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.

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

(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...

Associated publications:

, with**Converting Level Set Gradients to Shape Gradients**__Siqi Chen__and Richard J. Radke,*European Conference on Computer Vision***ECCV 2010**. [bibtex] [poster]

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