Developability of Heightfields via Rank Minimization SIGGRAPH 2020

SILVIA SELLÁN1, NOAM AIGERMAN2, ALEC JACOBSON1,2

1University of Toronto, 2Adobe Research

Our method, inspired by compressed sensing and the problem of rank minimization, takes as input a heightfield describing a surface (left) and outputs a heightfield describing a piecewise developable surface (right) which approximates the input. 3D model by Lloyd Chidgzey under CC BY-SA 3.0.

Abstract

This work concerns the computation and approximation of developable surfaces — surfaces that are locally isometric to the two-dimensional plane. These surfaces are heavily studied in differential geometry, and are also of great interest to fabrication, architecture and fashion. We focus specifically on developability of heightfields. Our main observation is that developability can be cast as a rank constraint, which can then be plugged into theoretically-grounded rank-minimization techniques from the field of compressed sensing. This leads to a convex semidefinite optimization problem, which receives an input heightfield and recovers a similar heightfield which is developable. Due to the sparsifying nature of compressed sensing, the recovered surface is piecewise developable, with creases emerging between connected developable pieces. The convex program includes one user-specified parameter, balancing adherence to the original surface with developability and number of patches. We moreover show, that in contrast to previous techniques, our discretization does not introduce a bias and the same results are achieved across resolutions and orientations, and with no limit on the number of creases and patches. We solve this convex semidefinite optimization problem efficiently, by devising a tailor-made ADMM solver which leverages matrix-projection observations unique to our problem. We employ our method on a plethora of experiments, from denoising 3D scans of developable geometry such as documents and buildings, through approximating general heightfields with developable ones, and up to interpolating sparse annotations with a developable heightfield.

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BibTeX

@article{Sellan:Developability:2020,
  title = {Developability of Heightfields via Rank Minimization},
  author = {Silvia Sell{\'{a}}n and Noam Aigerman and Alec Jacobson},
  year = {2020},
  journal = {ACM Transactions on Graphics},
}

Acknowledgements

This project is funded in part by NSERC Discovery (RGPIN2017–05235, RGPAS–2017–507938), New Frontiers of Research Fund (NFRFE–201), the Ontario Early Research Award program, the Canada Research Chairs Program, the Fields Centre for Quantitative Analysis and Modelling and gifts by Adobe Systems, Autodesk and MESH Inc. We thank Mirela Ben-Chen, Oded Stein and Derek Liu for their insightful conversation and advice; Herng Yi Cheng and Abhishek Madan for producing origami examples; Wenzheng Chen for his help with the 3D scanner setup in Fig. 10; Josh Holinaty, Naseem Hrab and Owlkids for allowing us to use their book in Fig. 3; and Honglin Chen, John Kanji, Ruiqi Wang and Rahul Arora for proofreading.