Yiheng Zhang

Spatio-Temporal Preprocessing and Regularization for Dynamic Linear Inverse Problems with Application to Diffuse Optical Imaging and Tomography

Abstract

Dynamic inverse problems are very common in many biomedical imaging applications, where the interior distributions in both space and time of physiological quantities are of interest, and we want to reconstruct them from measurements. The background application for this work is functional brain imaging of cerebral hemodynamics with Diffuse Optical Tomography (DOT).

Traditional methods solve the dynamic inverse problem at each time point independently, and only employ spatial regularization methods to address the ill-posedness problem, without use of any temporal dynamical information. In contrast, spatio-temporal regularization methods bring more flexibility in modeling prior knowledge in both space and time for purposes of regularization, and thus can improve reconstruction results.

The work in this thesis falls into three separate but closely related parts. We first investigate the theoretical effect of the joint spatio-temporal regularizations on dynamic, linear ill-posed inverse problems. We formulate three distinct previously-reported representative joint regularization methods in a common least-square-solution framework. These methods are the Kalman smoother algorithm based on a state space model, the isotropy condition of F. Greensite, and a method using multiple regularization constraints. Based on this formulation we elucidate relationships among the three approaches, compare their properties, point out the conditions under which they are equivalent, and discuss some practical considerations. This analysis provides insight to understand the distinct utilization of temporal information by different methods and their effects on the common joint regularization matrix.

We then focus on Diffuse Optical Tomography for functional brain imaging. One of the major concerns in using optical techniques in functional brain studies is to suppress physiological interference, systemic fluctuations which can obscure or even overwhelm the activation-evoked response. We develop a PCA-type spatial eigenfiltering algorithm to reduce interference for diffuse optical measurement and show experimentally, on real data, significant improvement of the localization of evoked oxygenation response.

To improve the spatial resolution by employing spatio-temporal regularization methods for diffuse optical tomography, we consider a Hemodynamic Response Function (HRF) model, similar to a technique widely used in functional MRI, for DOT reconstruction. We incorporate prior assumptions on the temporal behavior of the hemoglobin concentration in a HRF model and derive a joint spatio-temporal reconstruction via an ordinary least squares solution, using a linearized forward model. We jointly employ spatial regularization, spectral information, and temporal assumptions in the reconstruction. In simulations of brain activation, we show significant improvements in spatial resolution and contrast-to-noise ratio with appropriate temporal information.

Committee:
Dr. David Boas, MGH
Prof. Dana Brooks, NU (Advisor)
Prof. Eric Miller, NU
Prof. Gilead Tadmor, NU