Extraction of Hierarchical Surface Networks from Bilinear Surface Patches

Surface networks capture the topological relations between passes of a continuous surface, the paths of steepest descent and ascent starting at the passes, and the pits and peaks where the steepest paths end. Surface networks represent the topology of surfaces in a compressed form and allow fast investigation of the surfaces' convex and concave shapes. They are applied, for instance, for enhancing algorithms for surface analysis, for surface model simplification, and for surfaces visualization. Furthermore, they are themselves subjects of analysis as they are closely coupled to the intrinsic geometrical concepts and rules of continuous surfaces. This article extends the topology of surface networks in four ways: (i) objects at the edge of the surface model are introduced; (ii) intersections between valley and ridge lines are found to be possible, and such intersections are incorporated into the topology; (iii) horizontal areas may represent passes, pits, or peaks, and therefore must be detected and explicitly incorporated; and (iv) valley and ridge line hierarchies are recognized as inherent components of the surface network. They are extracted and explicitly represented. To ensure consistency and completeness of the surface network, a zero‐order continuous surface is specified from the raster data prior to the extraction. This article presents a method to represent and derive valley and ridge line hierarchies. The results are illustrated with two examples. The extracted networks are found to be consistent and complete. However, the extraction method tends to produce spurious pits, peaks, and passes, which form a drawback if the surface data are affected by noise.