I Introduction
Consider the problem of estimating an unknown signal
from a set of noisy measurements measurements . This task is often formulated as an optimization problem(1) |
where is the data-fidelity term and is the regularizer promoting solutions with desirable properties. Some popular regularizers in imaging include nonnegativity, sparsity, and self-similarity.
Two common algorithms for solving optimization problem (1) are fast iterative shrinkage/thresholding algorithm (FISTA) [2] and alternating direction method of multipliers (ADMM) [3]. These algorithms are suitable for solving large-scale imaging problems due to their low-computational complexity and ability to handle the non-smoothness of . Both ISTA and ADMM have modular structures in the sense that the prior on the image is only imposed via the proximal operator defined as
(2) |
The mathematical equivalence of the proximal operator to regularized image denoising has recently inspired Venkatakrishnan et al. [4] to replace it with a more general denoising operator of controllable strength . The original formulation of this plug-and-play priors (PnP) framework relies on ADMM, but it has recently been shown that it can be equally effective when used with other proximal algorithms. For example,
(3a) | |||
(3b) |
where and lead to PnP-ISTA and PnP-FISTA, respectively, and is the step-size [5] .
In many applications, the data-fidelity term consists of a large number of component functions
(4) |
where each and depend only on a subset of the measurements . Hence, when is large, traditional batch PnP algorithms may become impractical in terms of speed or memory requirements. The central idea of the recent PnP-SGD method [1] is to approximate the full gradient in (3a) with an average of component gradients
(5) |
where
are independent random variables that are distributed uniformly over
. The minibatch size parameter controls the number of per-iteration gradient components.We denote the denoiser-gradient operator by
(6) |
and its set of fixed points by
(7) |
The convergence of both batch PnP-FISTA and online PnP-SGD was extensively discussed in [1]. In particular, when is convex and smooth, and is an averaged operator (as well as other mild assumptions), it is possible to theoretically establish that the iterates of PnP-ISTA (without acceleration) gets arbitrarily close to with rate . Additionally, it is possible to establish that, in expectation, the iterates of PnP-SGD can also be made arbitrarily close to by increasing the minibatch parameter . As illustrated in Fig. 1, this makes PnP-SGD useful and mathematically sound alternative to the traditional batch PnP algorithms.
References
- [1] Y. Sun, B. Wohlberg, and U. S. Kamilov, “An online plug-and-play algorithm for regularized image reconstruction,” arXiv:1809.04693 [cs.CV], 2018.
- [2] A. Beck and M. Teboulle, “Fast gradient-based algorithm for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process., 2009.
- [3] M. V. Afonso, J. M.Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process., 2010.
- [4] S. V. Venkatakrishnan, C. A. Bouman, and B. Wohlberg, “Plug-and-play priors for model based reconstruction,” in Proc. GlobalSIP, 2013.
- [5] U. S. Kamilov, H. Mansour, and B. Wohlberg, “A plug-and-play priors approach for solving nonlinear imaging inverse problems,” IEEE Signal. Proc. Let., 2017.
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