## Section: New Results

Keywords : uncompressible fluid motion, optical flow, quasi-interpolation, partition of unity, radial basis functions, vector splines, quadtree.

### Robust uncompressible fluid flow estimation using a partition of unity

Participants : Till Isambert, Jean-Paul Berroir, Isabelle Herlin, Christine Graffigne [Université Paris V].

This study is part of a Phd thesis on fluid apparent motion estimation with application to oceanography and meteorology. We estimate sea surface streams using divergence free conservation equation coupled with a second order div-curl regularisation. The problem is solved in a vector splines framework and follows the work of Suter [30] : the vector spline is uniquely defined given a set of control points, defined by local image criteria such as the motion index.

However, the basis functions introduced in Suter's work grow arbitrary large whenever r , r being the distance to control points, with two consequences: first, evaluation of the solution at a single point depends on the whole set of control points, and second, computation of the coefficients of the solution involves inverting a dense and ill-conditionned matrix. These problems lead to prohibitive computational costs in case of large data sets.

A partition of unity is proposed to circumvent these problems. The idea is to first subdivise the image into smaller regions. The motion is then computed in each small region using vector splines: as the regions are small and contain a small number of control points, the vector spline estimation is numerically stable. Finally, all motion fields, defined in small regions, are merged into a single one defined on the whole domain.

The definition of smaller regions is based on an iterative quadtree subdivision approach: starting from an unique region (the image domain), each region is subdivided if it contains more than a fixed number of control points.

In order to minimize aliasing effects at the limits between regions, the vector splines are computed in balls centered on the quadtree cells, so that the vector splines of adjacent regions intersect. The overall solution is obtained by summing up every vector spline, computed in individual regions, weighted by local coefficients depending on the distance to the centre of the associated quadtree cell.

Figure 1 displays a result from the model on a synthetic oceanographic image sequence: left, a 193 cells quadtree is built from 6085 control points, defined according to local criteria (gradient norm and motion index) in the image. The subdvision is controlled in such a way that each cell contains no more than 200 points. The resulting motion field is displayed on the right side of figure 1 .