In this paper we show that a classic optical ow technique by Nagel and Enkelmann (1986) can be regarded as an early anisotropic diusion method with a diusion tensor. We introduce three improvements into the model formulation that (i) avoid inconsistencies caused by centering the brightness term and the smoothness term in dierent images, (ii) use a linear scale-space focusing strategy from coarse to ne 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 diusion{reaction equations. We prove that this system has a unique solution under realistic assumptions on the initial data, and we present an ecient linear implicit numerical scheme in detail. Our method creates ow elds with 100 % density over the entire image domain, it is robust under a large range of parameter variations, and it c...

Many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy. Typically, these energy functionals consist of two terms: a data term, which requires e.g. that a brightness constancy assumption holds, and a regularizer that encourages global or piecewise smoothness of the flow field. In this paper we present a systematic classification of rotation invariant convex regularizers by exploring their connection to diffusion filters for multichannel images. This taxonomy provides a unifying framework for data-driven and flow-driven, isotropic and anisotropic, as well as spatial and spatio-temporal regularizers. While some of these techniques are classic methods from the literature, others are derived here for the first time. We prove that all these methods are well-posed: they posses a unique solution that depends in a continuous way on the initial data. An interesting structural relation between isotropic and anisotropic flow-driven regularizers is identified, and a design criterion is proposed for constructing anisotropic flow-driven regularizers in a simple and direct way from isotropic ones. Its use is illustrated by several examples.

Multiple moving objects, occluded objects, or even a moving object against the static background give rise to discontinuities in the optical flow field in corresponding image sequences. Uniform global smoothing-based regularization cannot provide accurate estimates if the optical flow is discontinuous. A `weighted anisotropic smoothness'-based regularization technique is proposed for accurately estimating dense optical flow field. Weighted sum of the magnitude and first-order spatial derivatives of the optic flow field is used for regularization. Less regularization is performed at points where strong intensity gradient information is available. The solution at any point is interpolated more from those at neighboring points along the weaker intensity gradient-component. A new scalable algebraic multigrid algorithm is used to efficiently solve resulting second-order elliptic partial differential equations with discontinuities and anisotropies in coefficients. This multigrid tech...

A method for the interpolation of parameter fields estimated by total least squares is presented. This is applied to the study of dynamic processes where the motion and further values such as divergence or brightness changes are parameterised in a partial differential equation.

A cooperative integration of stereopsis and optic ow computation is presented. Central to our approach is the modelling of the visual processes as a sequence of coupled MRF's by dening suitable inter-process interactions based on some natural constraints. The integration makes each of the individual processes better constrained and more reliable. Further, as a result of the integration, it becomes possible to accurately preserve the discontinuities in both the ow and the disparity elds along with the regions of stereo occlusion. Results on both noisy synthetic image data and real images are presented. D.S.P. Laboratory, Core Technology Research Center, Samsung Advanced Institute of Technology, Suwon, South Korea 440-600. Email: sudhir@dspsun.sait.samsung.co.kr y Dept. Computer Science and Engineering, Indian Institute of Technology, New Delhi 110016, India. Email: suban@cse.iitd.ernet.in z Centre for Applied Research in Electronics, Indian Institute of Technology, New Delhi ...

Abstract—Variational methods for optical flow estimation are known for their excellent performance. The method proposed by Brox et al. [5] exemplifies the strength of that framework. It combines several concepts into single energy functional that is then minimized according to clear numerical procedure. In this paper we propose a modification of that algorithm starting from the spatiotemporal gradient constancy assumption. The numerical scheme allows to establish the connection between our model and the CLG(H) method introduced in [18]. Experimental evaluation carried out on synthetic sequences shows the significant superiority of the spatial variant of the proposed method. The comparison between methods for the realworld sequence is also enclosed. Keywords—optical flow, variational methods, gradient constancy assumption. I.