Image reconstruction in optical tomography: Solution of the perturbation equation

This dissertation is devoted to the solution of a linear perturbation equation encountered in optical tomography. Generally speaking, in the perturbation equation, both the weight matrix (operator) and the data are subject to noise or errors. To overcome the noise in the data, a RLS approach is employed to solve the perturbation equation. The regularization parameter is obtained by the Miller criterion. This RLS method is first applied to continuous wave (CW) data. It is then incorporated into a progressive expansion algorithm for time resolved (TR) data and a regularized progressive expansion algorithm is developed.; To alleviate the error effect existing in the weight matrix, a Rayleigh quotient form TLS (RQF-TLS) approach is explored. Statistical properties of the TLS solution are derived based on this formulation. It is shown that the RQF-TLS estimator is equivalent to the maximum likelihood estimator when the noise terms in both data and operator elements are identically and independently distributed (i.i.d.) Gaussian. A perturbation analysis of the RQF-TLS solution is conducted to derive the covariance matrix and the mean squares error (MSE) of the TLS estimate. It is shown that the MSE of the RQF-TLS is smaller than that of the least squares (LS) estimator when both the operator and data are subject to i.i.d noise.; In order to alleviate the ill-posed nature of the RQF-TLS solution, a regularized TLS (RTLS) approach is proposed, by adding a regularization term in the Rayleigh quotient function. Both the TLS and RTLS methods obtain their solutions using a conjugate gradient method which is particularly suitable for large-scale systems.; One challenging problem in solving the perturbation equation is that the computation complexity is usually very high due to the extremely large dimension of the weight matrix. To improve the efficiency of the RLS and TLS, wavelet based multigrid RLS and TLS methods are developed. Based on the transformed equation, wavelet based RLS and TLS are solved from coarse to fine resolutions in successive approximations. The computational complexity is reduced significantly under the same reconstruction quality criterion compared to the previous one-grid method which solves the equation in the finest resolution directly. This scheme also enables one to quickly identify regions of interest (ROI) from a coarse level reconstruction and restrict the reconstruction in the following fine resolutions to those regions.