Shock Scaffold Segregation and Surface Recovery

The shock scaffold is a hierarchical organization of the medial axis ( ) in 3D consisting of special medial points, and curves connecting these points, thereby forming a geometric directed graph [7]. We will describe a new method for segregating the shock scaffold ( ) for an unorganized cloud of points in 3D in two sub-groups, one of which is used to mesh the point generators into a surface interpolant.1 In the practical scenario where point generators are sampling the surfaces of 3D objects, we expect intuitively that part of the symmetry structure in the resulting closely approximates the original surface symmetries, while the remaining part arises from symmetries pertaining to the interaction of nearby sampled points. (e.g., compare Figure 1 (c) and (e-f)). We segregate an initial in two sub-groups in three main steps: (S1) rank shock curves by a geometric measure of the triangle (e.g., area) interpolating their associated triplet of generators; (S2) select a threshold to obtain a first segregation of the in the “initial surface and medial scaffolds;” (S3) examine the topology of each surface interpolant and clean-up the resulting surface (Fig.1). The construction of the initial segregation of in S2 is a “one shot,” i.e., non-iterative, process. We march through the sorted list of shock curves of the until an end condition is reached, e.g., all generators are interpolated as vertex of at least one surface triangle. In general, whichever (global) threshold we pick, we will have certain triangles which are not part of the desired interpolant to the surface. Typically, these extraneous triangles occur near concave, saddle-like regions, necks of the shape, i.e., where remote surface patches can be arbitrarily near each other. Our strategy is then to seek in S3 triangles which if removed do not change the connectivity of adjacent triangles, i.e., lead to no new holes in the mesh (Figure 2). Once this triangle removal step is completed, we obtain a final surface scaffold, , and a final medial scaffold, . Each shock curve of has an associated triangular surface interpolant; together, these triangles constitute the final surface interpolant, . An optional fourth step (S4) uses to construct coarse-scale axial and and rib curves (Figure