Abstruct-Zstimating the three-dimensional motion of an object from a sequence of projections is of paramount importance in a variety of applications in control and robotics, such as autonomous navigation, manipulation, servo, tracking, docking, planning, and surveillance. Although "visual motion estimation" is an old problem (the first formulations date back to the beginning of the century), only recently have tools from nonlinear systems estimation theory hinted at acceptable solutions.In this paper we formulate the visual motion estimation lproblem in terms of identification of nonlinear implicit systems with parameters on a topological manifold and propose a dynamic solution either in the local coordinates or in the embedding space of the parameter manifold. Such a formulation has structural advantages over previous recursive schemes, since the estimation of motion is decoupled from the estimation of the structure of the object being viewed, and therefore it is possible to handle occlusions in a principled way. I. INTRODUC~ONNDERSTANDING the geometry and kinematics of the U envizonment is a basic requirement for humans to successfully accomplish tasks such as walking, driving, and recognizing and grasping objects, It has been one of the principal goals of artificial intelligence, starting from the early 1970's, to build machines that recognize the shape and motion of objects within the environment. The goal is far from being reached and, indeed, it opens a new and exciting avenue of research in nonlinear systems theory.Although the first formulations of the visual motion estimation problem date back to the beginning of the century [ 191, [37], vision-based planning [ 141, and active sensing [69].As the reliability and the performance of the algorithms improves, vision starts being acknowledged in the automatic control community as a powerful and versatile sensor to measure motion, position, and structure of the environment, and the appropriate tools from nonlinear estimationhdentification theory start being exploited [161, [301, 1-59], [601. The implementation of sophisticated vision algorithms running in real time is not too far from becoming reality and, due also to the evolution of computer hardware, vision will be soon included "in the loop" of many control systems."Vision in the loop" raises new and interesting problems of system theoretic flavor, ranging from distributed filtering and processing of large amounts of sensory data to the analysis and control of new classes of dynamical systems. Crucial issues in the use of vision as a sensor in control systems are, for example, nonlinear observability and identifiability in a projective geometric framework as well as estimation and control on peculiar topological manifolds.In this paper we will be mainly concerned with the "visual motion estimation" problem: Given a sequence of images taken from a moving camera, reconstruct the relative threedimensional (3-D) motion between the camera and the environment (or scene).Since our goal is that of posing the vi...

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Stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of Markov random fields restricted to one dimension. Nonstationary reciprocal processes have been extensively studied in the past especially by Jamison et al. The specialization of the nonstationary theory to the stationary case, however, does not seem to have been pursued in sufficient depth in the literature. Stationary reciprocal processes (and reciprocal stochastic models) are potentially useful for describing signals which naturally live in a finite region of the time (or space) line. Estimation or identification of these models starting from observed data seems still to be an open problem which can lead to many interesting applications in signal and image processing. In this paper, we discuss a class of reciprocal processes which is the acausal analog of auto-regressive (AR) processes, familiar in control and signal processing. We show that maximum likelihood identification of these processes leads to a covariance extension problem for block-circulant covariance matrices. This generalizes the famous covariance band extension problem for stationary processes on the integer line. As in the usual stationary setting on the integer line, the covariance extension problem turns out to be a basic conceptual and practical step in solving the identification problem. We show that the maximum entropy principle leads to a complete solution of the problem

The rational covariance extension problem to determine a rational spectral density given a finite number of covariance lags can be seen as a matrix completion problem to construct an infinite-dimensional positive-definite Toeplitz matrix the northwest corner of which is given. The circulant rational covariance extension problem considered in this paper is a modification of this problem to partial stochastic realization of periodic stationary process, which are better represented on the discrete unit circle rather than on the discrete real line . The corresponding matrix completion problem then amounts to completing a finite-dimensional Toeplitz matrix that is circulant. Another important motivation for this problem is that it provides a natural approximation, involving only computations based on the fast Fourier transform, for the ordinary rational covariance extension problem, potentially leading to an efficient numerical procedure for the latter. The circulant rational covariance extension problem is an inverse problem with infinitely many solutions in general, each corresponding to a bilateral ARMA representation of the underlying periodic process. In this paper, we present a complete smooth parameterization of all solutions and convex optimization procedures for determining them. A procedure to determine which solution that best matches additional data in the form of logarithmic moments is also presented.