Filtered Iterative Denoising for Linear Inverse Problems

Iterative denoising algorithms (IDAs) have been tremendously successful in a range of linear inverse problems arising in signal and image processing. The classic instance of this is the famous Iterative Soft-Thresholding Algorithm (ISTA), based on soft-thresholding of wavelet coefficients. More modern approaches to IDAs replace soft-thresholding with a black-box denoiser, such as BM3D or a learned deep neural network denoiser. These are often referred to as ``plug-and-play" (PnP) methods because, in principle, an off-the-shelf denoiser can be used for a variety of different inverse problems. The problem with PnP methods is that they may not provide the best solutions to a specific linear inverse problem; better solutions can often be obtained by a denoiser that is customized to the problem domain. A problem-specific denoiser, however, requires expensive re-engineering or re-learning which eliminates the simplicity and ease that makes PnP methods attractive in the first place. This paper proposes a new IDA that allows one to use a general, black-box denoiser more effectively via a simple linear filtering modification to the usual gradient update steps that accounts for the specific linear inverse problem. The proposed Filtered IDA (FIDA) is mathematically derived from the classical ISTA and wavelet denoising viewpoint. We show experimentally that FIDA can produce superior results compared to existing IDA methods with BM3D.