  Blind deconvolution involves the estimation of a sharp signal or image given
only a blurry observation. Because this problem is fundamentally ill-posed,
strong priors on both the sharp image and blur kernel are required to
regularize the solution space. While this naturally leads to a standard MAP
estimation framework, performance is compromised by unknown trade-off parameter
settings, optimization heuristics, and convergence issues stemming from
non-convexity and/or poor prior selections. To mitigate some of these problems,
a number of authors have recently proposed substituting a variational Bayesian
(VB) strategy that marginalizes over the high-dimensional image space leading
to better estimates of the blur kernel. However, the underlying cost function
now involves both integrals with no closed-form solution and complex,
function-valued arguments, thus losing the transparency of MAP. Beyond standard
Bayesian-inspired intuitions, it thus remains unclear by exactly what mechanism
these methods are able to operate, rendering understanding, improvements and
extensions more difficult. To elucidate these issues, we demonstrate that the
VB methodology can be recast as an unconventional MAP problem with a very
particular penalty/prior that couples the image, blur kernel, and noise level
in a principled way. This unique penalty has a number of useful characteristics
pertaining to relative concavity, local minima avoidance, and scale-invariance
that allow us to rigorously explain the success of VB including its existing
implementational heuristics and approximations. It also provides strict
criteria for choosing the optimal image prior that, perhaps
counter-intuitively, need not reflect the statistics of natural scenes. In so
doing we challenge the prevailing notion of why VB is successful for blind
deconvolution while providing a transparent platform for introducing
enhancements.
