Overview of Propagating Interfaces Donald Tanguay October 30, 2002

Outline Propagating Interfaces Example: motion under curvature Functional Formulation Parametric Formulation Level Set Formulation Relationship to Snakes

Interfaces An interface (or front) is a boundary between two regions: “inside” and “outside.” In 2-D, an interface is a simple closed curve:

Propagating Interfaces How does an interface evolve over time? At a specific moment, the speed function F (L, G, I) describes the motion of the interface in the normal direction.

Propagating Interfaces Local – depend on local geometric information (e.g., curvature and normal direction) Global – depend on the shape and position of the front (e.g., integrals along the front, heat diffusion) Independent – do not depend on the shape of the front (e.g., an underlying fluid velocity that passively transports the front) Speed F(L,G,I) is a function of 3 types of properties:

Motion Under Curvature Example: Motion by curvature. Each piece moves perpendicular to the curve with speed proportional to the local curvature. large positive motion small negative motion

Motion Under Curvature Curvature κ is the inverse of the radius r of the osculating circle.

Motion Under Curvature

Functional Representation Eulerian framework: define fixed coordinate system on the world. For every world point x, there is (at most) one value y = f t (x). Falling snow example:

Functional Representation However, many simple shapes are multivalued; they are not functions regardless of the orientation of the coordinate system.

Parametric Representation Spatially parameterize the curve x by s so that at time t the curve is x t (s), where 0  s  S and the curve is closed: x t (0) = x t (S). Points on initial curve. Gradient (wrt time) is the speed in normal direction. Normal is perpendicular to curve, as is curvature.

Parametric Representation For motion under curvature, speed F depends only on local curvature κ – the equation of motion is thus: where curvature is and the normal is

Particle Methods In order to compute, discretize the parameterization into moving particles which reconstruct the front. Known under a variety of names: marker particle techniques, string methods, nodal methods. = # mesh particles = time step = parameterization step = location of point iΔs at time nΔt ΔsΔs

Particle Methods To construct a numerical algorithm, the derivatives are approximated as central differences based on the Taylor series:

Particle Methods

Numerical Instability Because Δs has dropped from Eq. 4.4, as neighboring particles move closer together the quotient approaches 0/0, which is numerically unstable! Uncontrollable oscillations stem from a feedback loop: 1.Small errors in particle positions produce 2.Local variations in approximated derivatives leading to 3.Variations in computed particle velocities causing 4.Uneven advancement of particles, yielding 5.Larger errors in particle positions.

Numerical Instability Decreasing time steps overcomes the instability but at additional computational cost. Varying the time step affects the final solution to an example problem: BADBETTERBEST

Numerical Instability Typical, unappealing remedies that alter the motion equations in non-obvious ways: “Smooth” the speed function to keep particles apart Redistribute particles periodically Introduce a filter to remove the oscillations in particle positions

Changing Topology Example: two fires merge into a single fire.

Changing Topology Buoys! In particle methods: Difficult (and expensive) to detect and change the particle chains Much more difficult as dimensionality increases

Difficulties With Particle Methods Instability Local singularities Management of particles: remove, redistribute, connect

Level Set Formulation Recast problem with one additional dimension – the distance from the interface.

Changing Topology

Level Set Formulation The interface always lies at the zeroth level set of the function , i.e., the interface is defined by the implicit equation  t (x, y) = 0.

Initial Value Formulation Define F as speed in normal direction: A particle on the front with path x(t) is on the zero level set: (1) by chain rule: (2) (1) into (2): which is the continuous level set equation.

Simple Computational Scheme h = spacing of uniform mesh t = time step (i, j) = grid nodes Discrete grid in x-y domain Finite difference approximations to derivatives The discrete level set equation:

Behavior at Singularities Naive formulation has bad behavior at “corner” singularities. BAD GOOD

Viscosity Solutions The straightforward speed function (e.g., F = 1) causes swallowtail. Add a little curvature term (viscosity), and the curve is mathemagically well-behaved!  -ly small

Level Set Benefits Straightforward in higher dimensions Topological changes are natural Accurate computational schemes exist Intrinsic geometric properties are easy to determine (e.g., normal, curvature) Adaptive computational strategies improve efficiency

Functional explicit: –Inadequate model for most problems Parametric: –Adequate model, but –Numerical instabilities –Explicit topology makes changes difficult Level Set, functional implicit: –Adequate model, and –Naturally handles changing topology, but –Slow; needs efficiency improvements Summary

Level sets is a general framework for accurately advancing an interface. Much of the challenge in particular interface problems is producing an adequate model of F.

Relation to Snakes Snakes (active contours) have evolved greatly from the initial 1988 paper of Kass, Witkin, and Terzopoulos. The term “snake” is now overloaded, and apparently the difference from level sets is often exaggerated. Two characteristics of snake formulations: Neighbors connected into a chain Additional “rod” or “spring” term in energy function attempts to keep neighbors apart to prevent degenerate chains

Further Topics Level Set implementation Narrow Band Technique – optimization for the initial value formulation Fast Marching Methods – optimization for boundary value formulation, F > 0 Art of Designing F Many example applications

References M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active Contour Models,” IJCV, Jan. 1988, pp J. A. Sethian. Level Set Methods and Fast Marching Methods. Cambridge University Press, J. A. Sethian, “Tracking Interfaces with Level Sets,” American Scientist, May-June 1997.