Speckle or multiplicative noise is a critical issue in coherence-based imaging systems, such as synthetic aperture radar and optical coherence tomography. Existence of speckle noise considerably limits the applicability of such systems by degrading their performance. On the other hand, the sophistications that arise in the study of multiplicative noise have so far impeded theoretical analysis of such imaging systems. As a result, the current acquisition technology relies on heuristic solutions, such as oversampling the signal and converting the problem into a denoising problem with multiplicative noise. This paper attempts to bridge the gap between theory and practice by providing the first theoretical analysis of such systems. To achieve this goal the log-likelihood function corresponding to measurement systems with speckle noise is characterized. Then employing compression codes to model the source structure, for the case of under-sampled measurements, a compression-based maximum likelihood recovery method is proposed. The mean squared error (MSE) performance of the proposed method is characterized and is shown to scale as $O\left({\sqrt {\frac{k \log n }{ m}}}\right)$ , where $k$ , $m$ and $n$ denote the intrinsic dimension of the signal class according to the compression code, the number of observations, and the ambient dimension of the signal, respectively. This result, while in contrast to imaging systems with additive noise in which MSE scales as $O\left({{\frac{k \log n }{ m}}}\right)$ , suggests that if the signal class is structured (i.e., $k \ll n$ ), accurate recovery of a signal from under-determined measurements is still feasible, even in the presence of speckle noise. Simulation results are presented that suggest image recovery under multiplicative noise is inherently more challenging than additive noise, and that the derived theoretical results are sharp.

中文翻译：

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存在散斑噪声时的压缩感知

散斑或相乘噪声是基于相干的成像系统中的一个关键问题，例如合成孔径雷达和光学相干断层扫描。散斑噪声的存在通过降低它们的性能而极大地限制了这些系统的适用性。另一方面，乘性噪声研究中出现的复杂性迄今为止阻碍了对此类成像系统的理论分析。因此，当前的采集技术依赖于启发式解决方案，例如对信号进行过采样并将问题转换为具有乘性噪声的去噪问题。本文试图通过对此类系统进行首次理论分析来弥合理论与实践之间的差距。为了实现这一目标，对具有散斑噪声的测量系统对应的对数似然函数进行了表征。然后采用压缩码对源结构进行建模，针对欠采样测量的情况，提出了一种基于压缩的最大似然恢复方法。所提出方法的均方误差 (MSE) 性能被表征并显示为 $O\left({\sqrt {\frac{k \log n }{ m}}}​​\right)$ ， 在哪里 $k$ , $m$和 $n$分别根据压缩码、观察次数和信号的环境维度表示信号类的内在维度。该结果与具有加性噪声的成像系统相比，其中 MSE 缩放为 $O\left({{\frac{k \log n }{ m}}}​​\right)$ ，表明如果信号类是结构化的（即， $k \ll n$ )，即使在存在散斑噪声的情况下，从未确定的测量值中准确恢复信号仍然是可行的。仿真结果表明，乘性噪声下的图像恢复本质上比加性噪声更具挑战性，并且推导出的理论结果很清晰。