This paper describes a method to estimate parametric motion models. Motivations for the use of such models are on one hand their efficiency, which has been demonstrated in numerous contexts such as estimation, segmentation, tracking and interpretation of motion, and on the other hand, their low computational cost compared to optical flow estimation. However, it is important to have the best accuracy for the estimated parameters, and to take into account the problem of multiple motion. We have therefore developed two robust estimators in a multiresolution framework. Numerical results support this approach, as validated by the use of these algorithms on complex sequences. 1

Two-dimensional image motion is the projection of the three-dimensional motion of objects, relative to a visual sensor, onto its image plane. Sequences of time-ordered images allow the estimation of projected two-dimensional image motion as either instantaneous image velocities or discrete image displacements. These are usually called the optical flow field or the image velocity field. Provided that optical flow is a reliable approximation to two-dimensional image motion, it may then be used to recover the three-dimensional motion of the visual sensor (to within a scale factor) and the three-dimensional surface structure (shape or relative depth) through assumptions concerning the structure of the optical flow field, the three-dimensional environment and the motion of the sensor. Optical flow may also be used to perform motion detection, object segmentation, time-to-collision and focus of expansion calculations, motion compensated encoding and stereo disparity measurement. We investiga...

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This paper presents a new approach to the estimation of two-dimensional motion vector fields from time-varying images. The approach is stochastic, both in its formulation and in the solution method. The formulation involves the specification of a deterministic structural model, along with stochastic observation and motion field models. Two motion models are proposed: a globally smooth model based on vector Markov random fields and a piecewise smooth model derived from coupled vector-binary Markov random fields. Two estimation criteria are studied. In the Maximum A Posteriori Probability (MAP) estimation the a posteriori probability of motion given data is maximized, while in the Minimum Expected Cost (MEC) estimation the expectation of a certain cost function is minimized. The MAP estimation is performed via simulated annealing, while the MEC algorithm performs iteration-wise averaging. Both algorithms generate sample fields by means of stochastic relaxation implemented via the Gibbs s...

Absfruet-A new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical flow fields in an image sequence. Standard formulations of this problem require the computationally intensive solution of an elliptic partial differential equation that arises from the often used “smoothness constraint” ’yl”. regularization. The interpretation of the smoothness constraint is utilized as a “fractal prior ” to motivate regularization based on a recently introduced class of multiscale stochastic models. The solution of the new problem formulation is computed with an efficient multiscale algorithm. Experiments on several image sequences demonstrate the substantial computational savings that can be achieved due to the fact that the algorithm is noniterative and in fact has a per pixel computational complexity that is independent of image size. The new approach also has a number of other important advantages. Specifically, multiresolution flow field estimates are available, allowing great flexibility in dealing with the tradeoff between resolution and accuracy. Multiscale error covariance information is also available, which is of considerable use in assessing the accuracy of the estimates. In particular, these error statistics can be used as the basis for a rational procedure for determining the spatially-varying optimal reconstruction resolution. Furthermore, if there are compelling reasons to insist upon a standard smoothness constraint, our algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation. Finally, the usefulness of our approach should extend to a wide variety of ill-posed inverse problems in which variational techniques seeking a “smooth ” solution are generally Used. I.

This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts–in particular making ties to topics such as wavelets and multigrid methods. A third is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for self-similar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden

In this paper we address the issue of recovering and segmenting the apparent velocity field in sequences of images. As for motion estimation, we minimize an objective function involving two robust terms. The first one cautiously captures the optical flow constraint, while the second (a priori) term incorporates a discontinuity-preserving smoothness constraint. To cope with the nonconvex minimization problem thus defined, we design an efficient deterministic multigrid procedure. It converges fast toward estimates of good quality, while revealing the large discontinuity structures of flow fields. We then propose an extension of the model by attaching to it a flexible object-based segmentation device based on deformable closed curves (different families of curve equipped with different kinds of prior can be easily supported). Experimental results on synthetic and natural sequences are presented, including an analysis of sensitivity to parameter tuning. INdex Terms--- Closed segmenting cu...

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.

This thesis describes some new approaches to the representation and analysis of visual motion, as perceived by a biological or machine visual system. We begin by discussing the computation of image motion fields, the projection of motion in the three-dimensional world onto the two-dimensional image plane. This computation is notoriously difficult, and there are a wide variety of approaches that have been developed for use in image processing, machine vision, and biological modeling. We show that a large number of the basic techniques are quite similar in nature, differing primarily in conceptual motivation, and that they each fail to handle a set of situations that occur commonly in natural scenery. The central theme of the thesis is that the failure of these algorithms is due primarily to the use of vector fields as a representation for visual motion. We argue that the translational vector field representation is inherently impoverished and error-prone. Furthermore, there is evidence that a ...