# Image Processing

## Shape warping and statistics

(Version française ?)

Remark: all the available reports on this page are ps.gz formatted. If you cannot read this format, do not hesitate to contact me...

### PhD-Thesis

My PhD-thesis deals with non-supervised image correspondances, means and statistics (ie without giving manually clues to the program). For example, let us consider a set of pictures of various people, and wonder what is their « mean » face (the mean of their faces), and which are the typical deformations we have to apply on this mean face in order to build other pictures of « the same kind »... The approach we study consists in defining a similarity criterium CS(A, B) between two images A and B (for example, the local cross-correlation), and a regularity criterium CR(h) on deformations h. Our aim is to find a mean image M, such that, if we note I_i the images which we want to compute the mean of, and h_i the associated deformations, that each I_i o h_i looks like M, with h_i regular enough. Id est: we minimize the sum of the CS(M, I_i o h_i) + CR(hi).

I will soon add on this web page a paper with more details, and also results from the computation of faces mean and statistics. But now, I list my previous works on similar subjects (curve statistics).

During the year 2001-2002, I attended to the courses of DEA MVA (Mathématiques/Vision/Apprentissage) in Cachan, and consequently had to deal with several fields of image processing, particularly with the problem of planar curve deformation (or 2D surface in 3D space, etc.). Here are my miscellaneous works, realized within the framework of this DEA.

### Curve deformation (DEA's four month work)

How to match two planar curves by minimizing the Hausdorff distance, shape statistics
Here is my DEA's report (in french) on the subject, directed by Olivier Faugeras and Renaud Keriven. After a quick presentation of the « Level-Sets » method, thanks to which we deal easily with curve evolution, we get interested into the curve stastistics problem: how to, for example, define the « mean » curve of a set of curves, and how to define their « characteristic deformations » ? The approach we study here consists in considering natural distances on the space of the curves (Hausdorff distance, W(1,2) norm) and in regularizing them so as to compute gradient descents (with respect to the curves). The presentation of the theory is completed by computed results.

Continuity of the « Hausdorff energy » with respect to the Hausdorff metric enriched by the length difference
Continuation of the mathematical study (in french) on the « Hausdorff energy » (introduced in the previous report), in particular on its continuity.

Approximations of shape metrics and application to shape warping and empirical shape statistics
INRIA research report (in english), which prolongs the work made during the DEA; one article accepted in ICIP 2003, another in FOCM.

Here are several examples of the evolution of one curve to another:

• two hands,
• two faces: a first evolution with a standard precision, a second evolution with a higher precision (i.e. with an initial important zoom - the evolution is not invariant by change of scale, because we fixed all the parameters once for all times, without image size dependance).
• two humped-polygon transformations, interesting because qualitatively different:

Poster
I attended to the « Designing Tomorrow's Category-Level 3D Object Recognition Systems: An International Workshop » in 2003, and I presented there a poster, here available (in english).

### Others

The « elastica » problem and image completion
Here is a report on the very badly posed problem of « pretty » curves building, in the framework of the extrapolation of a binary image. Given an image, which pixels only take two values (black and white), how to complete it in a zone where it has been unhappily erased ? We express the problem in the one of the optimal place of a (level) curve, which we solve by minimizing a criterion, depending on the integral of a power of the curvature along the curve. This report is supposed to be very easily readable (go trough the theoretical part, which is not very useful), you can at least have a look on the images :-)
(this report has been realized for J.-M. Morel's course, the subject being suggested by L. Moisan).

Radon transform of an image, ridgelet transform
Definition and computation of the Radon transform (in french) of an image, which leads to the « ridgelet » transform.
(from a work with Vincent Feuvrier, for Yves Meyer's course, the subject being suggested and framed by Jacques Froment).

Talk at the ENS sudents seminar
I presented in March 2003 a « talk » about my DEA report at the « séminaire des élèves du département d'informatique de l'ENS ».

To contact me...