Efficient marginal likelihood optimization in blind deconvolution

Abstract

In blind deconvolution one aims to estimate from an input blurred image y a sharp image x and an unknown blur kernel k. Recent research shows that a key to success is to consider the overall shape of the posterior distribution p(x, k\y) and not only its mode. This leads to a distinction between MAP <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x, k</sub> strategies which estimate the mode pair x, k and often lead to undesired results, and MAP <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> strategies which select the best k while marginalizing over all possible x images. The MAP <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> principle is significantly more robust than the MAP <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x, k</sub> one, yet, it involves a challenging marginalization over latent images. As a result, MAP <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> techniques are considered complicated, and have not been widely exploited. This paper derives a simple approximated MAP <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> algorithm which involves only a modest modification of common MAP <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x, k</sub> algorithms. We show that MAP <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> can, in fact, be optimized easily, with no additional computational complexity.