Expectation Minimum (EM): A New Principle for the Solution of Ill-Posed Problems in Photothermal Science

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Abstract
The expectation-minimum (EM) principle is a new strategy for recovering robust solutions to the ill-posed inverse problems of photothermal science. The expectation-minimum principle uses the addition of well-characterized random noise to a model basis to be fitted to the experimental response by linear minimization or projection techniques. The addition of noise to the model basis improves the conditioning of the basis by many orders of magnitude. Multiple projections of the data onto the basis in the presence of noise are averaged, to give the solution vector as an expectation value which reliably estimates the global minimum solution for general cases, while the conventional approaches fail. This solution is very stable in the presence of random error on the data. The expectation-minimum principle has been demonstrated in conjunction with several projection algorithms. The nature of the solutions recovered by the expectation minimum principle is nearly independent of the minimization algorithms used and depends principally on the noise level set in the model basis.