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

• 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

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