Variational correlation and decomposition methods for particle image velocimetry

Particle Image Velocimetry (PIV) is a non-intrusive optical measurement technique for industrial fluid flow questions. Small particles are introduced into liquids or gases and act as indicators for the movement of the investigated substance around obstacles or in regions where fluids mix. For the two-dimensional variant of the PIV method, a thin plane is illuminated by laser light rendering the particles therein visible. A high speed camera system records an image sequence of the highlighted area. The analysis of this data allows to determine the movement of the particles, and in this way to measure the speed, turbulence or other derived physical properties of the fluid. In state-of-the-art implementations, correspondences between regions of two subsequent image frames are determined using cross-correlation as similarity measurement. In practice it has proven to be robust against disturbances typically found in PIV data. Usually, an exhaustive search over a discrete set of velocity vectors is performed to find the one which describes the data best. In our work we consider a variational formulation of this problem, motivated by the extensive work on variational optical flow methods which allows to incorporate physical priors on the fluid. Furthermore, we replace the usually square shaped correlation window, which defines the image regions whose correspondence is investigated, by a Gaussian function. This design drastically increases the flexibility of the process to adjust to features in the experimental data. A sound criterion is proposed to adapt the size and shape of the correlation window, which directly formulates the aim to improve the measurement accuracy. The velocity measurement and window adaption are formulated as an interdependent variational problem. We apply continuous optimisation methods to determine a solution to this non-linear and non-convex problem. In the experimental section, we demonstrate that our approach can handle both synthetic and real data with high accuracy and compare its performance to state-of-the-art methods. Furthermore, we show that the proposed window adaption scheme increases the measurement accuracy. In particular, high gradients in motion fields are resolved well. In the second part of our work, we investigate an approach for solving very large convex optimisation problems. This is motivated by the fact that a variational formulation on the one hand allows to easily incorporate prior knowledge on data and variables to improve the quality of the solution. Furthermore, convex problems often occur as subprograms of solvers for non-convex optimisation tasks, as it is the case in the first part of this work. However, the extension of two-dimensional approaches to 3D, or to the time axis, as well as the ever increasing resolution of sensors, let the number of variables virtually explode. For many interesting applications, e.g. in medical imaging or fluid mechanics, the problem description easily exceeds the memory limits of available, single computational nodes. Thus, we investigate a decomposition method for the class of unconstrained, convex and quadratic optimisation problems. Our approach is based on the idea of Dual Decomposition, or Lagrangian Relaxation, and splits up the problem into a couple of smaller tasks, which can be distributed to parallel hardware. Each subproblem is again quadratic and convex and thus can be solved efficiently using standard methods. Their interconnection is respected to ensure that we find a solution to the original, non-decomposed problem. Furthermore we propose a framework to modify the numerical properties of the subproblems, which enables us to improve their convergence rates. The theoretical part is completed by the analysis of convergence conditions and rate. Finally, we demonstrate our approach by means of three relevant variational problems from image processing. Error measurements in comparison to single-domain solutions are presented to assess the accuracy of the decomposition.