$T$表示。(2)我们还给出了$T$上的信号值的错误估计，由$(\hatμ)_T$表示。在实践中，这两种方法都可以从现有的现有知识中获得。可替换地，在递归重建应用中，如实时动态MRI，可以使用来自先前时刻的支持估计和信号值估计作为$T$和$(\hatμ)_T$。在这项工作中，我们介绍了正则化的修改BPDN（reg-mod-BPDN），并获得其重建误差的可计算范围。Reg-mod-BPDN尝试找到集合$T$之外最稀疏的信号，同时与$T$上的$(\hatμ)_T$“足够接近”并满足数据约束。相应的结果修改后的BPDN和BPDN遵循的直接推论。第二个关键的贡献是一种方法来获得可计算的错误界，没有任何充分条件。这使得比较各种方法的界限变得容易。还提供了与许多现有方法的经验重建误差比较。
英文摘要：
We study the problem of sparse reconstruction from noisy undersampled measurements when the following two things are available. (1) We are given partial, and partly erroneous, knowledge of the signal's support, denoted by $T$. (2) We are also given an erroneous estimate of the signal values on $T$, denoted by $(\hatμ)_T$. In practice, both these may be available from available prior knowledge. Alternatively, in recursive reconstruction applications, like real-time dynamic MRI, one can use the support estimate and the signal value estimate from the previous time instant as $T$ and $(\hatμ)_T$. In this work, we introduce regularized modified-BPDN (reg-mod-BPDN) and obtain computable bounds on its reconstruction error. Reg-mod-BPDN tries to find the signal that is sparsest outside the set $T$, while being "close enough" to $(\hatμ)_T$ on $T$ and while satisfying the data constraint. Corresponding results for modified-BPDN and BPDN follow as direct corollaries. A second key contribution is an approach to obtain computable error bounds that hold without any sufficient conditions. This makes it easy to compare the bounds for the various approaches. Empirical reconstruction error comparisons with many existing approaches are also provided.
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