Guillaume Charpiat's Projects in Images

Converting Level Set Gradients to Shape Gradients

Images are zoomed when clicked.

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:

with respect to the shape, and then extend the shape deformation to the whole image (i.e. convert it to a level-set evolution)
with respect to the level-set representation directly.

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.

Results

This is better visualized with videos (gif animations, click to zoom) :

	(level set gradient)
	(shape gradient)

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 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:

Converting Level Set Gradients to Shape Gradients, with Siqi Chen and Richard J. Radke, European Conference on Computer Vision ECCV 2010. [bibtex] [poster]

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