Progressive embedding

Abstract

Tutte embedding is one of the most common building blocks in geometry processing algorithms due to its simplicity and provable guarantees. Although provably correct in infinite precision arithmetic, it fails in challenging cases when implemented using floating point arithmetic, largely due to the induced exponential area changes. We propose Progressive Embedding, with similar theoretical guarantees to Tutte embedding, but more resilient to the rounding error of floating point arithmetic. Inspired by progressive meshes, we collapse edges on an invalid embedding to a valid, simplified mesh, then insert points back while maintaining validity. We demonstrate the robustness of our method by computing embeddings for a large collection of disk topology meshes. By combining our robust embedding with a variant of the matchmaker algorithm, we propose a general algorithm for the problem of mapping multiply connected domains with arbitrary hard constraints to the plane, with applications in texture mapping and remeshing. CCS Concepts: • Computing methodologies → Shape modeling.

Adaptive Block Coordinate Descent for Distortion Optimization

“…As observed in our experiment, Tutte embedding has certain implementation‐related limitations and, for challenging meshes, it can fail even if the target domain is convex. In particular, as demonstrated in [SJZP19], Tutte embedding may produce invalid triangles due to precision loss in the floating point arithmetic. We tested ABCD(PN) and PE on a challenging Tutte failure example, introduced in [SJZP19].…”

“…Mapping of the Hele‐Shaw polygon [SJZP19] (top) onto the square (bottom) using ABCD(PN) (left) and the PE method (right). The initial Tutte embedding produces 41 flipped and 845 degenerate triangles.…”

“…It therefore seems that mapping to domains that match the structure of a source mesh should be a much better starting point if the algorithm can fix occasional foldovers. Although there is a number of recent studies on embedding triangle mesh onto non‐convex planar domains, our algorithm has unique properties: it operates both in 2D and 3D; it employs hard positional constraints, and therefore, unlike boundary first flattening (BFF) [SC18], it interpolates the exactly prescribed boundary shape; ABCD updates only vertex coordinates and, in contrast to the recently introduced progressive embedding (PE) [SJZP19], our algorithm does not involve mesh re‐triangulation.…”

Memory‐Efficient Bijective Parameterizations of Very‐Large‐Scale Models

“…Tutte's embedding and its variants [SJZP19, AL15, Flo03, Tut63] theoretically guarantee bijection. They have two main limitations.…”

“…To theoretically guarantee bijection, most methods start from a valid initialization and then reduce distortion while ensuring bijection. The bijective initialization can be obtained via the Tutte's embedding and its variants [Tut63,Flo03,SJZP19]. To effectively optimize distortion, there are two typical strategies.…”