Section: New Results
Multiscale fluid flow estimation using Voronoï centered radial basis functions
Oceanographic (current) and meteorological (winds) fluid flows are characterised by turbulence and hence the presence of structures of different temporal and spatial scales. The vector splines approach to motion estimation is adapted to oceanography (brightness conservation) and meteorology (mass transport equation) and preserves the rotatioanal patterns of the flow owing to its div-curl regularity constraint. But it needs to be formulated in a multiscale framework.
The multiscale approach is tackled by using compactly supported radial basis functions centered at Voronoï points of X. The set of Voronoï points VX defines an irregular grid which is connected to the spatial distribution of the original control points set. We define various scales by applying a thinning algorithm  to the set VX . The thinning algorithm provides a decomposition of VX into a nested sequence of subsets in such a way that the points in each subset are distributed as evenly as possible and their spatial density increases smoothly.
At each scale, we solve a least square problem to find coefficients of the solution. The system is large and consists in two terms: a term for the conservation equation and a regularization term which consists in a discretisation of partial differential operators applied to the radial basis functions.
The support of the basis functions are adapted to the grids at each scale and are a function of the ``fill distance'', which measures the sparsity of the grid. Since the basis functions are defined on a compact domain, the system is sparse and computing the solution is fast, especially with the use of iterative methods such as conjugate gradient. As a matter of fact, the complexity of the algorithm solely depends on the number of Voronoï centers and not on the number of selected control points used for motion estimation.
Model estimates motion at several levels of details, from global to local, explicitly taking into account the spatial scales related to turbulence. Figure 2 displays an example of motion estimation at two different scales from sea surface temperatures measurements in the Black Sea.