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The paper describes methods of image segmentation and edge detection based on graph-theoretic representations of images. The image is mapped onto a weighted graph and a spanning tree of this graph is used to describe regions or edges in the image. Edge detection is shown to be a dual problem to segmentation. A number of methods are developed, each providing a different segmentation or edge detection technique. The simplest of these uses the shortest spanning tree (SST), a notion that forms the basis of the other improved methods. These further methods make use of global pictorial information, removing many of the problems of the SST segmentation in its simple form and of other pixel linking algorithms. An important feature in all of the proposed methods is that regions may be described in a hierarchical way.