1 PDE Methods are Not Necessarily Level Set Methods Allen Tannenbaum Georgia Institute of Technology Emory University

2 PDE Methods in Computer Vision and Imaging  Image Enhancement  Segmentation  Edge Detection  Shape-from-Shading  Object Recognition  Shape Theory  Optical Flow  Visual Tracking  Registration

3 Scale in Biological Systems

4 Micro/Macro Models-Scale I

5 Micro/Macro Models-Scale II

6 How to Move Curves and Surfaces  Parameterized Objects: methods dominate control and visual tracking; ideal for filtering and state space techniques.  Level Sets: implicitly defined curves and surfaces. Several compromises; narrow banding, fast marching.  Minimize Directly Energy Functional: conjugate gradient on triangulated surface (Ken Brakke).

7 Level Sets-A History Independently: Peter Olver (1976), Ph.D. thesis Sigurd Angenent (Leiden University Report, 1982) Mathematical Justification: Chen-Giga-Goto (1991) Evans and Spruck (1991)

8 When Do They Work

9 Parameterized Curve Description infinite dimensional parameterization for derivations only, evolution should be geometric

10 Generic Curve Evolution The closed curve C evolves according to moves “particles” along the curve influences the curve’s shape How is the speed determined?

11 Classification of Curve Evolutions

12 Classification of Curve Evolutions Kass, Witkin, Terzopoulos, "Snakes: Active Contour Models," International Journal of Computer Vision, pp , 1988.

13 Classification of Curve Evolutions Terzopoulos, Szeliski, Active Vision, chapter Tracking with Kalman Snakes, pp. 3-20, MIT Press, 1992.

14 Classification of Curve Evolutions Kichenassamy, Kumar, Olver, Tannenbaum, Yezzi, "Conformal curvature flows: From phase transitions to active vision," Archive for Rational Mechanics and Analysis, vol. 134, no. 3, pp , Caselles, Kimmel, Sapiro, "Geodesic active contours," International Journal of Computer Vision, vol. 22, no. 1, pp , 1997.

15 Classification of Curve Evolutions

16 Static Approaches Kass snake (parametric) Geodesic active contour (geometric) using the functionals Minimize

17 leads to the Euler-Lagrange equations Minimizing Static Approaches Kass snake (parametric) Geodesic active contour (geometric)

18 Static Approaches Kass snake (parametric) Geodesic active contour (geometric) results in the gradient descent flow Minimizing is an artificial time parameter

19 Dynamic Approach Minimize the action integral whereis the Lagrangian, is the kinetic energy andis the potential energy.

20 Dynamic Approach using the functional Minimizing Terzopoulos and Szeliski (parametric)

21 results in the Euler-Lagrange equation Terzopoulos and Szeliski (parametric) Minimizing Dynamic Approach Here, is physical time

22 Dynamic Approach But what about a geometric formulation? results in the Euler-Lagrange equation Terzopoulos and Szeliski (parametric) Minimizing

23 Geometric Dynamic Approach Minimize using the Lagrangian results in the Euler-Lagrange equation

24 Geometric Dynamic Approach We can write We then obtain the following two coupled PDEs for the tangential and the normal velocities: The tangential velocity matters.

25 PDE’s Without Level Sets: Some Examples

26 Cortical Surface Flattening-Normal Brain

27 White Matter Segmentation and Flattening

28 Conformal Mapping of Neonate Cortex

29 Surface Warping-Area Preserving

30 Flame Morphing

31 Anisotropic active contours Add directionality

32 Curve minimization  Calculus of variations Start with initial curve Deform to minimize energy Steady state is locally optimum  Dynamic programming Choose seed point s For any point t, determine globally optimal curve t  s Registration, Atlas-based segmentation Segmentation

33 Synthetic example (3D)

34 Stochastic Approximations

35 Curvature Driven Flows

36 Euclidean and Affine Flows

37 Euclidean and Affine Flows

38 Birth/Death Zero Range Processes-I  S: discrete torus T N, W=N  Particle configuration space: N T N  Markov generator:

39 Birth/Death Zero Range Processes-II  Markov generator:

40 Birth/Death Zero Range Process-III  Markov generator:  Each particle configuration defines a positive measure on the unit circle:  To make the curve zero barycenter, a corrected measure is used:  Reconstruct the curve with:

41 The Tangential Component is Important

42 Nonconvex Curves

43 Stochastic Interpretation-I

44 Stochastic Interpretation-II

45 Stochastic Interpretation-III

46 Stochastic Curve Shortening

47 Conclusions  Level sets are a way of implementing curvature driven flows.  Loss of information.  Modifications are necessary.  Do not work if no maximum principle.  Combination with other methods, e.g. Bayesian.