A Level Set Formulation of Geodesic Curvature Flow on Simplicial Surfaces

Curvature flow (planar geometric heat flow) has been extensively applied to image processing, computer vision, and material science. To extend the numerical schemes and algorithms of this flow on surfaces is very significant for corresponding motions of curves and images defined on surfaces. In this work, we are interested in the geodesic curvature flow over triangulated surfaces using a level set formulation. First, we present the geodesic curvature flow equation on general smooth manifolds based on an energy minimization of curves. The equation is then discretized by a semi-implicit finite volume method (FVM). For convenience of description, we call the discretized geodesic curvature flow as dGCF. The existence and uniqueness of dGCF are discussed. The regularization behavior of dGCF is also studied. Finally, we apply our dGCF to three problems: the closed-curve evolution on manifolds, the discrete scale-space construction, and the edge detection of images painted on triangulated surfaces. Our method works for compact triangular meshes of arbitrary geometry and topology, as long as there are no degenerate triangles. The implementation of the method is also simple.

[1]  Roman Goldenberg,et al.  Fast Geodesic Active Contours , 1999, Scale-Space.

[2]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[3]  Ron Kimmel,et al.  Geometric curve flows on parametric manifolds , 2007, J. Comput. Phys..

[4]  J. Sethian,et al.  FRONTS PROPAGATING WITH CURVATURE DEPENDENT SPEED: ALGORITHMS BASED ON HAMILTON-JACOB1 FORMULATIONS , 2003 .

[5]  M. Gage.  Curve shortening makes convex curves circular , 1984 .

[6]  J. Sponring.  The entropy of scale-space , 1996, Proceedings of 13th International Conference on Pattern Recognition.

[7]  Luc Florack,et al.  Understanding and Modeling the Evolution of Critical Points under Gaussian Blurring , 2002, ECCV.

[8]  L. Evans,et al.  Motion of level sets by mean curvature. II , 1992 .

[9]  Andrew P. Witkin,et al.  Uniqueness of the Gaussian Kernel for Scale-Space Filtering , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[10]  Baba C. Vemuri,et al.  Shape Modeling with Front Propagation: A Level Set Approach , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[11]  Joachim Weickert,et al.  A semidiscrete nonlinear scale-space theory and its relation to the Perona - Malik paradox , 1996, TFCV.

[12]  Tony Lindeberg,et al.  Scale-space behaviour of local extrema and blobs , 1992, Journal of Mathematical Imaging and Vision.

[13]  Andrew P. Witkin,et al.  Scale-space filtering: A new approach to multi-scale description , 1984, ICASSP.

[14]  James A. Sethian,et al.  Flow under Curvature: Singularity Formation, Minimal Surfaces, and Geodesics , 1993, Exp. Math..

[15]  Luc Florack,et al.  The Topological Structure of Scale-Space Images , 2000, Journal of Mathematical Imaging and Vision.

[16]  J. Sethian,et al.  Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains , 1998 .

[17]  J. Weickert.  Nonlinear diffusion scale-spaces: From the continuous to the discrete setting , 1996 .

[18]  Kaleem Siddiqi,et al.  Geometric heat equation and nonlinear diffusion of shapes and images , 1994, 1994 Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

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

[20]  Ron Kimmel,et al.  Intrinsic Scale Space for Images on Surfaces: The Geodesic Curvature Flow , 1997, CVGIP Graph. Model. Image Process..

[21]  M. Grayson.  The heat equation shrinks embedded plane curves to round points , 1987 .

[22]  K. Mikula,et al.  Evolution of curves on a surface driven by the geodesic curvature and external force , 2006 .

[23]  Tony Lindeberg,et al.  Scale-Space Theory in Computer Vision , 1993, Lecture Notes in Computer Science.

[24]  M. Gage,et al.  The heat equation shrinking convex plane curves , 1986 .

[25]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[26]  Tony Lindeberg,et al.  On the Axiomatic Foundations of Linear Scale-Space , 1997, Gaussian Scale-Space Theory.

[27]  T. O’Neil.  Geometric Measure Theory , 2002 .

[28]  Jiansong Deng,et al.  Scale-Space Analysis of Discrete Filtering over Arbitrary Triangulated Surfaces , 2009, SIAM J. Imaging Sci..

[29]  J. Weickert.  Scale-Space Properties of Nonlinear Diffusion Filtering with a Diffusion Tensor , 1994 .

[30]  Steven J. Gortler,et al.  Fast exact and approximate geodesics on meshes , 2005, ACM Trans. Graph..

[31]  Andrew P. Witkin,et al.  Scale-Space Filtering , 1983, IJCAI.

[32]  Tony Lindeberg,et al.  Scale-space theory : A framework for handling image structures at multiple scales , 1996 .

[33]  M. Grayson.  Shortening embedded curves , 1989 .

[34]  S. Osher,et al.  Motion of curves constrained on surfaces using a level-set approach , 2002 .

[35]  Guillermo Sapiro,et al.  Geodesic Active Contours , 1995, International Journal of Computer Vision.

[36]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[37]  Alan L. Yuille,et al.  Scaling Theorems for Zero Crossings , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[38]  Jon Sporring,et al.  The entropy of scale-space , 1996, ICPR.

[39]  Luiz Velho,et al.  Discrete scale spaces via heat equation , 2001, Proceedings XIV Brazilian Symposium on Computer Graphics and Image Processing.

[40]  Tony Lindeberg Kth.  Scale-space: A framework for handling image structures at multiple scales , 1996 .

[41]  J. Koenderink.  The structure of images , 2004, Biological Cybernetics.

[42]  Achim Hummel,et al.  Representations Based on Zero-Crossing in Scale-Space-M , 2018, CVPR 1986.

[43]  M. Gage,et al.  Curve shortening on surfaces , 1990 .

[44]  Dezhong Chen,et al.  Curve Shortening Flow in a Riemannian Manifold , 2003, math/0312463.

[45]  Robert Hummel,et al.  Reconstructions from zero crossings in scale space , 1989, IEEE Trans. Acoust. Speech Signal Process..

[46]  Jiansong Deng,et al.  Diffusion Equations over Arbitrary Triangulated Surfaces for Filtering and Texture Applications , 2008, IEEE Transactions on Visualization and Computer Graphics.

[47]  Max A. Viergever,et al.  Scale Space Hierarchy , 2003, Journal of Mathematical Imaging and Vision.

[48]  Demetri Terzopoulos,et al.  Snakes: Active contour models , 2004, International Journal of Computer Vision.