Shape approximation by differential properties of scalar functions

This paper presents a method of shape chartification suitable for surface approximation. The innovation of this approach lies on the definition of an iterative refinement of the shape into a set of patches that are automatically tiled and used to approximate the original shape up to a prescribed error. The coding of the patches is supported by the Reeb graph and contains the rules to properly tile, stitch them, and reconstruct the original shape while preserving its topology, using a technique which is also exploited for reconstructing an object from non-planar contours. The method is geometry-aware by definition, as the nodes of the Reeb graph are representative of the main shape features, which belong to the approximated shape already at the initial iteration steps. The points of the reconstructed shape belong to the original surface, their total number is highly reduced, and the original connectivity is replaced by a set of patches that preserves the global topology of the input shape.