Plane Embedding of Dually Contracted Graphs

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The use of plane graphs for the description of image structure and shape representation poses two problems : (1) how to obtain the set of vertices, the set of edges and the incidence relation of the graph, and (2) how to embed the graph into the plane image. Initially, the image is represented by an embedded graph G in a straight forward manner, i.e. the edges of G represent the 4-connectivity of the pixels. Let $ \overline G $ denote a (planar) abstract dual of G. Dual graph contraction is used to reduce the pair ( $ \left( {\overline G ,G} \right) $ ,G) to a pair ( $ \left( {\overline H ,H} \right) $ ,H) of planar abstract duals. Dual graph contraction is unsymmetric due to an extra condition on the choice of the contraction kernels in G. This condition is shown to be necessary and sufficient for H to be embedded onto G. The embedding is applied to the description of image structure and to shape representation.