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URN: urn:nbn:de:bsz:291-scidok-47246
URL: http://scidok.sulb.uni-saarland.de/volltexte/2012/4724/


On the application of projection methods for computing optical flow fields

Schuster, Thomas ; Weickert, Joachim

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Freie Schlagwörter (Englisch): optical flow , variational methods , projection methods , tensor product B-spline
Institut: Fachrichtung 6.1 - Mathematik
DDC-Sachgruppe: Mathematik
Dokumentart: Preprint (Vorabdruck)
Schriftenreihe: Preprint / Fachrichtung Mathematik, Universit├Ąt des Saarlandes
Bandnummer: 194
Sprache: Englisch
Erstellungsjahr: 2007
Publikationsdatum: 15.03.2012
Kurzfassung auf Englisch: 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.
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