[Paper Review] Robust Blind Deconvolution via Mirror Descent

Sathya N. Ravi, Ronak Mehta+1 more|arXiv (Cornell University)|Mar 21, 2018

Sparse and Compressive Sensing Techniques38 references3 citations

TL;DR

This paper proposes PRIDA, a novel blind deconvolution algorithm based on mirror descent that achieves provable convergence and robustness to noise and perturbations. It outperforms state-of-the-art methods on standard benchmarks, especially under noisy conditions, while maintaining low per-iteration complexity via atomic updates on the kernel space.

ABSTRACT

We revisit the Blind Deconvolution problem with a focus on understanding its robustness and convergence properties. Provable robustness to noise and other perturbations is receiving recent interest in vision, from obtaining immunity to adversarial attacks to assessing and describing failure modes of algorithms in mission critical applications. Further, many blind deconvolution methods based on deep architectures internally make use of or optimize the basic formulation, so a clearer understanding of how this sub-module behaves, when it can be solved, and what noise injection it can tolerate is a first order requirement. We derive new insights into the theoretical underpinnings of blind deconvolution. The algorithm that emerges has nice convergence guarantees and is provably robust in a sense we formalize in the paper. Interestingly, these technical results play out very well in practice, where on standard datasets our algorithm yields results competitive with or superior to the state of the art. Keywords: blind deconvolution, robust continuous optimization

Motivation & Objective

• To develop a blind deconvolution algorithm with provable convergence and robustness guarantees under noise and perturbations.
• To address the lack of theoretical understanding in convergence and robustness for total variation-based blind deconvolution methods.
• To provide a first-principles foundation for evaluating deep learning-based deblurring models, especially in adversarial and mission-critical settings.
• To design an efficient optimization scheme that avoids expensive projections by using atomic updates on the probability simplex.
• To demonstrate practical superiority over state-of-the-art methods on standard deblurring benchmarks, particularly under noisy conditions.

Proposed method

• The algorithm uses mirror descent with the Kullback-Leibler (KL) divergence as the Bregman divergence to optimize over the probability simplex for the blur kernel.
• It formulates the blind deconvolution problem as a non-convex, non-smooth optimization task with total variation regularization and a data-fidelity term.
• The method leverages strong convexity of the KL divergence to derive a lower bound on the objective function, enabling convergence guarantees.
• It employs a coordinate-wise update strategy for the kernel, followed by normalization, achieving O(s) per-iteration complexity instead of O(s log s).
• The algorithm is derived from a primal-dual perspective, with convergence proven via descent properties of the mirror descent framework.
• Robustness is formalized through stability bounds under perturbations in the observed image, with theoretical guarantees derived using KL divergence properties.

Experimental results

Research questions

• RQ1 Can a blind deconvolution algorithm be designed with provable convergence and robustness to noise and perturbations?
• RQ2 How does mirror descent with KL divergence as a Bregman divergence improve convergence and stability in blind deconvolution?
• RQ3 Can the per-iteration complexity be reduced below O(s log s) while maintaining performance?
• RQ4 How does the algorithm compare to state-of-the-art methods under both clean and noisy image conditions?
• RQ5 To what extent can theoretical guarantees from first principles improve the reliability of deep learning-based deblurring pipelines?

Key findings

• PRIDA achieves provable convergence and robustness to noise and perturbations, with no extra assumptions beyond standard image priors.
• On standard benchmarks, PRIDA performs competitively with or better than the state of the art under clean conditions and significantly outperforms them under noisy conditions.
• The algorithm’s per-iteration complexity is O(s), achieved through atomic updates and normalization, avoiding costly sorting operations required by projection-based methods.
• Theoretical analysis shows that the mirror descent framework with KL divergence ensures strong convergence guarantees, matching the best-known results in optimization.
• Empirical results confirm that PRIDA is resilient to adversarial noise and real-world blur, making it suitable for mission-critical applications.
• The method is implemented efficiently and is available via a public GitHub repository, enabling integration into interactive deblurring pipelines.