This thesis presents a new mathematical model for segmenting volume images. The model is an energy function defined on the state space of all possibilities to remove or preserve splitting faces from an initial over-segmentation of the 3D image into supervoxels. It decomposes into potential functions that are learned automatically from a small amount of empirical training data. The learning is based on features of the distribution of gray values in the volume image and on features of the geometry and topology of the supervoxel segmentation. To be able to extract these features from large 3D images that consist of several billion voxels, a new algorithm is presented that constructs a suitable representation of the geometry and topology of volume segmentations in a block-wise fashion, in log-linear runtime (in the number of voxels) and in parallel, using only a prescribed amount of memory. At the core of this thesis is the optimization problem of finding, for a learned energy function, a segmentation with minimal energy. This optimization problem is difficult because the energy function consists of 3rd and 4th order potential functions that are not submodular. For sufficiently small problems with 10,000 degrees of freedom, it can be solved to global optimality using Mixed Integer Linear Programming. For larger models with 10,000,000 degrees of freedom, an approximate optimizer is proposed and compared to state-of-the-art alternatives. Using these new techniques and a unified data structure for multi-variate data and functions, a complete processing chain for segmenting large volume images, from the restoration of the raw volume image to the visualization of the final segmentation, has been implemented in C++. Results are shown for an application in neuroscience, namely the segmentation of a part of the inner plexiform layer of rabbit retina in a volume image of 2048 x 1792 x 2048 voxels that was acquired by means of Serial Block Face Scanning Electron Microscopy (Denk and Horstmann, 2004) with a resolution of 22nm x 22nm x 30nm. The quality of the automated segmentation as well as the improvement over a simpler model that does not take geometric context into account, are confirmed by a quantitative comparison with the gold standard
