1. 1 Curvature Driven Flows Allen Tannenbaum

2. 2 Basic curve evolution: Invariant Flows  Planar curve:  General flow:  General geometric flow:

3. 3 Smoothing by classical heat flow  Linear (curve parameter p is independent of t)  Equivalent to Gaussian filtering  Unique linear scale-space  Non geometric  Shrinks the shape  Implementation problems

4. 4 Invariant differential geometry  For every Lie group we will consider, exists and invariant parametrization s, the group arc-length  For every such a group exists an invariant signature, the group curvature, k High curvature Low curvature Negative curvature

5. 5 What and why invariant Camera motion Deformation  Camera/object movement in the space  Transformations description (for “flat” objects): Euclidean  Motion parallel to the camera and planar projection Affine  Planar projection Projective

6. 6 Euclidean geometric heat flow  Use the Euclidean arc-length:  The deformation: Smoothly deforms to a circle (Gage-Hamilton, Grayson) Geometric smoothing Reduces length as fast as possible

7. 7 Affine geometric heat flow  Use the affine arc-length:  The flow:

8. 8 Affine geometric heat flow-(cont.) Theorem (Angenent-Sapiro-Tannenbaum): Let be a maximal classical solution of the affine heat flow. Then shrinks to an elliptically shaped point as. Equation also introduced by Alvarez, Guichard, Lions,and Morel in a viscosity framework.

9. 9 Affine geometric heat flow (cont.)  Nonconvex curve becomes convex and then deforms into an ellipse.  Decreases area as fast as possible (in an affine form)  Applications: Curvature computation for shape recognition: reduce noise Simplify curvature computation (Faugeras ‘95) Object recognition for robot manipulation (Cipolla ‘95)

10. 10 General invariant flows  Theorem: For every sub-group of the projective group the most general invariant curve deformation has the form  Theorem: In general dimensions, the most general invariant flow is given by u: graph locally representing the hypersurface g: invariant metric E(g): variational derivative of g I differential invariant

11. 11 From Curves to Smoothing Filters Embed initial curve as zero level set of surface: Want evolution of surface to track motion of curve as zero level set: For affine geometric heat equation this leads to filter: Here is interpreted as a gray-level image.

12. 12 Smoothing with Linear Heat Equation 256 by 256 MRI brain image smoothed by linear heat equation: t=2, 6, 32, 128

13. 13 Smoothing with Geometric Heat Equation Smoothing with kappa filter: t=0, 4, 16, 64, 256, 1024

14. 14 Smoothing with Affine Heat Equation-I Smoothing with kappa-shleesh: t=0, 16, 128, 1024

15. 15 Smoothing with Affine Heat Equation-II Magnification of original image and image after 256 iterations of kappa-shleesh filter.