Motion from flow along contours: A note on robustness and ambiguous cases

Abstract

The projected contours of a rigid body in motion define a velocity field, a flow in the image plane, where the tangential components are invisible. We investigate robustness when estimating motion parameters from the normal components of velocity along contours.

Ambiguous curves are contours for which no unique motion can be recovered from normal flow. We show that they are field lines of some extra velocity field, which causes an invisible motion when added on top of the actual velocity field. (These field lines coincide with the contour.) This is an analytical description of what Waxman and Wohn (1984) called “the aperture problem in the large,” which actually was studied by Lie (1893) under the concept of “self-projective curves,” if we specialize to a planar surface patch. We make a catalogue of the class of ambiguous curves.

We study the robustness of motion parameters, obtained by least-squares estimation, by calculating the a mount by which noise (in the normal flow) is magnified. For simple curves and a motion model with four unknowns-two translation components, rotation and expansion-there are explicit formulas relating the robustness of motion parameters to how large a portion of the contour we use. Sometimes it helps to use more global information (a larger portion of, e.g., a closed contour), and sometimes there is no way out, because we are working with contour segments that too closely resemble ambiguous curves. Neither is there any hope to recover the true velocity field from normal flow locally, since any contour falls into one of the “ill-conditioned contours.”