1. Level Set Formulation for Curve Evolution Ron Kimmel www.cs.technion.ac.il/~ron Computer Science Department Technion-Israel Institute of Technology Geometric Image Processing Lab

2. Consider a closed planar curve The geometric trace of the curve can be alternatively represented implicitly as Implicit representation

3. Properties of level sets The level set normal Proof. Along the level sets we have zero change, that is, but by the chain rule So,

4. Properties of level sets The level set curvature Proof. zero change along the level sets,, also So,

5. Optical flow Problem: find the velocity also known as `optical flow’ It’s an `inverse’ problem, Given I(t) find

6. Aperture Problem

7. q`Normal’ vertical flow qHorizontal flow can not becomputed differentially.

8. Normal flow Due to the `aperture problem’ only the `normal’ velocity can be locally computed for the normal flow we have

9. Level Set Formulation implicit representation of C Then, Proof. By the chain rule Then, Recall that, and y x C(t) C(t) level set x y

10. Level Set Formulation qHandles changes in topology qNumeric grid points never collide or drift apart. qNatural philosophy for dealing with gray level images.

11. Numerical Considerations qFinite difference approximation. qOrder of approximation, truncation error, stencil. q(Differential) conservation laws. qEntropy condition and vanishing viscosity. qConsistent, monotone, upwind scheme. qCFL condition (stability examples)

12. Numerical Considerations Central derivative Forward derivative Backward derivative

13. Truncation Error Taylor expansion about x=ih Stencils

14. Numerical Approximations

15. Conservation Law Rate of change of the amount in a fixed domain G = Flux across the boundaries of G Differential conservation law

16. Generalized Solution 1D In 1D Weak solution satisfies

17. Hamilton-Jacobi In 1D: HJ=Hyperbolic conservation laws In 2D: just the `flavor’… Vanishing viscosity, of The `entropy condition’ selected the `weak solution’ that is the `vanishing viscosity solution’ also known as `entropy solution’.

18. Numerical Schemes Conservation form Numerical flux The scheme is monotone, if F is non-decreasing. Theorem: A monotone, consistent scheme, in conservation form converges to the entropy solution. Yet, up to 1 st order accurate ;-( …

19. Upwind Monotone Upwind scheme For we have upwind-monotone schemes we define Then, and the final scheme is

20. CFL Stability Condition At the limit For 3-point scheme of we need for the numerical domain of dependence to include the PDE domain of dependence domain of dependence domain of influence

21. CFL Stability Condition At the limit For 3-point scheme of we need for the numerical domain of dependence to include the PDE domain of dependence

22. 1D Example Solution Characteristics dx/dt=1 CFL condition Numeric scheme

23. 1D Example where Characteristics Numeric scheme CFL condition Numerical viscosity

24. 2D Example Numeric scheme CFL condition

25. 2D Examples Some flows require upwind/monotone schemes