# Total Variation Blind Deconvolution:

The Devil is in the Details

Institute of Computer Science and Applied Mathematics, University of Bern, Switzerland

In this paper we study the problem of blind deconvolution. Our
analysis is based on the algorithm of Chan and Wong[1] which
popularized the use of sparse gradient priors via total variation.
We use this algorithm because many methods in the literature are
essentially adaptations of this framework.

Such algorithm is an iterative alternating energy minimization where
at each step either the sharp image or the blur function are
reconstructed. Recent work of Levin et al.[2] showed that
any algorithm that tries to minimize that same energy would fail, as
the desired solution has a higher energy than the no-blur solution,
where the sharp image is the blurry input and the blur is a Dirac
delta. However, experimentally one can observe that Chan and Wong's
algorithm converges to the desired solution even when initialized
with the no-blur one. We provide both analysis and experiments to
resolve this paradoxical conundrum. We find that both claims are
right.

The key to understanding how this is possible lies in the details of
Chan and Wong's implementation and in how seemingly harmless choices
result in dramatic effects.

Our analysis reveals that the delayed scaling (normalization) in the
iterative step of the blur kernel is fundamental to the convergence
of the algorithm.

This then results in a procedure that eludes the no-blur solution,
despite it being a global minimum of the original energy.
We introduce an adaptation 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.

## Check out our latest work on blind deconvolution: Blind Deconvolution via Lower-Bounded Logarithmic Image Priors

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## Bibtex

@inproceedings{perrone_cvpr2014,

author = {Daniele Perrone and Paolo Favaro},

title = {Total Variation Blind Deconvolution: The Devil is in the Details},

booktitle = {EEE Conference on Computer Vision and Pattern Recognition (CVPR)},

year = {2014}

}

## References

[1] T. Chan and C. K. Wong. **Total variation blind deconvolution**. IEEE Transactions on Image Processing, 1998.

[2] A. Levin, Y. Weiss, F. Durand, and W. T. Freeman. **Understanding blind deconvolution algorithms**. IEEE Trans. Pattern Anal. Mach. Intell., 2011.

[3] L. Sun, S. Cho, J. Wang; and J. Hays: **Edge-based blur kernel
estimation using patch priors**, International Conference on
Computational Photography (ICCP), 2013.