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Simplicial Complex Augmentation Framework for Bijective Maps

Zhongshi Jiang, Scott Schaefer, Daniele Panozzo
ACM Transaction on Graphics (Proceedings of SIGGRAPH Asia 2017)
DOI: 10.1145/3130800.3130895


Bijective maps are commonly used in many computer graphics and scientific computing applications, including texture, displacement, and bump mapping. However, their computation is numerically challenging due to the global nature of the problem, which makes standard smooth optimization techniques prohibitively expensive. We propose to use a scaffold structure to reduce this challenging and global problem to a local injectivity condition. This construction allows us to benefit from the recent advancements in locally injective maps optimization to efficiently compute large scale bijective maps (both in 2D and 3D), sidestepping the need to explicitly detect and avoid collisions. Our algorithm is guaranteed to robustly compute a globally bijective map, both in 2D and 3D. To demonstrate the practical applicability, we use it to compute globally bijective single patch parametrizations, to pack multiple charts into a single UV domain, to remove self-intersections from existing models, and to deform 3D objects while preventing self-intersections. Our approach is simple to implement, efficient (two orders of magnitude faster than competing methods), and robust, as we demonstrate in a stress test on a parametrization dataset with over a hundred meshes.

Source Code

Source code is hosted on this GitHub repository. The program is built and tested on MacOS system with AppleClang.


git clone

Building Build Status

To build the program, you can make use of CMakeLists.txt in the root folder:

mkdir build
cd build
cmake -DCMAKE_BUILD_TYPE=Release ../


We provide three examples in this repo, parameterization of a camel, inflation of bunny, and self-intersection recovery of a leg.

./scaf_param_bin -m ../camel_b.obj
./scaf_flow_bin -m ../bunny.obj -t ../bunnyx30.obj
./scaf_flow_bin -m ../ -t ../models/

Press [space] to perform one iteration.

libigl version

A simplified 2D version is integrated into libigl (MPL v2 license)


This repository is an expolration groud for research purposes, therefore it contains files from multiple sources and copyright licenses. However, the author is in general happy to provide most of the code under MIT or MPL license. If you are interested in the commercial use of the algorithm, please contact the author.