Ziqi Qin (秦子祺)

Plug-and-Play (PnP) algorithms are a class of iterative methods for inverse imaging. Within an optimization algorithm, they combine a flexible fidelity term, encoding the forward operator, and a pretrained image denoiser, in order to deal with more severe corruptions such as blurring or downsampling when reconstructing an image. This work studies provably convergent PnP methods for mixed-noise forward processes by using the infimal convolution as a fidelity term, providing a statistical interpretation as a joint maximum a-posteriori estimator over both noises, and preserving the Bayesian MAP interpretation of PnP methods. Independently, we extend the PnP formulation to multiple prior terms using the Davis–Yin three-operator splitting. This extension can be combined with either standard fidelities or the proposed mixed-noise infimal-convolution fidelities. We verify that these generalized PnP methods are convergent under standard Kurdyka–Lojasiewicz conditions. Numerical experiments on Laplace-Gaussian and Poisson–Gaussian noise demonstrate stable convergence of single-prior and multiple-prior PnP methods, and divergence under fidelity mismatch. Furthermore, PnP with infimal convolution fidelities are able to scale to noise up to 38% standard deviation, with multiple priors reaching different stationary points that qualitatively preserve more textural properties.

Over the past decades, the concept "partial smoothness" has been playing as a powerful tool in several fields involving nonsmooth analysis, such as nonsmooth optimization, inverse problems and operation research, etc. The essence of partial smoothness is that it builds an elegant connection between the optimization variable and the objective function value through the subdifferential. Identifiability is the most appealing property of partial smoothness, as locally it allows us to conduct much finer or even sharp analysis, such as linear convergence or sensitivity analysis. However, currently the identifiability relies on non-degeneracy condition and exact dual convergence, which limits the potential application of partial smoothness. In this paper, we provide an alternative characterization of partial smoothness through only subdifferentials. This new perspective enables us to establish stronger identification results, explain identification under degeneracy and non-vanishing error. Moreover, we can generalize this new characterization to set-valued operators, and provide a complement definition of partly smooth operator proposed in [14].