- 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

- Intrisic and local terms

- Singularities
- Corners
- Branching
- Boundaries

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

- 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

- Take a binary representation of the "interior" and smooth it
- 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 d
**C**/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-t
**d**) - numerical simulation

- speed: F =
- 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

- Ignoring tangential speed

- 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

- The signed distance function to this
iso-surface is not defined on the initial manifold,
so epsilon must be

- 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

- Trick:
- What happens when points reach the axis
- Miracle 3: the explicit solution is correct even after

- 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

- 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

Last modified: Thu Dec 2 11:53:45 EST 1999