Affine invariant detection: edges, active contours, and segments

In this paper we undertake a systematic investigation of affine invariant object detection. Edge detection is first presented from the point of view of the affine invariant scale-space obtained by curvature based motion of the image level-sets. In this case, affine invariant edges are obtained as a weighted difference of images at different scales. We then introduce the affine gradient as the simplest possible affine invariant differential function which has the same qualitative behavior as the Euclidean gradient magnitude. These edge detectors are the basis both to extend the affine invariant scale-space to a complete affine flow for image denoising and simplification, and to define affine invariant active contours for object detection and edge integration. The active contours are obtained as a gradient flow in a conformally Euclidean space defined by the image on which the object is to be detected. That is, we show that objects can be segmented in an affine invariant manner by computing a path of minimal weighted affine distance, the weight being given by functions of the affine edge detectors. The geodesic path is computed via an algorithm which allows to simultaneously detect any number of objects independently of the initial curve topology.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  J. Davenport.  Editor , 1960 .

[3]  F. Bookstein.  Fitting conic sections to scattered data , 1979 .

[4]  J. Sethian.  Curvature and the evolution of fronts , 1985 .

[5]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[6]  Tomaso A. Poggio,et al.  On Edge Detection , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

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

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

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

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

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

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

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

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

[15]  S. Zucker,et al.  Toward a computational theory of shape: an overview , 1990, eccv 1990.

[16]  L. Rudin,et al.  Feature-oriented image enhancement using shock filters , 1990 .

[17]  S. Angenent.  Parabolic equations for curves on surfaces Part II. Intersections, blow-up and generalized solutions , 1991 .

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

[19]  Laurent D. Cohen,et al.  On active contour models and balloons , 1991, CVGIP Image Underst..

[20]  Laurent D. Cohen,et al.  Using Deformable Surfaces to Segment 3-D Images and Infer Differential Structures , 1992, ECCV.

[21]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

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

[23]  P. Lions,et al.  Image selective smoothing and edge detection by nonlinear diffusion. II , 1992 .

[24]  H. Soner.  MOTION OF A SET BY THE CURVATURE OF ITS BOUNDARY , 1993 .

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

[26]  Richard Szeliski,et al.  Modeling surfaces of arbitrary topology with dynamic particles , 1993, Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[27]  Guillermo Sapiro,et al.  Implementing continuous-scale morphology via curve evolution , 1993, Pattern Recognit..

[28]  D. Chopp.  Computing Minimal Surfaces via Level Set Curvature Flow , 1993 .

[29]  V. Caselles,et al.  A geometric model for active contours in image processing , 1993 .

[30]  Olivier D. Faugeras,et al.  Cartan's Moving Frame Method and Its Application to the Geometry and Evolution of Curves in the Euclidean, Affine and Projective Planes , 1993, Applications of Invariance in Computer Vision.

[31]  M. Gurtin.  Thermomechanics of Evolving Phase Boundaries in the Plane , 1993 .

[32]  Baba C. Vemuri,et al.  Evolutionary Fronts for Topology-Independent Shape Modeling and Recoveery , 1994, ECCV.

[33]  Guillermo Sapiro,et al.  Experiments on geometric image enhancement , 1994, Proceedings of 1st International Conference on Image Processing.

[34]  Ross T. Whitaker,et al.  Volumetric deformable models: active blobs , 1994, Other Conferences.

[35]  Guillermo Sapiro,et al.  Differential Invariant Signatures and Flows in Computer Vision: A Symmetry Group Approach , 1994, Geometry-Driven Diffusion in Computer Vision.

[36]  G. Sapiro,et al.  On affine plane curve evolution , 1994 .

[37]  Tai Sing Lee,et al.  Region competition: unifying snakes, region growing, energy/Bayes/MDL for multi-band image segmentation , 1995, Proceedings of IEEE International Conference on Computer Vision.

[38]  Luc Van Gool,et al.  Shape-extraction for curves using geometry-driven diffusion and functional optimization , 1995, Proceedings of IEEE International Conference on Computer Vision.

[39]  Guillermo Sapiro,et al.  Area and Length Preserving Geometric Invariant Scale-Spaces , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[40]  Emanuele Trucco,et al.  Geometric Invariance in Computer Vision , 1995 .

[41]  Alfred M. Bruckstein,et al.  Finding Shortest Paths on Surfaces Using Level Sets Propagation , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[42]  P. Olver.  Equivalence, Invariants, and Symmetry: References , 1995 .

[43]  P. Olver,et al.  Affine invariant edge maps and active contours , 1995 .

[44]  Anthony J. Yezzi,et al.  Gradient flows and geometric active contour models , 1995, Proceedings of IEEE International Conference on Computer Vision.

[45]  Demetri Terzopoulos,et al.  Topologically adaptable snakes , 1995, Proceedings of IEEE International Conference on Computer Vision.

[46]  P. Olver.  A ne Invariant Edge Maps and Active Contours , 1995 .

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

[48]  R. Keriven,et al.  Scale-spaces and a ne curvatureOlivier , 1995 .

[49]  Ross T. Whitaker,et al.  Algorithms for implicit deformable models , 1995, Proceedings of IEEE International Conference on Computer Vision.

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

[51]  P. Olver,et al.  Conformal curvature flows: From phase transitions to active vision , 1996, ICCV 1995.

[52]  P. Olver,et al.  Affine Geometry, Curve Flows, and Invariant Numerical Approximations , 1996 .

[53]  Vicent Caselles,et al.  Affine Invariant Segmentation by Variational Method , 1996, SIAM J. Appl. Math..

[54]  T. Lindeberg,et al.  Shape-adapted smoothing in estimation of 3-D depth cues from affine distortions of local 2-D structure , 1997 .

[55]  J. Vickers.  EQUIVALENCE, INVARIANTS AND SYMMETRY , 1997 .

[56]  R. Kimmel,et al.  Minimal surfaces: a geometric three dimensional segmentation approach , 1997 .

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

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