Optimal Transport and Image Multiphysics – TOMMI

Interpolating between two images is an old problem from image analysis, which finds applications for example to recover lost or damaged data in experiments films. It is also used in order to determine in which state the studied system could have been at some time when no observations were made. Image registration algorithms try to align images taken at different moments or from different angles. This allows tracking of regions of interest and has numerous applications in medical imaging.

Among approaches developed in the last years to tackle these problems, optimal transportation methods for image analysis have attracted a lot of attention [B03, BB00, BB01, CDPP10, AHT03, HRT10, PPC09], but to our knowledge without fully taking into account the physics of the represented objects. These methods amount to compute an optimal way of deforming one image to another, by minimizing the mean distance between each displaced pixel. Therefore no energy is attached to the cohesion of image regions which could represent physical objects, and some regions accordingly split into pieces along the optimal transportation path to merge back to the same object at its end. This is highly undesirable.

In parallel, Eulerian approaches, based on generalizations of the level-set method, have been successfully developed to model and mathematically study multiphysics couplings [CM04, B05, CM06, DR06, CMM08, BCM10, MMPR10]. For example, the coupling of an elastic membrane with a fluid into which it is immersed was rephrased as a complex fluid model of Korteweg type. This modeling not only opened the problem to mathematical analysis, but also allowed use of fast numerical methods due to the fact it could be discretized on a cartesian mesh.

As an image may be considered as a snapshot of an Eulerian field, a modeling in this system of coordinates looks appropriate in order to attach energy to physical objects composing the scene.

Our group gathers specialists in optimal transportation, image analysis and multiphysics methods, or in fields very connected to these themes, with which fruitful interactions are forecast. In this short introduction we stress the principal aims of our project:

1. to propose and study generalized optimal transport models which will attach a multiphysics model to the images to be interpolated or registrated.
2. to develop performant algorithms and numerical schemes for these models and benchmark them.
3. apply our new methods on image processing and image assimilation, especially in oceanography.

The achievement of this program will have great impact in research fields where computation of a distance between two images (functions) involving an underlying physics is crucial. For instance in data assimilation and image analysis (part of this project), but also in shape optimization, graphics, ...