Guillaume Charpiat's Projects in Images

Converting Level Set Gradients to Shape Gradients

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Aim : to perform gradient descents of energies depending on the signed distance function of a shape
Difficulties : shape gradient and level-set gradient are actually completely different
Approach : express mathematically the shape gradient, convert to a level-set-based evolution
Contribution : express the precise mathematical link between shape gradient and level-set gradient, and give a simple geometric algorithm converting the second one into the first one
Potential applications : segmentation of non-rigid objects with thin parts (roads, blood vessels, ...)

Shape gradient descent vs. Level set gradient descent
When a shape is represented by its signed distance function, and that the energy to be minimized depends on the signed distance function of the shape, they are two ways to perform gradient descent: It turns out that these two evolutions are completely different, and that generally one is rather more interested into the first one.
We express the two gradients (defined respectively on a shape and on the image), give a way to recover the shape gradient from the level-set gradient (with an integral and a change of coordinates), and propose a linear-complexity algorithm to perform this conversion.


This is better visualized with videos (gif animations, click to zoom) :
L2 gradient descent w.r.t. the level set function (level set gradient) L2 gradient descent w.r.t. the level set function
L2 gradient descent w.r.t. the shape (shape gradient) L2 gradient descent w.r.t. the shape

The energy minimized here is the L2 norm of the difference of the level-set representations (supposed to be the signed distance function) of the red shape and the black shape.

We also obtained very similar results for other level-set-representation-based energies that are common in the literature. See the article for details...

Converting gradients

skeleton a level-set variation constant along projection lines cutting cake in parts integration by cake part
Skeleton and projection lines Level-set variations should be constant along projection lines Change of coordinate system Idem Level-set gradient should be integrated over the set of points which share approximately the same projection point

Associated publications:

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