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.

This article describes a method for reducing the shape distortions due to scale-space smoothing that arise in the computation of 3-D shape cues using operators (derivatives) de ned from scale-space representation. More precisely, we are concerned with a general class of methods for deriving 3-D shape cues from 2-D image data based on the estimation of locally linearized deformations of brightness patterns. This class

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 14.2 Dierential formulation . . . . . . . . . . . . . . . . . . . . . 398 14.3 Uncertainty Model . . . . . . . . . . . . . . . . . . . . . . . . 400 14.4 Coarse-to- ne estimation . . . . . . . . . . . . . . . . . . . . 404 14.5 Implementation issues . . . . . . . . . . . . . . . . . . . . . . 411 14.5.1 Derivative lter kernels . . . . . . . . . . . . . . . . . 411 14.5.2 Averaging lter kernels . . . . . . . . . . . . . . . . . 412 14.5.3 Multi-scale warping . . . . . . . . . . . . . . . . . . . 412 14.5.4 Boundary handling . . . . . . . . . . . . . . . . . . . 414 14.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 14.6.1 Performance measures . . . . . . . . . . . . . . . . . 414 14.6.2 Synthetic sequences . . . . . . . . . . . . . . . . . . . 415 14.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 14.8 References . . . . . . . . . . . . . . . . . . . . . . . . .

Recent approaches to estimating two-dimensional image motion and to modeling the perception of image motion have achieved success with Bayesian formulations. With a Bayesian approach, the goal is to compute the posterior probability distribution of velocity, which is proportional to a likelihood function and a prior. The likelihood function describes the probability of observing the image data given the image velocity; surprisingly, there is still disagreement about the right likelihood function to use. Here we derive a likelihood function by starting from a generative model. We assume that the scene translates, conserving image brightness, while the image is equal to the projected scene plus noise. We discuss the connection between the resulting likelihood function and existing models of motion analysis. We show that the likelihood can be calculated by a population of units whose response properties are similar to \motion energy" units. This suggests that a population o...

In this paper we will present a least committed multi-scale method for computation of optic flow fields. We extract optic flow fields from normal flow, by fitting the normal components of a local polynomial model of the optic flow to the normal flow. This fitting is based on an analytically solvable optimization problem, in which an integration scalespace over the normal flow field regularizes the solution. An automatic local scale selection mechanism is used in order to adapt to the local structure of the flow field. The performance profile of the method is compared with that of existing optic flow techniques and we show that the proposed method performs at least as well as the leading algorithms on the benchmark image sequences proposed by Barron et al. [3]. We also do a performance comparison on a synthetic fire particle sequence and apply our method to a real sequence of smoke circulation in a pigsty. Both consist of highly complex non-rigid motion.

Algorithmic development of optical-ow routines is hampered by slow turnaround times (to iterate over testing, evaluation, and adjustment of the algorithm). To ease the problem, parallel implementation on a convenient general-purpose parallel machine is possible. A generic parallel pipeline structure, suitable for distributed-memory machines, has enabled parallelisation to be quickly achieved. Gradient, correlation, and phase-based methods of optical-ow detection have been constructed to demonstrate the approach.