Reliable estimation of dense optical flow fields with large displacements

In this paper we show that a classic optical flow technique by Nagel and Enkelmann (1986, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 8, pp. 565-593) can be regarded as an early anisotropic diffusion method with a diffusion tensor. We introduce three improvements into the model formulation that (i) avoid inconsistencies caused by centering the brightness term and the smoothness term in different images, (ii) use a linear scale-space focusing strategy from coarse to fine scales for avoiding convergence to physically irrelevant local minima, and (iii) create an energy functional that is invariant under linear brightness changes. Applying a gradient descent method to the resulting energy functional leads to a system of diffusion-reaction equations. We prove that this system has a unique solution under realistic assumptions on the initial data, and we present an efficient linear implicit numerical scheme in detail. Our method creates flow fields with 100% density over the entire image domain, it is robust under a large range of parameter variations, and it can recover displacement fields that are far beyond the typical one-pixel limits which are characteristic for many differential methods for determining optical flow. We show that it performs better than the optical flow methods with 100% density that are evaluated by Barron et al. (1994, Int. J. Comput. Vision, Vol. 12, pp. 43-47). Our software is available from the Internet.