Testing that a Local Optimum of the Likelihood is Globally Optimum using Reparameterized Embeddings

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. If the problem has a statistical maximum likelihood formulation, a local test of global optimality can be constructed. In this paper, we develop an improved test, based on a global maximum validation function proposed by Biernacki, under the assumption that the statistical distribution is in the generalized location family, a condition often satisfied in imaging problems. In addition, a new reparameterization and embedding procedure is presented that exploits knowledge about the forward operator to improve the global maximum validation function. Finally, the reparameterized embedding technique is applied to a physically-motivated joint-inverse problem arising in camera blur estimation. The advantages of the proposed global optimum testing techniques are numerically demonstrated in terms of increased detection accuracy and reduced computation.