Ill-posed linear inverse problems appear in many image processing applications, such as deblurring, superresolution and compressed sensing. Many restoration strategies involve minimizing a cost function, which is composed of fidelity and prior terms, balanced by a regularization parameter. While a vast amount of research has been focused on different prior models, the fidelity term is almost always chosen to be the least squares (LS) objective, that encourages fitting the linearly transformed optimization variable to the observations. In this paper, we examine a different fidelity term, which has been implicitly used by the recently proposed iterative denoising and backward projections (IDBP) framework. This term encourages agreement between the projection of the optimization variable onto the row space of the linear operator and the pseudoinverse of the linear operator ("back-projection") applied on the observations. We analytically examine the difference between the two fidelity terms for Tikhonov regularization and identify cases (such as a badly conditioned linear operator) where the new term has an advantage over the standard LS one. Moreover, we demonstrate empirically that the behavior of the two induced cost functions for sophisticated convex and non-convex priors, such as total-variation, BM3D, and deep generative models, correlates with the obtained theoretical analysis.

中文翻译：

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不适定线性反问题的基于反投影的保真项。

不适定的线性逆问题出现在许多图像处理应用中，例如去模糊，超分辨率和压缩感测。许多恢复策略涉及使成本函数最小化，该成本函数由保真度和先验项组成，并由正则化参数进行平衡。尽管大量研究集中在不同的先前模型上，但保真度术语几乎总是被选为最小二乘（LS）的目标，这鼓励将线性变换的优化变量拟合到观测值。在本文中，我们研究了一个不同的保真度术语，该术语已被最近提出的迭代去噪和反向投影（IDBP）框架隐式使用。该术语鼓励优化变量在线性算子的行空间上的投影与应用于观测值的线性算子的伪逆（“反投影”）之间的一致。我们分析地检查了两个保真项之间的差异，以进行Tikhonov正则化，并确定了新项比标准LS项具有优势的情况（例如条件差的线性算子）。此外，我们从经验上证明，对于复杂的凸和非凸先验而言，两个诱导成本函数的行为（例如总变异，BM3D和深生成模型）与所获得的理论分析相关。我们分析地检查了两个保真项之间的差异，以进行Tikhonov正则化，并确定了新项比标准LS项具有优势的情况（例如条件差的线性算子）。此外，我们从经验上证明，对于复杂的凸和非凸先验而言，两个诱导成本函数的行为（例如总变异，BM3D和深生成模型）与所获得的理论分析相关。我们分析地检查了两个保真项之间的差异，以进行Tikhonov正则化，并确定了新项比标准LS项具有优势的情况（例如条件差的线性算子）。此外，我们从经验上证明，对于复杂的凸和非凸先验而言，两个诱导成本函数的行为（例如总变异，BM3D和深生成模型）与所获得的理论分析相关。