Energy functional in Geodesic Active Contours

Question

I have read some papers about Geometric active contours of the author C.Gout and Le Guyader

[1] Segmentation under geometrical conditions using geodesic active contours and interpolation using level set methods

[2] Geodesic active contours under geometrical conditions: Theory and 3D applications

and I found out something called Energy functional to minimized, define as follow $E(\Gamma) = \int_{\Gamma} dg(|\nabla I|)ds$

where $d$ is a distance function and $I$ is the intial image, with $\Gamma$ is the initial contour.

Now I know the meaning of energy functional in Active contours, but I don't know where the formula above come from and how to derive it ?

Please help me by giving some detail or references to get to know clearly that energy functional. Thanks in advance !

Answer 1

Such energy functionals are not derived, they are proposed. One comes up with a model, guided by intuition and prior experience, then tests it experimentally... if the model shows promise, one puts some theory around it to make the thing publishable.

Section 2 of the first paper to which you linked explains the origin of the functional quite well. We want to find an "edge" in the given image that lies close to a few marked points. So, we need a curve $\Gamma$ such that

1. the gradient of brightness is large on $\Gamma$
2. $\Gamma$ stays near the marked points

So, one needs a functional that penalizes deviations from 1 and 2.

1. $g(|\nabla I|)$, where $g $ is decreasing, is a penalty for small gradient
2. $d$, the distance to marked points, is penalty for going far from them

The authors chose to use the product of penalties, $ d\cdot g(|\nabla I|) $, so that one being zero offsets the other. One could conceivably use the sum instead of product, but the sum is less natural because the two penalties do not have the same units. Multiplying two quantities with different units makes better sense than adding them.

Finally, we integrate along $\Gamma$ to determine the total penalty: $\int_\Gamma d\cdot g(|\nabla I|)$, and then seek the curve that minimizes this functional.