Convergent regularization in inverse problems and linear plug-and-play
  denoisers
Andreas Hauptmann Subhadip Mukherjee Carola-Bibiane Schönlieb Ferdia Sherry
Plug-and-play (PnP) denoising is a popular iterative framework for solving
imaging inverse problems using off-the-shelf image denoisers. Their empirical
success has motivated a line of research that seeks to understand the
convergence of PnP iterates under various assumptions on the denoiser. While a
significant amount of research has gone into establishing the convergence of
the PnP iteration for different regularity conditions on the denoisers, not
much is known about the asymptotic properties of the converged solution as the
noise level in the measurement tends to zero, i.e., whether PnP methods are
provably convergent regularization schemes under reasonable assumptions on the
denoiser. This paper serves two purposes: first, we provide an overview of the
classical regularization theory in inverse problems and survey a few notable
recent data-driven methods that are provably convergent regularization schemes.
We then continue to discuss PnP algorithms and their established convergence
guarantees. Subsequently, we consider PnP algorithms with linear denoisers and
propose a novel spectral filtering technique to control the strength of
regularization arising from the denoiser. Further, by relating the implicit
regularization of the denoiser to an explicit regularization functional, we
rigorously show that PnP with linear denoisers leads to a convergent
regularization scheme. More specifically, we prove that in the limit as the
noise vanishes, the PnP reconstruction converges to the minimizer of a
regularization potential subject to the solution satisfying the noiseless
operator equation. The theoretical analysis is corroborated by numerical
experiments for the classical inverse problem of tomographic image
reconstruction.