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The estimation of motion in an image sequence is a fundamental task in image processing. Frequently, the image sequence is corrupted by noise and one simultaneously asks for the underlying motion field and a restored sequence. In smoothly shaded regions of the restored image sequence the brightness constancy assumption along motion paths leads to a pointwise differential condition on the motion field. At object boundaries which are edge discontinuities both for the image intensity and for the motion field this condition is no longer well defined. In this paper a total-variation type functional is discussed for joint image restoration and motion estimation. This functional turns out not to be lower semicontinuous, and in particular fine-scale oscillations may appear around edges. By the general theory of vector valued functionals its relaxation leads to the appearance of a singular part of the energy density, which can be determined by the solution of a local minimization problem at edges. Based on bounds for the singular part of the energy and under appropriate assumptions on the local intensity variation one can exclude the existence of microstructures and obtain a model well-suited for simultaneous image restoration and motion estimation. Indeed, the relaxed model incorporates a generalized variational formulation of the brightness constancy assumption. The analytical findings are related to ambiguity problems in motion estimation such as the proper distinction between foreground and background motion at object edges.
Conti, Sergio 1 ; Ginster, Janusz 1 ; Rumpf, Martin 1, 2
@article{M2AN_2015__49_5_1463_0,
author = {Conti, Sergio and Ginster, Janusz and Rumpf, Martin},
title = {A $BV$ functional and its relaxation for joint motion estimation and image sequence recovery},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1463--1487},
publisher = {EDP-Sciences},
volume = {49},
number = {5},
year = {2015},
doi = {10.1051/m2an/2015036},
zbl = {1326.49019},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015036/}
}
TY - JOUR AU - Conti, Sergio AU - Ginster, Janusz AU - Rumpf, Martin TI - A $BV$ functional and its relaxation for joint motion estimation and image sequence recovery JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 1463 EP - 1487 VL - 49 IS - 5 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015036/ DO - 10.1051/m2an/2015036 LA - en ID - M2AN_2015__49_5_1463_0 ER -
%0 Journal Article %A Conti, Sergio %A Ginster, Janusz %A Rumpf, Martin %T A $BV$ functional and its relaxation for joint motion estimation and image sequence recovery %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 1463-1487 %V 49 %N 5 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015036/ %R 10.1051/m2an/2015036 %G en %F M2AN_2015__49_5_1463_0
Conti, Sergio; Ginster, Janusz; Rumpf, Martin. A $BV$ functional and its relaxation for joint motion estimation and image sequence recovery. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1463-1487. doi: 10.1051/m2an/2015036
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