Oded Stein
Skip Abstract Section
Skip Supplemental Material SectionAbstract
Current quadratic smoothness energies for curved surfaces either exhibit distortions near the boundary due to zero Neumann boundary conditions or they do not correctly account for intrinsic curvature, which leads to unnatural-looking behavior away from the boundary. This leads to an unfortunate trade-off: One can either have natural behavior in the interior or a distortion-free result at the boundary, but not both. We introduce a generalized Hessian energy for curved surfaces, expressed in terms of the covariant one-form Dirichlet energy, the Gaussian curvature, and the exterior derivative. Energy minimizers solve the Laplace-Beltrami biharmonic equation, correctly accounting for intrinsic curvature, leading to natural-looking isolines. On the boundary, minimizers are as-linear-as-possible, which reduces the distortion of isolines at the boundary. We discretize the covariant one-form Dirichlet energy using Crouzeix-Raviart finite elements, arriving at a discrete formulation of the Hessian energy for applications on curved surfaces. We observe convergence of the discretization in our experiments.
Supplemental Material
Available for Download
zip
stein.zip (31.3 MB)
Supplemental movie, appendix, image and software files for, A Smoothness Energy without Boundary Distortion for Curved Surfaces
- E. D. Andersen and K. D. Andersen. 2000. The <scp;>MOSEK</scp;> interior point optimizer for linear programming: An implementation of the homogeneous algorithm. In High Performance Optimization. Kluwer Academic Publishers, 197--232.Google Scholar
- Omri Azencot, Maks Ovsjanikov, Frédéric Chazal, and Mirela Ben-Chen. 2015. Discrete derivatives of vector fields on surfaces—an operator approach. ACM Trans. Graph. 34, 3 (May 2015), 29:1--29:13.Google ScholarDigital Library
- Ilya Baran and Jovan Popovic. 2007. Automatic rigging and animation of 3D characters. ACM Trans. Graph. 26, 3 (July 2007), 72--es.Google ScholarDigital Library
- Miklos Bergou, Max Wardetzky, David Harmon, Denis Zorin, and Eitan Grinspun. 2006. A quadratic bending model for inextensible surfaces. In Proceedings of the 4th Eurographics Symposium on Geometry Processing. 227--230.Google Scholar
- Miklós Bergou, Max Wardetzky, Stephen Robinson, Basile Audoly, and Eitan Grinspun. 2008. Discrete elastic rods. ACM Trans. Graph. 27, 3 (Aug. 2008), 63:1--63:12.Google ScholarDigital Library
- S. Bochner. 1946. Vector fields and Ricci curvature. Bull. Amer. Math. Soc. 52, 9 (1946), 776--797.Google ScholarCross Ref
- Daniele Boffi, Franco Brezzi, and Michel Fortin. 2013. Mixed Finite Element Methods and Applications. Springer-Verlag Berlin.Google Scholar
- Mario Botsch and Leif Kobbelt. 2004. An intuitive framework for real-time freeform modeling. ACM Trans. Graph. 23, 3 (Aug. 2004), 630--634.Google ScholarDigital Library
- Dietrich Braess. 2007. Finite Elements (3rd ed.). Cambridge University Press.Google Scholar
- Christopher Brandt, Leonardo Scandolo, Elmar Eisemann, and Klaus Hildebrandt. 2018. Modeling n-symmetry vector fields using higher-order energies. ACM Trans. Graph. 37, 2 (Mar. 2018), 18:1--18:18.Google ScholarDigital Library
- Susanne C. Brenner. 2015. Forty years of the Crouzeix-Raviart element. Numer. Meth. Part. Differ. Equat. 31, 2 (2015), 367--396.Google ScholarCross Ref
- Etienne Corman and Maks Ovsjanikov. 2019. Functional characterization of deformation fields. ACM Trans. Graph. 38, 1 (Jan. 2019), 8:1--8:19.Google ScholarDigital Library
- R. Courant and D. Hilbert. 1924. Methoden der Mathematischen Physik—Erster Band. Julius Springer, Berlin.Google Scholar
- Keenan Crane. 2018. Keenan’s 3D Model Repository. Retrieved from https://www.cs.cmu.edu/kmcrane/Projects/ModelRepository/.Google Scholar
- Keenan Crane, Mathieu Desbrun, and Peter Schröder. 2010. Trivial connections on discrete surfaces. Comput. Graph. For. 29, 5 (2010), 1525--1533.Google Scholar
- Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2013a. Robust fairing via conformal curvature flow. ACM Trans. Graph. 32, 4 (July 2013), Article 61,10 pages.Google ScholarDigital Library
- Keenan Crane, Clarisse Weischedel, and Max Wardetzky. 2013b. Geodesics in heat: A new approach to computing distance based on heat flow. ACM Trans. Graph. 32, 5 (Oct. 2013), 152:1--152:11.Google ScholarDigital Library
- Michel Crouzeix and P.-A. Raviart. 1973. Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. ESAIM: Math. Modell. Numer. Anal.—Modél. Math. Anal. Numér. 7, R3 (1973), 33--75.Google Scholar
- Mathieu Desbrun, Mark Meyer, Peter Schröder, and Alan H. Barr. 1999. Implicit fairing of irregular meshes using diffusion and curvature flow. In Proceedings of the 26th Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’99). ACM Press/Addison-Wesley Publishing Co., New York, NY, 317--324.Google Scholar
- Stephan Didas, Joachim Weickert, and Bernhard Burgeth. 2009. Properties of higher order nonlinear diffusion filtering. J. Math. Imag. Vis. 35, 3 (2009), 208--226.Google ScholarDigital Library
- David L. Donoho and Carrie Grimes. 2003. Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data. Proc. Nat. Acad. Sci. 100, 10 (2003), 5591--5596. DOI:https://doi.org/10.1073/pnas.1031596100Google ScholarCross Ref
- R. J. Duffin. 1959. Distributed and lumped networks. J. Math. Mech. 8, 5 (1959), 793--826.Google Scholar
- Gerhard Dziuk. 1988. Finite Elements for the Beltrami Operator on Arbitrary Surfaces. Springer Berlin, 142--155.Google Scholar
- Elliot English and Robert Bridson. 2008. Animating developable surfaces using nonconforming elements. ACM Trans. Graph. 27, 3 (Aug. 2008), 66:1--66:5.Google ScholarDigital Library
- Lawrence C. Evans. 2010. Partial Differential Equations: (2nd ed.). American Mathematical Society.Google Scholar
- P. J. Federico. 1982. Descartes on Polyhedra. Springer-Verlag New York Inc.Google Scholar
- Matthew Fisher, Peter Schröder, Mathieu Desbrun, and Hugues Hoppe. 2007. Design of tangent vector fields. ACM Trans. Graph. 26, 3 (2007).Google ScholarDigital Library
- Filippo Gazzola, Hans-Christoph Grunau, and Guido Sweers. 2010. Polyharmonic Boundary Value Problems. Springer-Verlag Berlin.Google Scholar
- Todor Georgiev. 2004. Photoshop healing brush: A tool for seamless cloning. In Proceedings of the European Conference on Computer Vision.Google Scholar
- Stanford Graphics. 2014. The Stanford 3D Scanning Repository. Retrieved from http://graphics.stanford.edu/data/3Dscanrep/.Google Scholar
- Eitan Grinspun, Mathieu Desbrun, Konrad Polthier, Peter Schröder, and Ari Stern. 2006. Discrete differential geometry: An applied introduction. In Proceedings of the ACM SIGGRAPH 2006 Courses (SIGGRAPH’06). ACM, New York, NY.Google ScholarDigital Library
- Gaël Guennebaud, Benoît Jacob et al. 2010. Eigen v3. Retrieved from http://eigen.tuxfamily.org.Google Scholar
- Klaus Hildebrandt, Konrad Polthier, and Max Wardetzky. 2006. On the convergence of metric and geometric properties of polyhedral surfaces. Geom. Dedic. 123, 1 (Dec. 2006), 89--112.Google Scholar
- Alec Jacobson. 2013. Algorithms and Interfaces for Real-time Deformation of 2D and 3D Shapes. Ph.D. Dissertation. ETH Zürich.Google Scholar
- Alec Jacobson. 2019. gptoolbox—Geometry Processing Toolbox. Retrieved from https://github.com/alecjacobson/gptoolbox.Google Scholar
- Alec Jacobson, Ilya Baran, Jovan Popović, and Olga Sorkine. 2011. Bounded biharmonic weights for real-time deformation. ACM Trans. Graph. 30, 4 (July 2011), 78:1--78:8.Google ScholarDigital Library
- Alec Jacobson, Daniele Panozzo et al. 2019a. libigl: A Simple C++ Geometry Processing Library. Retrieved from http://libigl.github.io/libigl/.Google Scholar
- Alec Jacobson, Daniele Panozzo et al. 2019b. Libigl Tutorial Data. Retrieved from https://github.com/libigl/libigl-tutorial-data.Google Scholar
- Alec Jacobson, Elif Tosun, Olga Sorkine, and Denis Zorin. 2010. Mixed finite elements for variational surface modeling. Comput. Graph. For. 29, 5 (2010), 1565--1574.Google Scholar
- Alec Jacobson, Tino Weinkauf, and Olga Sorkine. 2012. Smooth shape-aware functions with controlled extrema. Comput. Graph. For. 31, 5 (Aug. 2012), 1577--1586.Google Scholar
- jansentee3d. 2018. Dragon Tower. Retrieved from https://www.thingiverse.com/thing:3155868.Google Scholar
- Javo. 2016. Big Gladiator Helmet. Retrieved from https://www.thingiverse.com/thing:1345281.Google Scholar
- Pushkar Joshi, Mark Meyer, Tony DeRose, Brian Green, and Tom Sanocki. 2007. Harmonic coordinates for character articulation. ACM Trans. Graph. 26, 3 (July 2007), 71--es.Google ScholarDigital Library
- Jürgen Jost. 2011. Riemannian Geometry and Geometric Analysis. Springer-Verlag Berlin.Google Scholar
- Kwang I. Kim, Florian Steinke, and Matthias Hein. 2009. Semi-supervised regression using Hessian energy with an application to semi-supervised dimensionality reduction. In Advances in Neural Information Processing Systems 22, Y. Bengio, D. Schuurmans, J. D. Lafferty, C. K. I. Williams, and A. Culotta (Eds.). Curran Associates, Inc., 979--987.Google Scholar
- Felix Knöppel, Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2013. Globally optimal direction fields. ACM Trans. Graph. 32, 4 (July 2013), 59:1--59:10.Google ScholarDigital Library
- Felix Knöppel, Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2015. Stripe patterns on surfaces. ACM Trans. Graph. 34, 4 (July 2015), 39:1--39:11.Google ScholarDigital Library
- John M. Lee. 1997. Riemannian Manifolds. Springer-Verlag New York, Inc.Google Scholar
- Stamatis Lefkimmiatis, Aurélien Bourquard, and Michael Unser. 2011. Hessian-based norm regularization for image restoration with biomedical applications. IEEE Trans. Image Proc. 21 (09 2011), 983--95.Google Scholar
- Anat Levin, Dani Lischinski, and Yair Weiss. 2004. Colorization using optimization. In ACM SIGGRAPH 2004 Papers (SIGGRAPH ’04). Association for Computing Machinery, New York, NY, 689--694.Google ScholarDigital Library
- Yaron Lipman, Raif M. Rustamov, and Thomas A. Funkhouser. 2010. Biharmonic distance. ACM Trans. Graph. 29, 3 (July 2010), 27:1--27:11.Google ScholarDigital Library
- Beibei Liu, Yiying Tong, Fernando De Goes, and Mathieu Desbrun. 2016. Discrete connection and covariant derivative for vector field analysis and design. ACM Trans. Graph. 35, 3 (Mar. 2016), 23:1--23:17.Google ScholarDigital Library
- X. Y. Liu, C. H. Lai, and K. A. Pericleous. 2015. A fourth-order partial differential equation denoising model with an adaptive relaxation method. Int. J. Comput. Math. 92, 3 (Mar. 2015), 608--622.Google ScholarDigital Library
- Charles Loop. 1987. Smooth Subdivision Surfaces Based on Triangles. Master’s Thesis. University of Utah.Google Scholar
- M. Lysaker, A. Lundervold, and Xue-Cheng Tai. 2003. Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Proc. 12, 12 (Dec. 2003), 1579--1590.Google ScholarDigital Library
- Richard H. MacNeal. 1949. The Solution of Partial Differential Equations by Means of Electrical Networks. Ph.D. Dissertation. California Institute of Technology.Google Scholar
- MATLAB. 2019. version 9.6 (R2019a). The MathWorks Inc.Google Scholar
- Jorge Nocedal and Stephen J. Wright. 2006. Numerical Optimization (2nd ed.). Springer Science + Business Media, LLC.Google Scholar
- Pascal Peter, Sebastian Hoffmann, Frank Nedwed, Laurent Hoeltgen, and Joachim Weickert. 2016. Evaluating the true potential of diffusion-based inpainting in a compression context. Image Commun. 46, C (Aug. 2016), 40--53.Google Scholar
- Peter Petersen. 2006. Riemannian Geometry (2nd ed.). Springer Science + Business Media, LLC.Google Scholar
- Konrad Polthier and Markus Schmies. 1998. Straightest Geodesics on Polyhedral Surfaces. Springer Berlin, 135--150. DOI:https://doi.org/10.1007/978-3-662-03567-2_11Google Scholar
- Alessio Quaglino. 2012. Membrane Locking in Discrete Shell Theories. Ph.D. Dissertation. Universität Göttingen.Google Scholar
- Nicolas Ray, Bruno Vallet, Laurent Alonso, and Bruno Levy. 2009. Geometry-aware direction field processing. ACM Trans. Graph. 29, 1 (Dec. 2009), 1:1--1:11.Google ScholarDigital Library
- John William Strutt Rayleigh. 1894. The Theory of Sound. Vol. 1. Macmillan and Co.Google Scholar
- Reinhard Scholz. 1978. A mixed method for 4th order problems using linear finite elements. RAIRO. Anal. numér. 12, 1 (1978), 85--90.Google Scholar
- Nicholas Sharp, Yousuf Soliman, and Keenan Crane. 2018. The vector heat method. CoRR abs/1805.09170 (2018). Retrieved from http://arxiv.org/abs/1805.09170.Google Scholar
- O. Sorkine, D. Cohen-Or, Y. Lipman, M. Alexa, C. Rössl, and H.-P. Seidel. 2004. Laplacian surface editing. In Proceedings of the Eurographics/ACM SIGGRAPH Symposium on Geometry Processing. ACM, New York, NY, 175--184.Google ScholarDigital Library
- Oded Stein, Eitan Grinspun, Max Wardetzky, and Alec Jacobson. 2018. Natural boundary conditions for smoothing in geometry processing. ACM Trans. Graph. 37, 2 (May 2018), 23:1--23:13.Google ScholarDigital Library
- Daniel Sýkora, Ladislav Kavan, Martin Čadík, Ondřej Jamriška, Alec Jacobson, Brian Whited, Maryann Simmons, and Olga Sorkine-Hornung. 2014. Ink-and-ray: Bas-relief meshes for adding global illumination effects to hand-drawn characters. ACM Trans. Graph. 33, 2 (Apr. 2014), Article 16, 15 pages.Google ScholarDigital Library
- Amir Vaxman, Marcel Campen, Olga Diamanti, Daniele Panozzo, David Bommes, Klaus Hildebrandt, and Mirela Ben-Chen. 2016. Directional field synthesis, design, and processing. Comput. Graph. For. 35, 2 (2016), 545--572.Google Scholar
- Yu Wang, Alec Jacobson, Jernej Barbič, and Ladislav Kavan. 2015. Linear subspace design for real-time shape deformation. ACM Trans. Graph. 34, 4 (2015), 57:1--57:11.Google ScholarDigital Library
- Yu Wang, Alec Jacobson, Jernej Barbič, and Ladislav Kavan. 2017. Error in “Linear Subspace Design for Real-time Shape Deformation.” Technical Report.Google Scholar
- Max Wardetzky. 2006. Discrete Differential Operators on Polyhedral Surfaces—Convergence and Approximation. Ph.D. Dissertation. FU Berlin.Google Scholar
- Max Wardetzky, Saurabh Mathur, Felix Kälberer, and Eitan Grinspun. 2007. Discrete laplace operators: No free lunch. In Proceedings of the 5th Eurographics Symposium on Geometry Processing (SGP’07). Eurographics Association, 33--37.Google Scholar
- Ofir Weber, Roi Poranne, and Craig Gotsman. 2012. Biharmonic coordinates. Comput. Graph. For. 31, 8 (2012), 2409--2422.Google ScholarDigital Library
- Ofir Weber, Olga Sorkine, Yaron Lipman, and Craig Gotsman. 2007. Context-aware skeletal shape deformationss. Comput. Graph. For. 26, 3 (2007), 265--274.Google Scholar
- Tino Weinkauf, Yotam Gingold, and Olga Sorkine. 2010. Topology-based smoothing of 2D scalar fields with C1-continuity. Comput. Graph. For. 29, 3 (2010).Google Scholar
- Eric W. Weisstein. 2019. Monge Patch. From MathWorld--A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/MongePatch.html.Google Scholar
- Roland Weitzenböck. 1885. Invariantentheorie. P. Noordhoff.Google Scholar
- YahooJAPAN. 2013. Plane. Retrieved from https://www.thingiverse.com/thing:182252.Google Scholar
Index Terms
- A Smoothness Energy without Boundary Distortion for Curved Surfaces
Recommendations
Curvature formulas for implicit curves and surfaces
Special issue: Geometric modelling and differential geometryCurvature formulas for implicit curves and surfaces are derived from the classical curvature formulas in Differential Geometry for parametric curves and surfaces. These closed formulas include curvature for implicit planar curves, curvature and torsion ...
Lines of curvature and umbilical points for implicit surfaces
This paper develops a method to analyze and compute the lines of curvature and their differential geometry defined on implicit surfaces. With our technique, we can explicitly obtain the analytic formulae of the associated geometric attributes of an ...
Concentrated curvature for mean curvature estimation in triangulated surfaces
CTIC'12: Proceedings of the 4th international conference on Computational Topology in Image ContextWe present a mathematical result that allows computing the discrete mean curvature of a polygonal surface from the so-called concentrated curvature generally used for Gaussian curvature estimation. Our result adds important value to concentrated ...
Comments