Integral invariants for robust geometry processing

Differential invariants of curves and surfaces such as curvatures and their derivatives play a central role in Geometry Processing. They are, however, sensitive to noise and minor perturbations and do not exhibit the desired multi-scale behavior. Recently, the relationships between differential invariants and certain integrals over small neighborhoods have been used to define efficiently computable integral invariants which have both a geometric meaning and useful stability properties. This paper considers integral invariants defined via distance functions, and the stability analysis of integral invariants in general. Such invariants proved useful for many tasks where the computation of shape characteristics is important. A prominent and recent example is the automatic reassembling of broken objects based on correspondences between fracture surfaces.

[1]  Stanley Osher,et al.  Fast Sweeping Methods for Static Hamilton-Jacobi Equations , 2004, SIAM J. Numer. Anal..

[2]  Leonidas J. Guibas,et al.  Robust global registration , 2005, SGP '05.

[3]  Reinhard Klette,et al.  Surface Registration Markers from Range Scan Data , 2006, IWCIA.

[4]  Dominique Hulin,et al.  Mean Curvature and Asymptotic Volume of Small Balls , 2003, Am. Math. Mon..

[5]  A. Aleksandrov,et al.  Intrinsic Geometry of Surfaces , 1967 .

[6]  Robert Schrader,et al.  On the curvature of piecewise flat spaces , 1984 .

[7]  P. Danielsson.  Euclidean distance mapping , 1980 .

[8]  J. Mitani,et al.  Making papercraft toys from meshes using strip-based approximate unfolding , 2004, SIGGRAPH 2004.

[9]  Pierre Alliez,et al.  Anisotropic polygonal remeshing , 2003, ACM Trans. Graph..

[10]  Konrad Polthier,et al.  Anisotropic Filtering of Non‐Linear Surface Features , 2004, Comput. Graph. Forum.

[11]  L. Gool,et al.  Semi-differential invariants , 1992 .

[12]  Johannes Wallner,et al.  Geometric Modeling with Conical Meshes and Developable Surfaces , 2006, ACM Trans. Graph..

[13]  U. Pinkall,et al.  Discrete isothermic surfaces. , 1994 .

[14]  Helmut Pottmann,et al.  Geometry of the Squared Distance Function to Curves and Surfaces , 2002, VisMath.

[15]  Martin Rumpf,et al.  Robust feature detection and local classification for surfaces based on moment analysis , 2004, IEEE Transactions on Visualization and Computer Graphics.

[16]  Andrew Zisserman,et al.  Geometric invariance in computer vision , 1992 .

[17]  Stefano Soatto,et al.  Integral Invariant Signatures , 2004, ECCV.

[18]  H. Seidel,et al.  Ridge-valley lines on meshes via implicit surface fitting , 2004, SIGGRAPH 2004.

[19]  Luigi Ambrosio,et al.  Curvature and distance function from a manifold , 1998 .

[20]  Ulrich Pinkall,et al.  Computing Discrete Minimal Surfaces and Their Conjugates , 1993, Exp. Math..

[21]  Stanley Osher,et al.  Fast Sweeping Algorithms for a Class of Hamilton-Jacobi Equations , 2003, SIAM J. Numer. Anal..

[22]  Mark Meyer,et al.  Discrete Differential-Geometry Operators for Triangulated 2-Manifolds , 2002, VisMath.

[23]  Y. Tsai.  Rapid and accurate computation of the distance function using grids , 2002 .

[24]  Chi-Keung Tang,et al.  Robust estimation of adaptive tensors of curvature by tensor voting , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[25]  S. Rusinkiewicz.  Estimating curvatures and their derivatives on triangle meshes , 2004 .

[26]  Konrad Polthier,et al.  Smooth feature lines on surface meshes , 2005, SGP '05.

[27]  H. Pottmann,et al.  Computational Line Geometry , 2001 .

[28]  A. Adamson,et al.  Ray tracing point set surfaces , 2003, 2003 Shape Modeling International..

[29]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[30]  Marc Pouget,et al.  Estimating differential quantities using polynomial fitting of osculating jets , 2003, Comput. Aided Geom. Des..

[31]  Gabriel Taubin,et al.  Estimating the tensor of curvature of a surface from a polyhedral approximation , 1995, Proceedings of IEEE International Conference on Computer Vision.

[32]  Peter Schröder,et al.  Discrete Willmore flow , 2005, SIGGRAPH Courses.

[33]  Shi-Min Hu,et al.  Robust principal curvatures on multiple scales , 2006, SGP '06.

[34]  Frédéric Chazal,et al.  Molecular shape analysis based upon the morse-smale complex and the connolly function , 2002, SCG '03.

[35]  Hans-Christian Hege,et al.  Visualization and Mathematics III , 2011 .

[36]  Martin Rumpf,et al.  Feature sensitive multiscale editing on surfaces , 2004, The Visual Computer.

[37]  Hongkai Zhao,et al.  High Order Fast Sweeping Methods for Static Hamilton–Jacobi Equations , 2006, J. Sci. Comput..

[38]  Heping Ma,et al.  Optimal Error Estimates of the Chebyshev-Legendre Spectral Method for Solving the Generalized Burgers Equation , 2003, SIAM J. Numer. Anal..

[39]  Alla Sheffer,et al.  D‐Charts: Quasi‐Developable Mesh Segmentation , 2005, Comput. Graph. Forum.

[40]  Helmut Pottmann,et al.  Reassembling fractured objects by geometric matching , 2006, ACM Trans. Graph..

[41]  Tao Ju.  Robust repair of polygonal models , 2004, SIGGRAPH 2004.

[42]  Tao Ju,et al.  Robust repair of polygonal models , 2004, ACM Trans. Graph..

[43]  Chandrajit L. Bajaj,et al.  Anisotropic diffusion of surfaces and functions on surfaces , 2003, TOGS.

[44]  Shi-Min Hu,et al.  Principal curvatures from the integral invariant viewpoint , 2007, Comput. Aided Geom. Des..

[45]  Naokazu Yokoya,et al.  Range Image Segmentation Based on Differential Geometry: A Hybrid Approach , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[46]  Ronald N. Perry,et al.  Adaptively sampled distance fields: a general representation of shape for computer graphics , 2000, SIGGRAPH.

[47]  Anshuman Razdan,et al.  Curvature estimation scheme for triangle meshes using biquadratic Bézier patches , 2005, Comput. Aided Des..

[48]  K. Polthier.  Polyhedral Surfaces of Constant Mean Curvature , 2002 .

[49]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[50]  L. Santaló.  Integral geometry and geometric probability , 1976 .

[51]  Hongkai Zhao,et al.  A fast sweeping method for Eikonal equations , 2004, Math. Comput..

[52]  Alfred M. Bruckstein,et al.  Subpixel distance maps and weighted distance transforms , 1993, Optics & Photonics.

[53]  Greg Turk,et al.  Simplification and Repair of Polygonal Models Using Volumetric Techniques , 2003, IEEE Trans. Vis. Comput. Graph..

[54]  H. Piaggio.  Differential Geometry of Curves and Surfaces , 1952, Nature.

[55]  M. L. Connolly.  Measurement of protein surface shape by solid angles , 1986 .

[56]  Chang-Hun Kim,et al.  Finding ridges and valleys in a discrete surface using a modified MLS approximation , 2005, Comput. Aided Des..

[57]  Victoria Interrante,et al.  A novel cubic-order algorithm for approximating principal direction vectors , 2004, TOGS.

[58]  Helmut Pottmann,et al.  Registration of point cloud data from a geometric optimization perspective , 2004, SGP '04.

[59]  Alfred M. Bruckstein,et al.  Sub-pixel distance maps and weighted distance transforms , 1996, Journal of Mathematical Imaging and Vision.

[60]  David Cohen-Steiner,et al.  Restricted delaunay triangulations and normal cycle , 2003, SCG '03.

[61]  Szymon Rusinkiewicz,et al.  Estimating curvatures and their derivatives on triangle meshes , 2004, Proceedings. 2nd International Symposium on 3D Data Processing, Visualization and Transmission, 2004. 3DPVT 2004..