  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.
