Tracking regions in an image sequence is a challenging and di cult problem in image processing and computer vision, and at the same time, one that has many important applications: Automated video surveillance, video database search and retrieval, automated video editing, ... So far, numerous approaches to region tracking have been proposed. Many of them su er from excessive constraints imposed on the motion of the region being tracked and need an explicit motion model (e.g. a ne, euclidean, ...). Some, which do not need a parametrized motion model, rely instead on a dense motion eld. By and large, most rely on some kind or other of motion information. Those which d o n o t u s e a n y motion information use instead a model of the region being tracked, typically by assuming strong intensity boundaries, or constraining the shape of the region to belong to a parametrized family of shapes. In this paper, we propose a novel approach to region tracking that derives from a Bayesian formulation. The novelty o f the approach i s t wofold: First, no motion eld or motion parameters need to be computed. This removes a major burden since accurate motion computation has been and remains a challenging problem and the quality of region tracking algorithms based on motion critically depends on the computed motion elds and parameters. The second novelty of this approach i s t h a t v ery little a priori information about the region being tracked is used in the algorithm. In particular, unlike numerous tracking algorithms, no assumption is made on the strength of the intensity edges of the boundary of the region being tracked, nor is its shape assumed to be of a certain parametric form. The problem of region tracking is formulated as a Bayesian estimation problem, and the resulting tracking algorithm is expressed as a level set partial di erential equation. We present further extensions to this partial di erential equation, allowing the possibility of including additional information in the tracking process, such as priors on the region's intensity boundaries, and we brie y discuss the issue of numerical implementation. Very promising experimental results are provided using numerous real image sequences with natural object and camera motion.

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Segmentation of motion in an image sequence is one of the most challenging problems in image processing, while at the same time one that finds numerous applications. To date, a wealth of approaches to motion segmentation have been proposed. Many of them suffer from the local nature of the models used. Global models, such as those based on Markov random fields, perform, in general, better. In this paper, we propose a new approach to motion segmentation that is based on a global model. The novelty of the approach is twofold. First, inspired by recent work of other researchers we formulate the problem as that of region competition, but we solve it using the level set methodology. The key features of a level set representation, as compared to active contours, often used in this context, are its ability to handle variations in the topology of the segmentation and its numerical stability. The second novelty of the paper is the formulation in which, unlike in many other motion segmentation algorithms, we do not use intensity boundaries as an accessory; the segmentation is purely based on motion. This permits accurate estimation of motion boundaries of an object even when its intensity boundaries are hardly visible. Since occasionally intensity boundaries may prove beneficial, we extend the formulation to account for the coincidence of motion and intensity boundaries. In addition, we generalize the approach to multiple motions. We discuss possible discretizations of the evolution (PDE) equations and we give details of an initialization scheme so that the results could be duplicated. We show numerous experimental results for various formulations on natural images with either synthetic or natural motion.

Abstract-We present a novel approach to constraining the evolution of active contours used in image analysis. The proposed approach constrains the final curve obtained at convergence of curve evolution to be related to the initial curve from which evolution begins through an element of a desired Lie group of plane transformations. Constraining curve evolution in such a way is important in numerous tracking applications where the contour being tracked in a certain frame is known to be related to the contour in the previous frame through a geometric transformation such as translation, rotation, or affine transformation, for example. It is also of importance in segmentation applications where the region to be segmented is known up to a geometric transformation. Our approach is based on suitably modifying the Euler-Lagrange descent equations by using the correspondence between Lie groups of plane actions and their Lie algebras of infinitesimal generators, and thereby ensures that curve evolution takes place on an orbit of the chosen transformation group while remaining a descent equation of the original functional. The main advantage of our approach is that it does not necessitate any knowledge of nor any modification to the original curve functional and is extremely straightforward to implement. Our approach therefore stands in sharp contrast to other approaches where the curve functional is modified by the addition of geometric penalty terms. We illustrate our algorithm on numerous real and synthetic examples.

The purpose of this study is to prove convergence results for the Horn and Schunck optical-flow estimation method. Horn and Schunck stated optical-flow estimation as the minimization of a functional. When discretized, the corresponding Euler-Lagrange equations form a linear system of equations We write explicitly this system and order the equations in such a way that its matrix is symmetric positive definite. This property implies the convergence Gauss-Seidel iterative resolution method, but does not afford a conclusion on the convergence of the Jacobi method. However, we prove directly that this method also converges. We also show that the matrix of the linear system is block tridiagonal. The blockwise iterations corresponding to this block tridiagonal structure converge for both the Jacobi and the Gauss-Seidel methods, and the Gauss-Seidel method is faster than the (sequential) Jacobi method.