  In this paper, we are interested in the application to video segmentation of
the discrete shape optimization problem involving the shape weighted perimeter
and an additional term depending on a parameter. Based on recent works and in
particular the one of Darbon and Sigelle, we justify the equivalence of the
shape optimization problem and a weighted total variation regularization. For
solving this problem, we adapt the projection algorithm proposed recently for
solving the basic TV regularization problem. Another solution to the shape
optimization investigated here is the graph cut technique. Both methods have
the advantage to lead to a global minimum. Since we can distinguish moving
objects from static elements of a scene by analyzing norm of the optical flow
vectors, we choose the optical flow norm as initial data. In order to have the
contour as close as possible to an edge in the image, we use a classical edge
detector function as the weight of the weighted total variation. This model has
been used in one of our former works. We also apply the same methods to a video
segmentation model used by Jehan-Besson, Barlaud and Aubert. In this case, only
standard perimeter is incorporated in the shape functional. We also propose
another way for finding moving objects by using an a contrario detection of
objects on the image obtained by solving the Rudin-Osher-Fatemi Total Variation
regularization problem.We can notice the segmentation can be associated to a
level set in the former methods.
