Sharpening and Sparsifying with Surface Hessians

SIGGRAPH Asia 2024

Dylan Rowe, University of Southern California

Alec Jacobson, University of Toronto and Adobe Research Toronto

Oded Stein, University of Southern California

Figure 1. Our intrinsic L1 smoothness energy constructs piecewise smooth functions. Unlike L2 energies, which fit a smooth function without any ridges to solve an interpolation problem (and thus have to hallucinate a saddle), we construct a piecewise flat function (left). When denoising, our discetization produces clean isolines compared to previous, non-intrinsic approaches (center). Our L1 energy can construct sparse cut indicator functions to segment meshes (right).

Abstract

The L1 Hessian energy measures the norm of the Hessian of a function on a surface (and NOT the squared norm, as is common with many geometry applications that employ L2). Its minimizers tend to be locally linear with a sparse set of curved ridges. We introduce a fully-intrinsic discretization of this energy for triangle meshes and show that it can be optimized using off-the-shelf conic program solvers. We apply it to stylization, denoising, interpolation, hole-filling, and segmentation tasks. Our L1 approach exhibits multiple important differences from its more-familiar L2 counterpart: it preserves ridge-like features in the input, it naturally incorporates a flatness prior for reconstruction, and, at its extreme, it distills its input to an abstract, angular form.

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Acknowledgements

Our research is funded in part by NSERC Discovery (RGPIN–2022–04680), the Ontario Early Research Award program, the Canada Research Chairs Program, a Sloan Research Fellowship, the DSI Catalyst Grant program and gifts by Adobe Inc.

We thank Silvia Sellán and Yuta Noma for proofreading.

We acknowledge and thank the authors of the 2D and 3D assets used in this project. Please see the paper for more details.