Affine Invariant Flows in the Beltrami Framework

We analyze the role of different invariant principles in image processing and analysis. A distinction between the passive and active principles is emphasized, and the geometric Beltrami framework is shown to incorporate and explain some of the known invariant flows e.g. the equi-affine invariant flow for hypersurfaces. It is also demonstrated that the new concepts put forward in this framework enable us to suggest new invariants namely the case where the codimension is greater than one.

[1]  P. Olver,et al.  Affine Invariant Detection: Edge Maps, Anisotropic Diffusion, and Active Contours , 1999 .

[2]  Ron Kimmel,et al.  A general framework for low level vision , 1998, IEEE Trans. Image Process..

[3]  Guillermo Sapiro,et al.  Affine invariant detection: edges, active contours, and segments , 1996, Proceedings CVPR IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[4]  Alfred M. Bruckstein,et al.  On Projective Invariant Smoothing and Evolutions of Planar Curves and Polygons , 1997, Journal of Mathematical Imaging and Vision.

[5]  Luc Van Gool,et al.  An Extended Class of Scale-Invariant and Recursive Scale Space Filters , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[6]  A. Polyakov.  Quantum Geometry of Bosonic Strings , 1981 .

[7]  Yehoshua Y. Zeevi,et al.  Representation of colored images by manifolds embedded in higher dimensional non-Euclidean space , 1998, Proceedings 1998 International Conference on Image Processing. ICIP98 (Cat. No.98CB36269).

[8]  Ron Kimmel,et al.  From High Energy Physics to Low Level Vision , 1997, Scale-Space.

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

[10]  A. Bruckstein,et al.  Invariant signatures for planar shape recognition under partial occlusion , 1993 .

[11]  P. Lions,et al.  Axioms and fundamental equations of image processing , 1993 .

[12]  H. Pollaczek-Geiringer,et al.  W. Blaschke, Vorlesungen über Differentialgeometrie I. 2. Auflage (Grundlehren der math. Wiss. in Einzeldarstellungen, Bd. I). Verlag J. Springer, Berlin 1924 , 1925 .

[13]  Olivier Faugeras,et al.  Three-Dimensional Computer Vision , 1993 .

[14]  David A. Forsyth,et al.  Invariant Descriptors for 3D Object Recognition and Pose , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[15]  Guillermo Sapiro,et al.  Invariant Geometric Evolutions of Surfaces and Volumetric Smoothing , 1997, SIAM J. Appl. Math..

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

[17]  Steven Haker,et al.  Differential and Numerically Invariant Signature Curves Applied to Object Recognition , 1998, International Journal of Computer Vision.

[18]  O. Faugeras.  Three-dimensional computer vision: a geometric viewpoint , 1993 .

[19]  On projective plane curve evolution , 1996 .

[20]  Ron Kimmel,et al.  Images as Embedded Maps and Minimal Surfaces: Movies, Color, Texture, and Volumetric Medical Images , 2000, International Journal of Computer Vision.

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

[22]  Guillermo Sapiro,et al.  Affine invariant scale-space , 1993, International Journal of Computer Vision.