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, those applications typically bene t 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 ac- cumulation. This work presents ABF++, a highly e cient extension of the ABF method that overcomes those drawbacks, while maintaining all the advantages of ABF. ABF++ robustly pa- rameterizes meshes of hundreds of thousands and milions of triangles within minutes. It is based on three main components : (1) a new numercial 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 u,v's from the angle space solution that avoids the numeri- cal instabilities which hindered the ABF reconstruction scheme ; and (3) an e cient hierarchical solution technique. The speedup at (1) does not come at the expence of greater distortion and the resulting parameterizations remain provably valid. The hirarchical technique (3) enables pa- rameterization of models with milions of faces in seconds, at the expence of a minor increase in parametric distortion. As a result of those extensions the ABF++ method is extremely suitable for robustly and e ciently parameterizing models for geometry proceesing applications.

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   |texte=   ABF++ : Fast and Robust Angle Based Flattening
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