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A wavelet multi-scale method for the inverse problem of diffuse optical tomography. (English) Zbl 1317.65194
Summary: This paper deals with the estimation of optical property distributions of participating media from a set of light sources and sensors located on the boundaries of the medium. This is the so-called diffuse optical tomography problem. Such a nonlinear ill-posed inverse problem is solved through the minimization of a cost function which depends on the discrepancy, in a least-square sense, between some measurements and associated predictions. In the present case, predictions are based on the diffuse approximation model in the frequency domain while the optimization problem is solved by the L-BFGS algorithm. To cope with the local convergence property of the optimizer and the presence of numerous local minima in the cost function, a wavelet multi-scale method associated with the L-BFGS method is developed, implemented, and validated. This method relies on a reformulation of the original inverse problem into a sequence of sub-inverse problems of different scales using wavelet transform, from the largest scale to the smallest one. Numerical results show that the proposed method brings more stability with respect to the ordinary L-BFGS method and enhances the reconstructed images for most of initial guesses of optical properties.

MSC:
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35R25 Ill-posed problems for PDEs
35R30 Inverse problems for PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
65T60 Numerical methods for wavelets
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Software:
FreeFem++; L-BFGS
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References:
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