This chapter presents a study at the interface of image processing and data assimilation: the assimilation of images. The numerical forecast of geophysical fluids is extremely difficult, mainly because they are governed by the general nonlinear equations of fluid dynamics. Over the past 20 years, observations of ocean and atmosphere circulation have become much more readily available, as a result of new satellite techniques. However, the huge amount of information provided by satellite images must therefore be exploited, as more and more space-borne observations of increasing quality are available.
Several ideas have been very recently developed to assimilate image data. A first idea consists of identifying some characteristic structures of the image and then in tracking them in time. This is currently developed in meteorology, using an adaptive thresholding technique for radiance temperatures in order to identify and track several cells [85]. Another idea is to consider a dual problem and to create some model images, coming from the numerical model itself, and to compare the satellite images with these model images, using for example a curvlet approach [82].
We propose here to define a fast and efficient way to identify, or extract, velocity fields from several images (or a complete sequence of images). Assuming this point, we would then be able to obtain billions of pseudo-observations, corresponding to the extracted velocity fields, that could be considered in the usual data assimilation processes. The main advantage of such an approach is to provide a lot of information on the velocity, which is a state variable of all geophysical models, as it is much more easy to assimilate data that are directly related to the state variables. We should mention that a satellite image can have a resolution of
pixels, and that some satellites transmit such images every 15 to 60 minutes [67]. We propose in this paper a way to identify one velocity vector for each pixel of the image. Of course, we will see that all the identified velocity fields are not reliable, mainly when there is no visible characteristic phenomenon, but we should be able to provide an amount of information that is comparable to the currently assimilated observations.
The hypothesis that is underlying this work is that the grey level of the points are preserved during the motion, this is known as the constant brightness hypothesis. The constant brightness hypothesis was introduced in [71], and the linearized equation derived from this hypothesis is the cornerstone of optical flow methods [80,9,30]. This hypothesis is sometimes replaced by an integrated continuity equation in order to take into account the spreading of intensity sources [60,61,51,76].
This hypothesis is justified here in the framework of oceanography, as the objet of interest, allowing us to track the fluid and identify its velocity, is usually a passive tracer, at least on relative short time periods: chlorophyll, sea surface temperature, chemical pollutants (e.g. hydrocarbons), ...All these tracers do not interact with the water on a short time period, and they are passively transported by the fluid.
We propose here to use an integrated version of the constant brightness hypothesis. Instead of linearizing the constant brightness hypothesis like in standard optical flow techniques, we define a nonlinear cost function that takes into account the fact that time sampling occurs at a finite rate. The cost function obtained from the integrated constant brightness assumption is minimized in nested subspaces of admissible displacement vector fields. Several regularization norms are considered.
We refer to [23] for the results of many numerical experiments, on both simulated and real data. These results show that our method provides very quickly full velocity fields, with an estimator of the quality of the results, while the PIV (Particle Imaging Velocimetry) method, currently considered as a reference method in fluid mechanics and oceanography, is unable to provide more than one pertinent velocity vector every
pixels.