1. An Active contour Model without Edges
Tony Chan and Luminita Vese
Reviewed by Jie Zhang – 12/30/2005

2. Abstract
This paper Proposes a new model for active contours to detect objects in a given image based on techniques of curve evolution using mean curvature motion techniques, Mumford-Shah functional for segmentation and level sets.
In the level set formulation, the problem becomes a “mean-curvature flow”-like evolving the active contour, which will stop on the desired boundary.
The paper minimizes an energy which can be seen as a particular case of the so-called minimal partition problem.
3) The stopping term does not depend on the gradient of the image, as in the classical active contour models, but Is instead related to a particular segmentation of the image, which is based on Mumford-Shah techniques.
4) Finally, it presents various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable.

3. Basic Notation Let be a bounded and open subset of , with its boundary
Let be a given image, as a bounded function defined on and with real values.
Usually, is a rectangle in the plane and takes values between 0 and 255.
Denote by a piecewise parameterized curve.

4. Edge function in classical models
Usually, the positive and regular edge-function is , decreasing such that
where is the convolution of the image with the Gaussian
The function will be strictly positive in homogeneous regions, and near zero on the edges.

5. Problems of classical snakes or active contour models
All the classical snakes or active contour models rely on this edge-function g, depending on the gradient of the image, to stop the curve evolution.
The discrete gradients are bounded and then the stopping function g is never zero on the edges, and the curve may pass through the boundary.
If the image is noisy, then the isotropic smoothing Gaussian has to be strong, which will smooth the edges too.
We’d better obtain a model which can detect contours both with or without gradient, for instance objects with very smooth boundaries or even with discontinuous boundaries.

6. Description of the model - Notation
Let be the evolving curve. We denote by and two constants, representing the averages of “inside” and “outside” the curve .
Assume the image is formed by two regions of approximately piecewise-constant intensities, of distinct values and . Assume further that the object to be detected is represented by the region with the value and let denote his boundary by .
Then, inside the object (inside ):
outside the object (outside ):

7. Description of the model - Idea
Fitting energy:
is variable curve. We say that the boundary of the object is the minimizer of the fitting energy.
Figure for illustration:

8. Description of the model – Developed Model
Add some regularizing terms, the length of and/or the area inside .
where and are constant unknowns, and are fixed parameters
In this paper,
* the measured units of and can be referred in this paper. A a

9. Description of the model – Relation with Mumford-Shah
The Mumford-Shah functional for segmentation is:
A reduced form of Mumford-Shah is simply the restriction of to piecewise constant functions , i.e With a constant, on each connected component of
. Therefore,, the constants are in fact the averages of on each .
The reduced case is called the minimal partition problem.
This active contour model is a particular case of the minimal partition problem, in which:
This particular case can be formulated and solved using the level set method.

10. Description of the model – level set formulation
In level set method, an evolving curve is represented by the zero level set of a Lipschitz continuous function:
We choose to be positive inside and negative outside .

11. Description of the model -

12. Experimental results