[Paper Review] A Clearer Picture of Blind Deconvolution

Daniele Perrone, Paolo Favaro|arXiv (Cornell University)|Nov 30, 2014

Advanced Image Processing Techniques35 references3 citations

TL;DR

This paper resolves a long-standing paradox in blind deconvolution by showing that total variation (TV)-based algorithms succeed not due to the TV prior itself, but because of the sequential enforcement of constraints—particularly blur kernel positivity and normalization—during optimization. Despite theoretical guarantees that such algorithms should remain stuck at the no-blur solution, the authors demonstrate that this implementation detail enables convergence to the true solution, achieving state-of-the-art performance with a simple, robust algorithm.

ABSTRACT

Blind deconvolution is the problem of recovering a sharp image and a blur kernel from a noisy blurry image. Recently, there has been a significant effort on understanding the basic mechanisms to solve blind deconvolution. While this effort resulted in the deployment of effective algorithms, the theoretical findings generated contrasting views on why these approaches worked. On the one hand, one could observe experimentally that alternating energy minimization algorithms converge to the desired solution. On the other hand, it has been shown that such alternating minimization algorithms should fail to converge and one should instead use a so-called Variational Bayes approach. To clarify this conundrum, recent work showed that a good image and blur prior is instead what makes a blind deconvolution algorithm work. Unfortunately, this analysis did not apply to algorithms based on total variation regularization. In this manuscript, we provide both analysis and experiments to get a clearer picture of blind deconvolution. Our analysis reveals the very reason why an algorithm based on total variation works. We also introduce an implementation of this algorithm and show that, in spite of its extreme simplicity, it is very robust and achieves a performance comparable to the state of the art.

Motivation & Objective

• To resolve the contradiction between empirical success and theoretical failure of TV-based blind deconvolution algorithms.
• To analyze why alternating minimization with total variation regularization converges to the correct solution despite theoretical predictions of failure.
• To demonstrate that the key to success lies not in the TV prior but in the sequential application of constraints during optimization.
• To show that a minimal, constraint-sequenced algorithm achieves performance comparable to state-of-the-art methods without advanced enhancements.

Proposed method

• The authors analyze the convergence behavior of TV-based blind deconvolution using a simplified one-dimensional model with step-function images and kernels.
• They prove analytically that if all constraints (positivity, normalization) are enforced simultaneously, the algorithm remains trapped at the no-blur solution for any regularization strength.
• They show that when constraints are applied sequentially—first on the blur kernel, then on the image—the algorithm escapes the no-blur solution and converges to the true solution.
• The method relies on a primal-dual algorithm (PAM) with alternating minimization, where constraints are applied in a staggered fashion rather than simultaneously.
• The authors implement a minimal TV-based blind deconvolution algorithm that omits filtering, edge enhancement, and blur kernel priors, focusing only on sequential constraint enforcement.
• They validate the method on standard datasets and compare it to state-of-the-art approaches, showing competitive performance despite simplicity.

Experimental results

Research questions

• RQ1 Why do TV-based blind deconvolution algorithms converge to the correct solution in practice, despite theoretical results predicting failure?
• RQ2 What role does the sequential enforcement of constraints play in enabling convergence away from the no-blur solution?
• RQ3 Can a simple algorithm without advanced priors or filtering achieve performance comparable to state-of-the-art methods?
• RQ4 How does the structure of the cost functional and constraint handling affect convergence in non-convex blind deconvolution problems?

Key findings

• Theoretical analysis confirms that joint minimization with all constraints applied simultaneously leads to convergence at the no-blur solution, regardless of regularization strength.
• In practice, algorithms like Chan and Wong's [8] succeed because constraints are applied sequentially, not simultaneously, which breaks the symmetry that traps the algorithm at the no-blur solution.
• The sequential enforcement of blur kernel positivity and normalization is sufficient to enable convergence to the true solution, even with only a TV prior and no additional priors or filtering.
• A minimal implementation of the algorithm—without filtering, edge enhancement, or blur kernel priors—achieves performance comparable to state-of-the-art methods on standard datasets.
• Exact boundary conditions are shown to improve results, while filtered versions of the input signal may be undesirable in this context.