Optimal recursive estimation techniques for dynamic medical image reconstruction

The focus of this thesis is to mathematically model and solve the inverse problem of reconstructing a dynamic medical image. We use a stochastic approach based on a Markov process to model the problem. We introduce a novel proximal approach based on a Bregman projection, and we apply it during the Kalman filter algorithm to ensure positivity and spatial regularization. We do not postulate precise a-priori information about the underlying dynamics of the physical process. We establish theoretical properties of our solution, and we test our method for the case of image reconstruction in time-dependent single photon emission computed tomography (SPECT). Static SPECT reconstruction algorithms assume that the activity does not vary in time. In many situations, however, physicians are interested in the dynamics of the underlying physiological process. For example, rate of uptake or wash-out of the pharmaceutical tracer will provide functional diagnosis information. Thus arises the need to explore time-varying SPECT which, mathematically, is an ill-posed inverse problem. In this thesis, we investigate a projected Kalman reconstruction approach to estimate the dynamic activity. We give a brief overview of imaging in general, and medical imaging in particular. We then describe some important aspects of SPECT imaging, one of the two main imaging modalities in nuclear medicine. We formulate a linear state-space model of the problem, and we introduce the optimal recursive Kalman filter (KF) and smoother. However, the Kalman output image is unidentifiable because of the presence of negative components in the activity. Setting negative values of the activity to zero or taking their absolute value does not lead to an acceptable solution. We thus incorporate a proximal method to induce a positive estimator, and then we establish a number of mathematical and statistical properties of our estimator. While KF does a temporal smoothing, it does not include a spatial regularization. We present spatial regularization schemes, and we give a detailed description on how to implement them. We provide numerical results to corroborate the effectiveness of our reconstruction method and to confirm our theoretical results. Finally, we summarize our findings and state directions of present and future work.