  Many mathematical imaging problems are posed as non-convex optimization
problems. When numerically tractable global optimization procedures are not
available, one is often interested in testing ex post facto whether or not a
locally convergent algorithm has found the globally optimal solution. When the
problem is formulated in terms of maximizing the likelihood function under a
statistical model for the measurements, one can construct a statistical test
that a local maximum is in fact the global maximum. A one-sided test is
proposed for the case that the statistical model is a member of the generalized
location family of probability distributions, a condition often satisfied in
imaging and other inverse problems. We propose a general method for improving
the accuracy of the test by reparameterizing the likelihood function to embed
its domain into a higher dimensional parameter space. We show that the proposed
global maximum testing method results in improved accuracy and reduced
computation for a physically-motivated joint-inverse problem arising in
camera-blur estimation.
