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Title: Mathematical And Numerical Studies On The Inverse Problems Associated With Propagation Of Field Correlation Through A Scattering Object
Authors: Varma, Hari M
Advisors: Vasu, R M
Keywords: Light Scattering - Measuring Instruments
Inverse Problems (Mathematics)
Diffuse Correlation Tomography (DCT)
Submitted Date: Feb-2010
Series/Report no.: G23653
Abstract: This thesis discusses the inverse problem associated with the propagation of field autocorrelation of light through a highly scattering object like tissue. In the first part of the thesis we consider the mathematical issues involved in inverting boundary measurements made from diffuse propagation of light through highly scattering objects for their optical and mechanical properties. We present the convergence analysis of the Gauss-Newton algorithm for the recovery of object properties applicable for both diffuse correlation tomography (DCT) and diffuse optical tomography (DOT). En route to this, we establish the existence of solution and Frechet differentiability of the forward propagation equation. The two cases of the delta source and the Gaussian source illuminations are considered separately and the smoothness of solution of the forward equation in these cases is established. Considering DCT as an example, we establish the feasibility of recovering the particle diffusion coefficient (DB ) through minimizing the data-model mismatch of the field autocorrelation at the boundary using the Gauss-Newton algorithm. Some numerical examples validating the theoretical results are also presented. In the second part of the thesis, we reconstruct optical absorption coefficient, µa, and particle diffusion coefficient, DB , from simulated measurements which are integrals of a quantity computed from the measured intensity and intensity autocorrelation g2(τ ) at the boundary. We also recovered the mean square displacement (MSD) distribution of particles in an inhomogeneous object from the sampled g2(τ ) measured on the boundary. From the MSD, we compute the storage and loss moduli distributions in the object. We have devised computationally easy methods to construct the sensitivity matrices which are used in the iterative reconstruction algorithms for recovering these parameters from these measurements. The results of reconstruction of inhomogeneities in µa, DB , MSD and the visco-elastic parameters, which are presented, show forth reasonably good positional and quantitative accuracy. Finally we introduce a self regularized pseudo-dynamic scheme to solve the above inverse problem, which has certain advantages over the usual minimization method employing a variant of the Newton algorithm. The computational difficulties involved in the inversion of ill-conditioned matrices arising in the nonlinear inverse DCT problem are avoided by introducing artificial dynamics and considering the solution to be the steady-state response (if it exists) of the artificially evolving dynamical system, represented by ordinary differential equations (ODE) in pseudo-time. We show that the asymptotic solution obtained through the pseudo-time marching converges to the optimal solution which minimizes a mean-square error functional, provided the Hessian of the forward equation is positive definite in the neighborhood of this optimal solution. The superior noise tolerance and regularization-insensitive nature of pseudo-dynamic strategy are proven through numerical simulations in the context of DCT.
Abstract file URL: http://etd.ncsi.iisc.ernet.in/abstracts/1584/G23653-Abs.pdf
URI: http://etd.iisc.ernet.in/handle/2005/1215
Appears in Collections:Instrumentation and Applied Physics (iap)

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