ABF++: fast and robust angle based flattening

Conformal parameterization of mesh models has numerous applicationsin geometry processing. Conformality is desirable for remeshing,surface reconstruction, and many other mesh processingapplications. Subject to the conformality requirement, theseapplications typically benefit from parameterizations with smallerstretch. The Angle Based Flattening (ABF) method, presented a fewyears ago, generates provably valid conformal parameterizationswith low stretch. However, it is quite time-consuming and becomeserror prone for large meshes due to numerical error accumulation.This work presents ABF++, a highly efficientextension of the ABF method, that overcomes these drawbacks whilemaintaining all the advantages of ABF. ABF++ robustly parameterizesmeshes of hundreds of thousands and millions of triangles withinminutes. It is based on three main components: (1) a new numericalsolution technique that dramatically reduces the dimension of thelinear systems solved at each iteration, speeding up the solution;(2) a new robust scheme for reconstructing the 2D coordinates fromthe angle space solution that avoids the numerical instabilitieswhich hindered the ABF reconstruction scheme; and (3) an efficienthierarchical solution technique. The speedup with (1) does not comeat the expense of greater distortion. The hierarchical technique(3) enables parameterization of models with millions of faces inseconds at the expense of a minor increase in parametricdistortion. The parameterization computed by ABF++ are provablyvalid, that is they contain no flipped triangles. As a result ofthese extensions, the ABF++ method is extremely suitable forrobustly and efficiently parameterizing models forgeometry-processing applications.