# Part I: Evolving manifolds

## Why ? What ?

• Segmentation
• Energy minimization
• Euler-Lagrange
• Active contours
• Front-evolution (interfaces)
• Speed: different types
• Intrisic and local terms
• Regularization: mean curvature vector
• Growing (balloon): constant normal speed (for hypersurfaces)
• Underlying vector field
• To pull the manifold towards features of an image

## Issues

• Singularities
• Corners
• Branching
• Boundaries
• Representation of the manifold
• Triangulation (graph of particles) => nodal methods (snakes, ...)
• Implicit representation => level-set methods

## Level-set methods, codim 1 (hypersurfaces)

• Implicit representation :
• The manifold is the 0-level set of a smooth function u defined near the manifold: { x : u(x) = 0 }
• For instance, we can use a signed distance function (thm: such a function always exists for an oriented hypersurface without boudaries)
• The technique is the same for any dimension
• Explicit => implicit : initialization techniques
• Take a binary representation of the "interior" and smooth it
• Not a distance function
• Advance front with speed 1 and compute crossing times
• Chicken vs. Egg
• Solve the Eikonal equation |grad u|=1 and u(x)=0 on the manifold
• Fast-marching initialization
• Implicit => explicit : extracting techniques
• Marching hyper-cubes (ambiguities)
• Marching simplices
• Inside/outside
• The implicit representation is flexible and the topology of level-sets can change automatically
• Overlapping => merging
• Shrinking
• Translating equations
• Ignore tangential speed (this corresponds to changing parametrization)
• The equation dC/dt = F N (F speed, N normal vector) becomes du/dt = - F |grad u|
• An example : translation
• speed: F = N.d (d constant and uniform vector)
• equation: du/dt = - (grad u).d
• solution: u(t,x) = u0(x-td)
• numerical simulation
• Problems
• Ignoring tangential speed
• Numerical problems
• Manifolds with boundary
• How to solve the equation where u is not differentiable
• This happens when merging or splitting the manifold
• For some applications, the interface model is not flexible enough
• The distinction between interior and exterior does not necessarily make sense
• We may want to handle overlapping without merging
• Computation time, memory storage
• Need to choose a grid resolution : difficult to change it during the evolution

## Level-set methods for arbitrary codimension

Let n be the dimension of the space, d the dimension of the manifold we want to evolve and k its codimension (d+k=n).
• Problem: the distance function to the manifold is not smooth on the manifold
• Idea: consider "tubes" or oriented iso-surfaces around the manifold (that is, level sets of the distance function) and evolve them using an extension of the speed function F.
• Usually, F is the sum of two terms
• an "exterior" speed: there is no problem to extend it
• a mean curvature speed: we can't just take the mean curvature of iso-surfaces because the (k-1) largest curvatures correspond to the bending of the tubes; the solution is the consider only the d smallest principal curvatures
• Putting this in the level-set framework gives an equation for an implicit representation u of the manifold, which is not defined on this manifold
• Theoretical solution (Ambrosio and Soner): use the viscosity solution theory to give sense to the equation (that is, to define what are the solutions)
• Problem: how to compute viscosity solution ?
• Implementation solution: the epsilon-level set method
• Choose an iso-surface (the epsilon level-set) and evolve it as a codim 1 manifold, but with the extension of the speed defined above
• Issue: choice of epsilon
• The signed distance function to this iso-surface is not defined on the initial manifold, so epsilon must be large enough to have a band around the iso-surface where the equation is well defined is the usual sense.
• epsilon must be small enough to localize the actual manifold

# Part II: Mean curvature flow of a circle in the space

It would be useful to have explicit solutions to test algorithms and adjust parameters

## Surface of revolution evolving under smallest curvature flow

• The equation
• Principal curvatures of the surface of revolution
• One corresponds to the curvature of the generating curve
• The other one corresponds to the revolution
• Evolution when the smallest curvature is the one we want
• Trick: add a tangential speed
• Miracle 1: we have the explicit solution in this case
• Miracle 2: if we are in this case at t=0, then this solution is correct for any t
• What happens when points reach the axis
• Miracle 3: the explicit solution is correct even after

## Evolution of the torus

• The whole scenario
• The interior pinches and the hole shrinks
• It disappears and the topology changes
• The surface evolves to look like a sphere
• Then it shrinks and disappears
• Description of the skeleton
• We have the viscosity solution for all t

## Issues

• The shape of the "tube" around the manifold is not really a tube any more
• There is an infinite curvature when the hole disappears
• How to simulate numerically the changement of topology
• The skeleton of the "tube" doesn't give precise informations about the position of the manifold

Alain Frisch