Computer Science Department • Technion-Israel Institute of Technology Level Set Formulationfor Curve Evolution Ron Kimmel www.cs.technion.ac.il/~ron • Geometric Image Processing Lab
Implicit representation Consider a closed planar curve The geometric trace of the curve can be alternatively represented implicitly as
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,
Properties of level sets The level set curvature Proof.zero changealong the level sets, , also So,
Optical flow Problem: find the velocity also known as `optical flow’ It’s an `inverse’ problem, Given I(t) find
Aperture Problem • `Normal’ vertical flow • Horizontal flow can not be computed differentially.
Normal flow Due to the `aperture problem’ only the `normal’ velocity can be locally computed for the normal flow we have Image analysis
Level Set Formulation y C(t) implicit representation of C Then, Proof. By the chain rule Then, Recall that , and x y C(t) level set x Image synthesis
Level Set Formulation • Handles changes in topology • Numeric grid points never collide or drift apart. • Natural philosophy for dealing with gray level images.
Numerical Considerations • Finite difference approximation. • Order of approximation, truncation error, stencil. • (Differential) conservation laws. • Entropy condition and vanishing viscosity. • Consistent, monotone, upwind scheme. • CFL condition (stability examples)
Numerical Considerations Central derivative Forward derivative Backward derivative
Truncation Error Taylor expansion about x=ih Stencils
Conservation Law Rate of change of the amount in a fixed domain G = Flux across the boundaries of G Differential conservation law
Generalized Solution 1D In 1D Weak solution satisfies
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’.
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 1st order accurate ;-( …
Upwind Monotone Upwind scheme For we have upwind-monotone schemes we define Then, and the final scheme is
domain of influence domain of dependence 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
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
1D Example Solution Characteristics dx/dt=1 CFL condition Numeric scheme
Numerical viscosity 1D Example where Characteristics Numeric scheme CFL condition
2D Example Numeric scheme CFL condition
2D Examples require upwind/monotone schemes Some flows