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We present in this paper a variational approach to accurately estimate simultaneously the velocity field and its derivatives directly from PIV image sequences. Our method differs from other techniques that have been presented in the literature in the fact that the energy minimization used to estimate the particles motion depends on a second order Taylor development of the flow. In this way, we are not only able to compute the motion vector field, but we also obtain an accurate estimation of their derivatives. Hence, we avoid the use of numerical schemes to compute the derivatives from the estimated flow that usually yield to numerical amplification of the inherent uncertainty on the estimated flow. The performance of our approach is illustrated with the estimation of the motion vector field and the vorticity on both synthetic and real PIV datasets.

Estimation of motion has many applications in fluid analysis. Lots of work has been carried out using Particle Image Velocimetry to design experiments which capture and measure the flow motion using 2D images. Recent technological advances allow capturing 3D PIV image sequences of moving particles. In this context, we propose a new threedimensional variational (energy-based) technique. Our technique is based on solenoidal projection to take into account the incompressibility of the real flow. It uses the result of standard flow motion estimation techniques like iterative cross-correlation or pyramidal optical flow as an initialization, and improves significantly their accuracies. The performance of the proposed technique is measured and illustrated using numerical simulations.

Currently, meteorological satellites provide multichannel image sequences including visible, temperature and water vapor channels. Based on a variational approach, we propose mathematical models to address some of the usual challenges in satellite image analysis such as: (i) the estimation and smoothing of the cloud structures by decoupling them into different layers depending on their altitudes, (ii) the estimation of the cloud structure motion by combining information from all the channels, and (iii) the 3D visualization of both the cloud structure and the estimated displacements. We include information of all the channels in a single variational motion estimation model. The associated Euler-Lagrange equations yield to a nonlinear system of partial differential equations that we solve numerically using finite-difference schemes. We illustrate the performance of the proposed models with numerical experiments on two multichannel satellite sequences of the North Atlantic, one of them from the Hurricane Vince. Based on a realistic synthetic ground truth motion, we show that our multichannel approach overcomes the single channel estimation for both the average Euclidean and angular errors.