Motion Estimation with Diffusion

This paper develops techniques for the implementation of motion estimation. Optical flow estimation has been proposed as a pre-processing step for many high-level vision algorithms. Gradient-based approaches compute the spatio-temporal derivatives, differentiating the image with respect to time and thus computing the optical flow field. Horn and Schunck's method in particular is considered a benchmarking algorithm of gradient-based differential methods, useful and powerful, yet simple and fast. They formulated an {\em optical flow constraint equation} from which to compute optical flow which cannot fully determine the flow but can give the component of the flow in the direction of the intensity gradient. An additional constraint must be imposed, introducing a supplementary assumption to ensure a smooth variation in the flow across the image. The brightness derivatives involved in the equation system were estimated by Horn and Schunck using first differences averaging. Gradient-based methods for optical flow computation can suffer from unreliability of the image flow constraint equation in areas of an image where local brightness function is non-linear or where there are rapid spatial or temporal changes in the intensity function. Little and Verri suggested regularisation to help the numerical stability of the solution. Usually this takes the form of smoothing of the function or surface by convolving before the derivative is taken. Smoothing has effects of surpressing noise and ensuring differentiability of discontinuities. The method proposed is a finite element method, based on a triangular mesh, in which {\em diffusion} is added into the system of equations. Thus the algorithm performs a type of smoothing while also retrieving the velocity. So the process involves diffusion with movement as opposed to the original Horn and Schunck process of movement only. In this proposed algorithm the derivatives of image intensity are approximated using a finite element approach. Quantitative and qualitative results are presented for real and synthetic images.