Dynamic Shapes of Arbitrary Dimension: The Vector Distance Functions

We present a novel method for representing and evolving objects of arbitrary dimension. The method, called the Vector Distance Function (VDF) method, uses the vector that connects any point in space to its closest point on the object. It can deal with smooth manifolds with and without boundaries and with shapes of different dimensions. It can be used to evolve such objects according to a variety of motions, including mean curvature. If discontinuous velocity fields are allowed the dimension of the objects can change. The evolution method that we propose guarantees that we stay in the class of VDFs and therefore that the intrinsic properties of the underlying shapes such as their dimension, curvatures can be read off easily from the VDF and its spatial derivatives at each time instant. The main disadvantage of the method is its redundancy: the size of the representation is always that of the ambient space even though the object we are representing may be of a much lower dimension. This disadvantage is also one of its strengths since it buys us flexibility.

[1]  Johan Montagnat,et al.  Representation, shape, topology and evolution of deformable surfaces. Application to 3D medical imag , 2000 .

[2]  O. Faugeras,et al.  Variational principles, surface evolution, PDE's, level set methods and the stereo problem , 1998, 5th IEEE EMBS International Summer School on Biomedical Imaging, 2002..

[3]  Benjamin B. Kimia,et al.  Image segmentation by reaction-diffusion bubbles , 1995, Proceedings of IEEE International Conference on Computer Vision.

[4]  G. Barles.  Solutions de viscosité des équations de Hamilton-Jacobi , 1994 .

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

[6]  T. Chan,et al.  A Variational Level Set Approach to Multiphase Motion , 1996 .

[7]  O. Faugeras,et al.  Representing and Evolving Smooth Manifolds of Arbitrary Dimension Embedded in Rn as the Intersection of n Hypersurfaces : The Vector Distance Functions , 1999 .

[8]  Guillermo Sapiro,et al.  3D active contours , 1996 .

[9]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

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

[11]  Yun-Gang Chen,et al.  Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations , 1989 .

[12]  James A. Sethian,et al.  Theory, algorithms, and applications of level set methods for propagating interfaces , 1996, Acta Numerica.

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

[14]  Olivier Faugeras,et al.  Shape Representation as the Intersection of n-k Hypersurfaces , 2000 .

[15]  Olivier D. Faugeras,et al.  Variational principles, surface evolution, PDEs, level set methods, and the stereo problem , 1998, IEEE Trans. Image Process..

[16]  Guillermo Sapiro,et al.  Minimal Surfaces Based Object Segmentation , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

[17]  J. Sethian.  Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws , 1990 .

[18]  Guillermo Sapiro,et al.  Region tracking on level-sets methods , 1999, IEEE Transactions on Medical Imaging.

[19]  Rachid Deriche,et al.  Geodesic Active Contours and Level Sets for the Detection and Tracking of Moving Objects , 2000, IEEE Trans. Pattern Anal. Mach. Intell..

[20]  V. Caselles,et al.  Snakes in Movement , 1996 .

[21]  Jan J. Koenderink,et al.  Solid shape , 1990 .

[22]  Andrew J. Majda,et al.  Wave Motion: Theory, Modelling, and Computation , 1987 .

[23]  Olivier D. Faugeras,et al.  Reconciling Distance Functions and Level Sets , 1999, Scale-Space.

[24]  Roberto Cipolla,et al.  The visual motion of curves and surfaces , 1998, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

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

[26]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

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

[28]  G. Barles,et al.  Front propagation and phase field theory , 1993 .

[29]  Michael Isard,et al.  Active Contours , 2000, Springer London.

[30]  Olivier Faugeras,et al.  Reconciling Distance Functions and Level Sets , 2000, J. Vis. Commun. Image Represent..

[31]  Jack Xin,et al.  Diffusion-Generated Motion by Mean Curvature for Filaments , 2001, J. Nonlinear Sci..

[32]  J. A. Sethian,et al.  HYPERSURFACES MOVING WITH CURVATURE-DEPENDENT SPEED: HAMILTON-JACOBI EQUATIONS, CONSERVATION LAWS AND NUMERICAL ALGORITHMS , 1989 .

[33]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

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

[35]  M. Gage,et al.  The Curve Shortening Flow , 1987 .

[36]  Demetri Terzopoulos,et al.  Constraints on Deformable Models: Recovering 3D Shape and Nonrigid Motion , 1988, Artif. Intell..

[37]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[38]  Olivier D. Faugeras,et al.  Co-dimension 2 Geodesic Active Contours for MRA Segmentation , 1999, IPMI.

[39]  H. Soner,et al.  Level set approach to mean curvature flow in arbitrary codimension , 1996 .