Level Set Formulation for Curve Evolution

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## Level Set Formulation for Curve Evolution

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**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