Jussi Tohka
Signal Processing Laboratory
Tampere University of Technology
A problem is well-posed if
Note that the exact shape of O is not known. That is, fitting rigid contour models (registration) is not possible. Also location and orientation of O in an image I is unknown.
As said object O does not usually have clear boundary, so straightforward approaches using edge detectors or other local operators would lead to trouble.
P2: Find the contour C_{O }that minimizes the energy function
where E_{image }is energy term that depends on underlying image, E_{internal} is the regularization term that depends only on the shape of the C, and is the regularization parameter that weights the two energy types.
Above formula captures the essence of deformable models that is quite the same than the idea of Bayesian Maximum A Posteriori (MAP) estimation. Indeed, deformable models can, with suitable assumptions, be put into a probabilistic framework and then it turns out that minimizing the energy of the contour is equivalent with finding the contour that maximizes certain a posteriori probability.
To actually do some computations we have to fix a representation for the arbitrary contour C. There are basically three types of representations available:
- Parametrized or explicit: C is defined as a parametrized curve c(s) = (x(s),y(s)), where s varies over a certain interval, say [0,1].
- Implicit: C is defined as trough level sets of implicit function i.e.
Discrete: C is just an ordered set of points, which are though to form a polygonal line.
Parametrized representation leads to `snakes' in their traditional formulation. They however rely on user interaction, and as such are not very suitable for fully automated target extraction.
Several methods have been used for minimization, for example greedy-type algorithms, dynamic programming, genetic algorithms, hopfield networks and simulated annealing with (when suitable) different kind of iteration techniques.