ABF++:: Fast and robust angle based flattening

Conformal parameterization of mesh models has numerous applications in geometry processing. Conformality is desirable for remeshing, surface reconstruction, and many other mesh processing applications. Subject to the conformality requirement, these applications typically benefit from parameterizations with smaller stretch. The Angle Based Flattening (ABF) method, presented a few years ago, generates provably valid conformal parameterizations with low stretch. However, it is quite time-consuming and becomes error prone for large meshes due to numerical error accumulation. This work presents ABF++, a highly efficient extension of the ABF method, that overcomes these drawbacks while maintaining all the advantages of ABE ABF++ robustly parameterizes meshes of hundreds of thousands and millions of triangles within minutes. It is based on three main components: (1) a new numerical solution technique that dramatically reduces the dimension of the linear systems solved at each iteration, speeding up the solution; (2) a new robust scheme for reconstructing the 2D coordinates from the angle space solution that avoids the numerical instabilities which hindered the ABF reconstruction scheme; and (3) an efficient hierarchical solution technique. The speedup with (1) does not come at the expense of greater distortion. The hierarchical technique (3) enables parameterization of models with millions of faces in seconds at the expense of a minor increase in parametric distortion. The parameterization computed by ABF++ are provably valid, that is they contain no flipped triangles. As a result of these extensions, the ABF++ method is extremely suitable for robustly and efficiently parameterizing models for geometry-processing applications.