Estimating the Shape of a Moving Contour

Most tracking problems in computer vision can be conceptualized as nonlinear estimation problems, with the gray level of each pixel being an observation. The difficulty in making use of this point of view is that i) we lack a plausible model for the underlying stochastic processes, and ii) the resulting problems would be of very high dimension as well as being nonlinear, making them all but intractable. The search for an approach that makes effective use of the existing spatial correlations has led to the study of various methods of simplification involving feature points, contours, blobs, etc. In particular there has been considerable work based on organizing visual evidence around parametrized contours called snakes [I]. It seems that the most geometrically natural evolutionary equations for curves in the plane, such as the smoothing flow for closed curves investigated by Gage and Hamilton [Z], are nonlinear. [3] have shown that linear models based on finite dimensional parametrization can also be effective and remain tractable even in a stochastic setting. This paper continues in a similar way. We introduce a coordinate system and represent the curve using its horizontal and vertical components (~1, LZ), expressed as functions of time and arc length. We introduce continuum models for the evolution of the coordinates and noisy observation models that allow us to formulate and solve a realistic class of estimation problems. One interesting question brought into focus by this work is that of determining how to optimize the use of spatial correlation along the curve, and temporal correlation in the evolution of the curve, so as to reduce the effects of the observation noise. Our formulas in Section 7 give an answer to this question.