The main focus of this work is the reconstruction of the signals f and gi, i=1,…,N, from the knowledge of their sums hi=f+gi, under the assumption that f and the gi's can be sparsely represented with respect to two different dictionaries Af and Ag. This generalizes the well-known “morphological component analysis” to a multi-measurement setting. The main result of the paper states that f and the gi's can be uniquely and stably reconstructed by finding sparse representations of hi for every i with respect to the concatenated dictionary [Af,Ag], provided that enough incoherent measurements gi are available. The incoherence is measured in terms of their mutual disjoint sparsity. This method finds applications in the reconstruction procedures of several hybrid imaging inverse problems, where internal data are measured. These measurements usually consist of the main unknown multiplied by other unknown quantities, and so the disjoint sparsity approach can be directly applied. As an example, we show how to apply the method to the reconstruction in quantitative photoacoustic tomography, also in the case when the Grüneisen parameter, the optical absorption and the diffusion coefficient are all unknown.
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机译:这项工作的主要重点是在假设f和gi可以相对稀疏表示的前提下,根据它们的和hi = f + gi的知识,重构信号f和gi,i = 1,…,N。到两个不同的字典Af和Ag。这将众所周知的“形态成分分析”概括为一个多测量设置。本文的主要结果表明,只要有足够的非相干测量gi可用,就可以找到i相对于级联字典[Af,Ag]的稀疏表示,从而可以唯一且稳定地重建f和gi。不相干性是根据它们相互不相交的稀疏性来衡量的。这种方法在几个混合成像逆问题的重建过程中找到了应用,其中测量了内部数据。这些测量通常由主要未知数乘以其他未知数组成,因此可以直接应用不相交的稀疏性方法。例如,我们展示了如何将这种方法应用于定量光声层析成像中的重建,以及在Grüneisen参数,光吸收和扩散系数都未知的情况下。
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