This work introduces a new, unified representation (termed GeoMap) for capturing both the geometrical and the topological aspects of a segmentation result. This representation is not static, but suitable for all intermediate working states of a segmentation process and serves as a common representation for a multitude of algorithms. Throughout a segmentation process, the GeoMap maintains the consistency between the topological and geometrical information and the duality of regions and boundaries. The main contribution is a new, sub-pixel precise GeoMap realization and a corresponding watershed segmentation algorithm, which is used to produce an oversegmentation into sub-pixel precise superpixels (data-dependent picture elements that are independent of the sampling grid). Furthermore, an integrated environment for image
segmentation based on a relevance filtering process that reduces the number of irrelevant boundaries within a GeoMap, using a combination of region- or boundary- based, automatic or (semi-)interactive segmentation tools in order to produce the final result, is introduced. Finally, the book shows that since the new sub-pixel GeoMap does not suffer from certain limitations imposed on previous representations by the pixel grid, it becomes flexible enough to be suitable for applications beyond image segmentation, e.g. triangulations, skeletonization, or a new boundary reconstruction method based on α-shapes.