After [10, 15, 12, 2, 4] minimum cut/maximum ow algorithms on graphs emerged as an increasingly useful tool for exact or approximate energy minimization in low-level vision. The combinatorial optimization literature provides many min-cut/max-ow algorithms with dierent polynomial time complexity. Their practical eciency, however, has to date been studied mainly outside the scope of computer vision.

Phase contrast magnetic resonance angiography (PC-MRA) is a non-invasive method for 3D vessel delineation, which for each voxel not only provides measurement of speed (conveyed as a speed image), but also gives a three-component estimate of flow direction (in the form of phase images). In this thesis, we present a new approach to reconstructing vessels and aneurysms from PC-MRA, and demonstrate how speed and flow coherence information extracted from a PC-MRA dataset can be combined for detecting and reconstructing normal vessels and aneurysms with relatively low flow rate and low signal-to-noise ratio (SNR). We propose to use a Maxwell-Gaussian mixture density to model the background signal and combine this with a uniform distribution for modelling vascular signal to give a Maxwell-Gaussian-uniform (MGU) mixture model of speed image intensity. The MGU model param-eters are estimated by the Expectation-Maximisation (EM) algorithm. It is shown that the Maxwell-Gaussian mixture distribution models the background signal more accurately than a Maxwell distribution. Although the MGU model works satisfactorily in classifying the back-