The applicability of Green's theorem to computation of rate of approach

The rate of approach (ROA) of a moving observer toward a scene point, as es timated at a given instant, is proportional to the component of the observe r'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 ar ound 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, fo r given assumptions about the variability of the distance to points on the surface patch. We illustrate our analysis quantitatively using synthetic da ta, and we also validate it qualitatively on real image sequences.