A PDE model for computing the optimal flow

In this paper we present a new model for optical ow calculation using a variational formulation which preserves discontinuities of the ow much better than classical methods We study the Euler Lagrange equations asociated to the variational problem In the case of quadratic energy we show the existence and uniqueness of the corresponding evolution problem Since our method avoid linearization in the optical ow constraint it can recover large displacement in the scene We avoid convergence to irrelevant local minima by embedding our method into a linear scale space framework and using a focusing strategy from coarse to ne scales Introduction Optical Flow computation is a key problem in arti cial vision It consists of nding the motion of objects in a sequence of images We shall consider images I x y and I x y de ned on IR to simplify the discussion which represent consecutive views in a sequence of images Determining the optical ow is then nding a function h x y u x y v x y such that I x y I x u x y y v x y x y IR In general this problem has an in nite number of solutions Take for example a sequence representing a black disk moving on a white background In this case any function x y h x y associating a point of the black disk in the rst image to a point on the black disk in the second image and a point of the background in the rst image to another point on the background in the second image satis es the last equality for all points x y Notice that we only best can compute the apparent motion i e the motion in the direction normal to the disk boundary The possible circular motion a rotation leaves the disk unchanged is totally undetectable To compute h x y the preceeding equality is usually linearized yielding the so called optical ow constraint I x I x hrI x h x i x y IR where x x y If at each point we suppose that the motion is only in the perpen dicular direction to the level line passing through this point i e h x k x rI x then we deduce from last equation that when hrI x rI x i then k x I x I x hrI x rI x i Unfortunately this last equality is too local and allows only for estimating motions of the order of one pixel Indeed in images sequences objects move with a priori unpredictable velocities and thus there can be an important displacement of the objects between two consecutive images In order to estimate these large displacements it is necessary to introduce a scale factor for the fusion of information before computing the ow h x A common way of doing this is to convolve both images with a gaussian kernel G x where is the standard deviation before computing the motion In other words we study equation G I I x hr G I x h x i x y R Notice that this model does not impose any regularity condition on the solution h x In this work we propose a new model for computing h x via the minimum of the following energy functional