## Abstract

Based on modern invariant theory and symmetry groups, a high- level way of defining invariant geometric flows for a given Lie group is first described in this work. We then analyze in more detail different subgroups of the projective group, which are of special interest for computer vision. We classify the corresponding invariant flows and show that the geometric heat flow is the simplest possible one. Results on invariant geometric flows of surfaces are presented in this paper as well. We then show how the planar curve flow obtained for the affine group can be used for geometric smoothing of planar shapes and edge preserving enhancement of MRI. We conclude the paper with the presentation of other applications of geometric flows in image processing, which include segmentation, anisotropic diffusion of color images, and contrast normalization.

### Keywords

Manifold Shrinkage Eter Convolution Zucker## Preview

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### References

- [1]L. Alvarez, F. Guichard, P. L. Lions, and J. M. Morel, “Axioms and fundamental equations of image processing, ”
*Arch. Rational Mechanics***123:3**, September 1993.MathSciNetGoogle Scholar - [2]L. Alvarez, P. L. Lions, and J. M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion, ”
*SIAM J. Numer. Anal***29**, pp. 845–866, 1992.MathSciNetMATHCrossRefGoogle Scholar - [3]S. Angenent, “Parabolic equations for curves on surfaces, Part II. Intersections, blow-up, and generalized solutions, ”
*Annals of Mathematics***133**, pp. 171–215, 1991.MathSciNetMATHCrossRefGoogle Scholar - [4]S. Angenent, G. Sapiro, and A. Tannenbaum, “On the affine heat equation for nonconvex curves, ” TR, November 1994, submitted.Google Scholar
- [5]V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic snakes, ”
*HP Labs TR*, September 1994.Google Scholar - [6]W. Blaschke,
*Vorlesungen*ü*ber Differentialgeometrie II*, Verlag Von Julius Springer, Berlin, 1923.MATHGoogle Scholar - [7]Y. G. Chen, Y, Giga, and S. Goto, “Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, ”
*J. Differential Geometry***33**, pp. 749–786, 1991.MathSciNetMATHGoogle Scholar - [8]M. G. Crandall, H. Ishii, and P. L. Lions, “User’s guide to viscosity solutions of second order partial linear differential equations, ”
*Bulletin of the American Mathematical Society***27**, pp. 1–67, 1992.MathSciNetMATHCrossRefGoogle Scholar - [9]C. L. Epstein and M. Gage, “The curve shortening flow, ” in
*Wave Motion: Theory, Modeling, and Computation*, A. Chorin and A. Majda, Editors, Springer-Verlag, New York, 1987.Google Scholar - [10]L. C. Evans and J. Spruck, “ Motion of level sets by mean curvature, I, ”
*J. Differential Geometry***33**, pp. 635–681, 1991.MathSciNetMATHGoogle Scholar - [11]O. Faugeras, “ On the evolution of simple curves of the real projective plane, ”
*Comptes rendus de l*’*Acad. des Sciences de Paris***317**, pp. 565–570, September 1993.MathSciNetMATHGoogle Scholar - [12]M. Gage, “On an area-preserving evolution equation for plane curves, ”
*Contemporary Mathematics***51**, pp. 51–62, 1986.MathSciNetGoogle Scholar - [13]M. Gage and R. S. Hamilton, “The heat equation shrinking convex plane curves, ”
*J. Differential Geometry***23**, pp. 69–96, 1986.MathSciNetMATHGoogle Scholar - [14]M. Grayson, “The heat equation shrinks embedded plane curves to round points, ”
*J. Differential Geometry***26**, pp. 285–314, 1987.MathSciNetMATHGoogle Scholar - [15]H. W. Guggenheimer,
*Differential Geometry*, McGraw-Hill Book Company, New York, 1963.MATHGoogle Scholar - [16]B. K. P. Horn and E. J. Weldon, Jr., “ Filtering closed curves, ”
*IEEE Trans. Pattern Anal Machine Intell***8**, pp. 665–668, 1986.CrossRefGoogle Scholar - [17]B. B. Kimia, A. Tannenbaum, and S. W. Zucker, “Shapes, shocks, and deformations, I” to appear in
*International Journal of Computer Vision*.Google Scholar - [18]D. G. Lowe, “Organization of smooth image curves at multiple scales, ”
*International Journal of Computer Vision***3**, pp. 119–130, 1989.CrossRefGoogle Scholar - [19]J. L. Mundy and A. Zisserman (Eds.),
*Geometric Invariance in Computer Vision*, MIT Press, 1992.Google Scholar - [20]J. Oliensis, “Local reproducible smoothing without shrinkage, ”
*IEEE Trans. Pattern Anal. Machine Intell*.**15**, pp. 307–312, 1993.CrossRefGoogle Scholar - [21]P. J. Olver,
*Equivalence, Invariants, and Symmetry*, preliminary version of book, 1994.Google Scholar - [22]P. J. Olver,
*Applications of Lie Groups to Differential Equations*, Second Edition, Springer-Verlag, New York, 1993.MATHGoogle Scholar - [23]P. J. Olver, “ Differential invariants, ” to appear in
*Acta Appl. Math.*Google Scholar - [24]P. J. Olver, G. Sapiro, and A. Tannenbaum, “ Differential invariant signatures and flows in computer vision: A symmetry group approach, ” in [28].Google Scholar
- [25]P. J. Olver, G. Sapiro, and A. Tannenbaum, “Invariant geometric evolutions of surfaces and volumetric smoothing, ” MIT Technical Report — LIDS, April 1994. To appear in
*SIAM-JAM*.Google Scholar - [26]S. J. Osher and J. A. Sethian, “Fronts propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, ”
*Journal of Computational Physics***79**, pp. 12–49, 1988.MathSciNetMATHCrossRefGoogle Scholar - [27]P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion, ”
*IEEE Trans. Pattern Anal Machine Intell***12**, pp. 629–639, 1990.CrossRefGoogle Scholar - [28]B. Romeny, Editor,
*Geometry Driven Diffusion in Computer Vision*, Kluwer, 1994.MATHGoogle Scholar - [29]L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms, ”
*Physica D***60**, pp. 259–268, 1992.MATHCrossRefGoogle Scholar - [30]G. Sapiro, R. Kimmel, D. Shaked, B. B. Kimia, and A. M. Bruckstein, “Implementing continuous-scale morphology via curve evolution, ”
*Pattern Recognition***26:9**, 1993.CrossRefGoogle Scholar - [31]G. Sapiro and A. Tannenbaum, “On affine plane curve evolution, ”
*Journal of Functional Analysis***119:1**, pp. 79–120, January 1994.MathSciNetMATHCrossRefGoogle Scholar - [32]G. Sapiro and A. Tannenbaum, “AfSne invariant scale-space, ”
*International Journal of Computer Vision***11**, pp. 25–44, 1993.CrossRefGoogle Scholar - [33]G. Sapiro and A. Tannenbaum, “Image smoothing based on an affine invariant flow, ”
*Proceedings of the Conference on Information Sciences arid Systems*, Johns Hopkins University, pp. 196–201, March 1993.Google Scholar - [34]G. Sapiro and A. Tannenbaum, “Formulating invariant heat-type curve flows, ”
*Proceedings of the SPIE Conference on Geometric Methods in Computer Vision II*, San Diego, July 1993.Google Scholar - [35]G. Sapiro and A. Tannenbaum, “On invariant curve evolution and image analysis, ”
*Indiana Journal of Mathematics***42:3**, pp. 985–1009, 1993.MathSciNetMATHCrossRefGoogle Scholar - [36]G. Sapiro and A. Tannenbaum, “ Area and length preserving geometric invariant scale-spaces, ” MIT Technical Report - LIDS-2200 (accepted for publication in IEEE-PAMI). Also in
*Proc. ECCV ’*;*94*, Stockholm, May 1994.Google Scholar - [37]G. Sapiro, and A. Tannenbaum, “Edge preserving geometric enhance¬ment of MRI data, ” TR, University of Minnesota, EE Dept., April 1994.Google Scholar
- [38]G. Sapiro and V. Caselles, “Histogram modification via partial differential equations, ”
*HP Labs TR*, October 1994.Google Scholar - [39]G. Sapiro and D. Ringach, “ Anisotropic diffusion in color space, ”
*HP Labs TR*, November 1994.Google Scholar - [40]R. Schwartz, “The pentagram map, ”
*Exp. Math*11, pp. 71–81, 1992.Google Scholar - [41]A. P. Witkin, “ Scale-space filtering, ”
*Int. Joint Conf. AI*, pp. 1019–1021, 1983.Google Scholar - [42]S. Kichenassamy, A. Kumar, R Olver, A. Tannenbaum, and A. Yezzi, “Grandient flows and geometric active contours, ” 1994.Google Scholar
- [43]R. Malladi, J.A. Sethian, and B.C. Vemuri, “Shape modeling with front propagation: A level set approach, ”
*IEEE-PAMI*, to appear.Google Scholar