author = {Sergio Conti and Janusz Ginster and Martin Rumpf},
  title = {A {BV} Functional and its Relaxation for Joint Motion Estimation
	and Image Sequence Recovery},
  journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
  year = {2015},
  volume = {49},
  pages = {1463--1487},
  number = {5},
  abstract = {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.},
  doi = {10.1051/m2an/2015036}