| Medical image analysis algorithms aim at increasing the speed, accuracy, and reliability by which medical images are processed and ultimately understood. Active contour and energy minimization techniques are commonly used in medical image analysis applications. The results of these techniques are optimal under certain assumptions and provide meaningful clinical insights. In this thesis, we develop energy minimization techniques for medical image analysis. The primary focus of this thesis is the construction of a theoretical and applied framework for: (1) Geodesic Tractography: In this work, we develop a mathematical framework for finding optimal paths in oriented domains. In oriented domains, image data depends both upon position and upon direction. In other words, for each position and direction in the domain there exists a unique voxel intensity. The use of a Finsler metric is shown to be particularly suited for this type of problem. In fact, we show that the Finsler condition is necessary to ensure that the flow is well-posed. The development of this theory is couched in an application to diffusion-weighted magnetic resonance imagery (DW-MRI). It is shown that representative or anchor tracts are found which optimally connect two regions of interest in the brain. (2) Tractography Segmentation: In this work, we show how these optimal paths may be used to initialize a volumetric segmentation which captures neural fiber bundles. We present a key problem for volumetric segmentation along with two approaches for overcoming this problem: via either a local constraining of statistics or a tensor warping preprocessing step. Also, in this work, we present medical image analysis algorithms using Bayesian segmentation frameworks. In the first, we present our work on the segmentation of brain MRI tissue into tissue classes. In the second, we present the construction of a model of colon haustra for use in computer-aided detection (CAD) within a Bayesian framework. The core software components of this thesis are being made available in the NAMIC toolkit (see http://www.na-mic.org). |
