I am also interested in the application of Differential Geometry to 3D image processing, mainly, how to extract stable geometric features such as lines or points from medical images, where objects have smoothed surfaces with very complex and a-priory unknown topology.
In collaboration with Alexis Gourdon (RR-1672), I have designed new methods to extract iso-surfaces and iso-surface intersections while maintaining the topological properties of reconstructed surfaces and curves (RR-1881-1). We have also shown how to compute their differential characteristics (RR-1881-2). The applications are the extraction of stable features like crest lines from 3D medical images and the automatic registration of those images.
In RR-1901 I have extended those differential geometry methods to feature points extraction, called extremal points, whereas in RR-2003, in collaboration with Serges Benayoun, I have shown how to compute the corresponding points using 4D differential geometry and the hypersurface corresponding to the 3D image intensity values. At last, I have unified those notions of extremal lines and extremal points in RR-2149, into the concept of extremal mesh, which is defined by a new differential invariant of 3D surfaces called the Gaussian extremality (Eg).
The precision of the registration method based on points has been theoretically assessed by Xavier Pennec and myself in RR-2470.
I am currently studying the behavior of those geometric features in scale space with the help of Martha Fidrich (see for example RR-2365).