  Non-uniqueness and instability are characteristic features of image
reconstruction processes. As a result, it is necessary to develop
regularization methods that can be used to compute reliable approximate
solutions. A regularization method provides of a family of stable
reconstructions that converge to an exact solution of the noise-free problem as
the noise level tends to zero. The standard regularization technique is defined
by variational image reconstruction, which minimizes a data discrepancy
augmented by a regularizer. The actual numerical implementation makes use of
iterative methods, often involving proximal mappings of the regularizer. In
recent years, Plug-and-Play image reconstruction (PnP) has been developed as a
new powerful generalization of variational methods based on replacing proximal
mappings by more general image denoisers. While PnP iterations yield excellent
results, neither stability nor convergence in the sense of regularization has
been studied so far. In this work, we extend the idea of PnP by considering
families of PnP iterations, each being accompanied by its own denoiser. As our
main theoretical result, we show that such PnP reconstructions lead to stable
and convergent regularization methods. This shows for the first time that PnP
is mathematically equally justified for robust image reconstruction as
variational methods
