The dimensionality of visual motion analysis can be reduced by analyzing directional components (projections) of flow fields. In contrast to vector fields, these lower-dimensional spaces exhibit two simple geometric properties which are invariant to the scene structure and depend only on the camera motion. Using these properties, structure and motion can be separated. The approach is closely related to the concept of direction-selective processing of visual motion fields, for which biological evidence has been discovered in mammalian visual systems. We use a recursive observer model where a collection of filters tuned to specific flow directions provide the directional motion parameters. The original motion parameter vector can be computed (if necessary) by combining the directional parameters as vector components. The model is applicable to general camera motion and to large camera field of view (FOV) and does not require point correspondence. In addition to the recursive model, the temporal integration of instantaneous measurements is extended to image sequences using tracking, which facilitates reconstruction of the camera motion trajectory, even if the velocity of the motion changes. The approach is highly scalable and efficiently parallelizable. We demonstrate it on long image sequences.

Keywords: Egomotion estimation, Fisher's linear discriminant, Projection of flow fields, Robust line fitting, Tracking.

It is shown that the problem of independent motion detection can be addressed by analyzing constraints on low-dimensional directional (projected) components of flow fields. We construct a robust algorithm, implemented as a recursive filter, to extract directional motion parameters from long image sequences. Based on this a qualitative approach is described to detect independent motion, involving a combination of robust line-fitting and a one-dimensional search. The low-dimensional subspaces of projections facilitate efficient dynamic self-adaptation of detection thresholds to achieve good performance under changing operational conditions. The analysis is extended to long image sequences by incorporating tracking and spatio-temporal filtering. The approach is applicable to general camera motion and cluttered scenes using a wide range of camera field of view. The method does not require point-correspondence and can be simply extended to stereo. We demonstrate the proposed approach on a variety of real image sequences.

Keywords: Signature analysis, Partial egomotion estimation, Detection of moving objects, Robust line fitting, Tracking, Spatio-temporal filtering.

In this paper the general concept of a migration process (MP) is introduced; it involves iterative displacement of each point in a set as function of a neighborhood of the point, and is applicable to arbitrary sets with arbitrary topologies. After a brief analysis of this relatively general class of iterative processes and of constraints on such processes, we restrict our attention to processes in which each point in a set is iteratively displaced to the average (centroid) of its equigeodesic neighborhood. We show that MP's of this special class can be approximated by ``reaction-diffusion''-type PDE's, which have received extensive attention recently in the contour evolution literature. Although we show that MP's constitute a special class of these evolution models, our analysis of migrating sets does not require the machinery of differential geometry. In Part I of the paper we characterize the migration of closed curves and extend our analysis to arbitrary connected sets in the continuous domain ($\bbR^m$) using the frequency analysis of closed polygons, which has been re-discovered recently in the literature. We show that migrating sets shrink, and also derive other geometric properties of MP's. In Part II we will reformulate the concept of migration in a discrete representation .

Keywords: Iterative processes, Discrete deformation of sets, Contour evolution.

Following the study of migration processes in the continuous domain in Part I of this paper, we reformulate the concept of migration in the discrete domain (Z^m) and define Discrete Migration Processes (DMP). We demonstrate that this model is a natural discrete representation of the continuous model and maintains the model's features in a qualitative sense. We show that under discrete migration any discrete set shrinks to a limit in finitely many iterations. The discrete representation provides an advantageous basis for digitally implementing the MP model. Using this implementation we illustrate the discrete migration of various types of sets under various types of constraints.

Keywords: Discrete iterative processes, Discrete deformation of sets, Contour evolution, Discrete active models.

Optimization processes based on ``active models'' play central roles in many areas of computational vision as well as computational geometry. Unfortunately, current models usually require highly complex and sophisticated mathematical machinery and at the same time they suffer from a number of limitations which impose restrictions on their applicability. In this paper a simple class of discrete active models, called migration processes (MPs), is presented. The processes are based on iterated averaging over neighborhoods defined by constant geodesic distance. It is demonstrated that the MP model - a system of self-organizing active particles - has a number of advantages over previous models, both parametric active models (``snakes'') and implicit (contour evolution) models. Due to the generality of the MP model the process can be applied to derive natural solutions to a variety of optimization problems, including defining (minimal) surface patches given their boundary curves; finding shortest paths joining sets of points; and decomposing objects into ``primitive'' parts.

Keywords: Discrete active models, optimization, geometric diffusion processes, minimal path, minimal surface, shape decomposition.