# American Institute of Mathematical Sciences

November  2007, 1(4): 673-690. doi: 10.3934/ipi.2007.1.673

## On the application of projection methods for computing optical flow fields

 1 Fakultät für Maschinenbau, Helmut--Schmidt--Universität, Holstenhofweg 85, 22043 Hamburg, Germany 2 Fakultät für Mathematik und Informatik, Universität des Saarlandes, Geb. E1.1, 66041 Saarbrücken, Germany

Received  May 2007 Published  October 2007

Detecting optical flow means to find the apparent displacement field in a sequence of images. As starting point for many optical flow methods serves the so called optical flow constraint (OFC), that is the assumption that the gray value of a moving point does not change over time. Variational methods are amongst the most popular tools to compute the optical flow field. They compute the flow field as minimizer of an energy functional that consists of a data term to comply with the OFC and a smoothness term to obtain uniqueness of this underdetermined problem. In this article we replace the smoothness term by projecting the solution to a finite dimensional, affine subspace in the spatial variables which leads to a smoothing and gives a unique solution as well. We explain the mathematical details for the quadratic and nonquadratic minimization framework, and show how alternative model assumptions such as constancy of the brightness gradient can be incorporated. As basis functions we consider tensor products of B-splines. Under certain smoothness assumptions for the global minimizer in Sobolev scales, we prove optimal convergence rates in terms of the energy functional. Experiments are presented that demonstrate the feasibility of our approach.
Citation: Thomas Schuster, Joachim Weickert. On the application of projection methods for computing optical flow fields. Inverse Problems and Imaging, 2007, 1 (4) : 673-690. doi: 10.3934/ipi.2007.1.673
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