The Applicability of Green‘s Theorem to Computation of Rate of Approach

Authors:
Zoran Duric;Azriel Rosenfeld;James Duncan
Affiliations:
Department of Computer Science, George Mason University, Fairfax, VA 22032, USA;Center for Automation Research, University of Maryland, College Park, MD 20742-3275, USA;Department of Mechanical Engineering, University of Maryland, College Park, MD 20742-3035, USA
Venue:
International Journal of Computer Vision
Year:
1999

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Abstract

The rate of approach (ROA) of a moving observer toward a scene point, as estimated at a given instant, is proportional to the componentof the observer‘s instantaneous velocity in the direction of the point. In this paper we analyze the applicability of Green‘s theorem to ROA estimation.We derive a formula which relates three quantities: the average value of the ROA for a surface patch in the scene;a surface integral that depends on the surface slant of the patch;and the contour integral of the normal motion field around the image of the boundary of the patch. We analyze how much larger the ROA on the surface patch can be than the value of the contour integral, for given assumptions about thevariability of the distance to points on the surface patch.We illustrate our analysis quantitatively using synthetic data, and we alsovalidate it qualitatively on real image sequences.