  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 $BV$ 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.
