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

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.

Photothermal imaging of dehydrochlorination in poly(vinyl chloride) films

“…quasi-uniqueness) of the recovered solutions is ensured when u is set at or above the relative standard deviation in h( t) , assuming that the experimental random error is constant and time invariant. This has also been shown for linear regularization techniques (46)(47)(48). In this work, (T = 0.03 (3%) was used because it is set just above the maximum level of relative rms error (random plus bias) observed in the experimental data, as analyzed by the data residuals associated with the fit of the model to the data (see below).…”

“…The depth profile reconstructions reported in this work arise from solutions to an ill-posed problem, which is intractable without some form of regularization (47). In the case of the stochastic algorithm, EM-NNLS (30), used in this work, such regularization is provided by the noise that is added to the basis set and which behaves as a nonlinear regularization parameter (30,48). Although a detailed discussion may be found elsewhere (47)(48)(49), the effect of regulaxization is to impose a filtering (smoothing) on the reconstruction of q(x), which stabilizes its recovery, providing a quasi-unique, albeit broadened or filtered, solution to J3q 5.…”

“…The resolving power of a given basis set may be assessed by plotting a "resolving kernel", S(x, 4, for the reconstruction (29)(30)46). The resolving kernel consists of the line spread obtained in the reconstruction of an infinitesimal plane flux source placed at increasing subsurface depth, x,, in the sample.…”

Depth Profiling of Optical Absorption in Thin Films via the Mirage Effect and a New Inverse Scattering Theory. Part I: Principles and Methodology

“…In this and our previous works, 26,27 we have used two different algorithms to evaluate G . The ® rst, zero-order…”

“…15 is provided by the expectation-minimum principle. 26,27 This algorithm uses seeding of the matrix G e with white noise of known, constant variance. A regularized solution is recovered by averaging a fam ily of individual solutions recovered in the presence of noise on G :…”