Block Coordinate Plug-and-Play Methods for Blind Inverse Problems

Summary

The paper proposes a block-coordinate optimization strategy for plug-and-play (PnP) methods to solve blind inverse problems in imaging. Empirical validation is performed with blind deblurring and parallel CS-MRI. Theoretical study on the convergence of the proposed BC-PnP was extended from the theoretical result of PnP with MMSE denoisers.

Strengths

1. To the best of my knowledge, this is one of the first works to extend PnP methods to blind inverse problems 2. Theory is sound and proves proper convergence properties of the proposed method. 3. The method seems to be robust and generally applicable to inverse problems arising in imaging, as validated in the experiments.

Weaknesses

1. I do not find much novelty in the theory of the paper. While the components are all sound, both components of the extension (extending to block coordinate update and extending to denoisers that are optimal only up to a constant error) seems to be quite trivial. I would appreciate it if the authors could elaborate on this point. 2. BC-PnP seems to rely quite heavily on initialization for the imaging operators (i.e. sensitivity maps, blur kernels) that are close to ground truth. Such initialization strategy is not always possible, and for many cases even when it is possible, is often computationally demanding. Does BC-PnP converge properly also for cases when there is no such initialization strategy? Adding on this point, it might not be fair to compare other blind inverse problem solver that do not leverage any initialization strategy. 3. For blind deblurring, it seems that the authors only tested on Gaussian blur kernels, which are known to be relatively easy to fit. Does the method scale to more complex blur kernels, such as in [1,2]? **Referneces** [1] Levin, Anat, et al. "Understanding and evaluating blind deconvolution algorithms." 2009 IEEE conference on computer vision and pattern recognition. IEEE, 2009. [2] Chung, Hyungjin, et al. "Parallel diffusion models of operator and image for blind inverse problems." Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2023.

Questions

Please see weaknesses

Rating

4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

3 good

Presentation

2 fair

Contribution

2 fair

Limitations

Yes.

Summary

This paper presents a new plug-and-play algorithm for solving blind inverse problems, i.e., where the forward sensing operator is not fully known. The method leverages pretrained denoisers for both the signals and the unknown parameters of the sensing model. A theoretical analysis of the proposed algorithm is presented, showing that under certain conditions on the denoisers and data fidelity functions, the algorithm is guaranteed to converge to a local minimizer.

Strengths

- The paper presents a novel plug-and-play algorithm for blind inverse problems, with some theoretical guarantees under relatively mild assumptions. - The experimental results show a good performance with respect to other competing methods in two different blind inverse problems.

Weaknesses

I find it a bit surprising that the PnP method outperfoms other unrolled algorithms trained for a specific inverse problem. Even the authors express that "While denoisers provide a convenient, principled, and flexible mechanism to specify priors, they are inherently self-supervised and their empirical performance can thus be suboptimal compared to priors trained in a supervised fashion for a specific inverse problem" I wonder if the comparison presented in the paper is fair, do the competing methods use architectures that have a similar expressive power than the DRUNet denoiser used for PnP? Moreover, there exist end-to-end architectures that can estimate both the forward operator parameters and the underlying image (e.g. "Deep algorithm unrolling for blind image deblurring" by Li et al.), which haven't been evaluated here.

Questions

- What is the performance of the proposed method on the blind deblurring problem with non-isotropic kernels? It would be good to show an example in this setting.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

The paper discusses some of the limitations of the proposed method. However, there is no discussion related to the inherent large computational complexity of PnP methods, that require many more iterations than unrolled networks to reach convergence.

Summary

The paper generalizes the plug-and-play framework from single-block to a multi-block case by embedding unknown inverse operators. It provides convergence analysis for the proposed method and conducted various experiments on two imaging applications, i.e., parallel MRI and blind image deblurring. Numerical results are sufficiently convincing to show that the proposed method works better than the other comparable PnP methods in the blind measurement operator case.

Strengths

The paper reads well and is well-organized from theoretical discussions to numerical experiments. The proposed method is original with clear motivation of unknown inverse operators from the realistic setting. Thus, it may also shed some light on other imaging applications.

Weaknesses

1. The computational cost of this framework is not analyzed or even mentioned. Likewise, the running times for all the experimental results could be listed and compared. Some other end-to-end methods may also be tested. 2. Some minor notational confusion exists, e.g., lines 198-199. 3. Theorems 1-2 require square-summable condition for the error sequences to ensure the convergence of gradients. Further discussions may be added about how this condition can be satisfied or checked in practice.

Questions

In all the numerical experiments, are the assumptions 1-5 and conditions in Theorems 1-2 verified or checked? If yes, the details could be included in the supplement.

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

The authors have listed the limitations and provided some potential solutions in future works.

Summary

Develops a plug-and-play technique for blind (unknown forward model) imaging inverse problems. Key idea is to use two denoising algorithms: One as a prior on the image and one as a prior model on the measurement operator (e.g., blur kernel). Also analyzes the convergence of the algorithm. Successfully applies (in simulation) the proposed algorithm to MRI with unknown coil sensitive maps and deblurring with unknown blur kernel.

Strengths

Investigates important and understudied problem Proposed technique is well-justified and seemingly effective I like the paper's approach to framing the blind image reconstruction problem (equation (10)). Easy to read Proposed method seems generally applicable to lots of problems and opens up and opportunity for lots of follow-up work Code will be shared publicly

Weaknesses

Lacks comparisons with closest competitors. Particulary [60]. Missing related work: There's a line of work on PnP for holography with unknown phase errors (e.g., [A]) that may be worth mentioning. [A] Pellizzari, Casey J., Mark F. Spencer, and Charles A. Bouman. "Coherent plug-and-play: digital holographic imaging through atmospheric turbulence using model-based iterative reconstruction and convolutional neural networks." IEEE Transactions on Computational Imaging 6 (2020): 1607-1621.

Questions

Diffusion models are essentially denoisers by a different name, therefore a comparison with [60] seems obligatory. They are essentially solving the same problem (though the current method has more theory). How do they compare?

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

5: You are absolutely certain about your assessment. You are very familiar with the related work and checked the math/other details carefully.

Soundness

4 excellent

Presentation

4 excellent

Contribution

4 excellent

Limitations

Adequately discussed

Discussion Period Response to All Reviewers and Area Chairs

Thank you all again for reviewing our work! An additional thanks to the reviewers who have already taken our rebuttal into consideration and the area chairs for managing the review of our paper. As we near the end of the discussion period, let us know if there is anything else we can do to improve your evaluation of our work.