Surface Structure and Three-Dimensional Motion from Image Flow Kinematics

Abstract

This study concerns a new formulation and method of solu tion of the image flow problem. It is relevant to the maneu vering of a robotic system through an environment containing other moving objects or terrain. The two-dimensional image flow is generated by the relative rigid-body motion of a smooth, textured object along the line of sight to a monocular camera. By analyzing this evolving image sequence, we hope to extract the instantaneous motion (described by six degrees of freedom) and local structure (slopes and curvatures) of the object along the line of sight. The formulation relates a new local representation of an image flow to object motion and structure by twelve nonlinear algebraic equations. The repre sentation parameters are given by the two components of image velocity, three components of rate of strain, spin, and six independent image gradients of rate of strain and spin, evaluated at the point on the line of sight. These kinematic variables are motivated by the deformation of a finite element of flowing continuum. A method for solving these equations was devised and successfully implemented on a VAX com puter. A number of examples were explored revealing two classes of ambiguous scenes (i.e., nonunique solutions are ob tained). A sensitivity analysis was conducted to estimate noise levels in the representation parameters that still yield acceptable solutions; indications are that the method is quite stable. Finally, an approach is suggested by which the kine matic variables may be extracted from evolving contours in an image sequence.

Center of mass perception and inertial frames of reference

“…Free variation in the perspective mapping is an unavoidable facet of the visual perception of shape. A more promising approach, recognizing that the observer is never motionless, is to study the regularities in the optical flow patterns that are generated over perspective transformations (Koenderink, 1975;Koenderink, 1986;Koenderink, 1990;Koenderink & van Doom, 1978;Nakayama, 1985;Thompson, 1989;Thompson, Mutch, & Berzins, 1984Waxman & Ullman, 1985). We must emphasize that we have not pursued this aspect of center of mass perception here.…”

Geometry and Photometry in 3D Visual Recognition

“…Waxman and Wohn (1985) and Bachelder and Uliman (1992) suggest methods for correspondence that account for the 3D to 2D geometry in a way that is limited to locally planar surfaces. Waxman and Wohn suggest an approach by which the surface is broken down into local planar patches, and they derive correspondence using the observation that planar surfaces under perspective projection give rise to a quadratic flow field (Waxman and Ullman, 1985). As with the method of Waxman and Ullman, the smaller the patch size the more unstable the system becomes because of narrowing of the field of view (see Adiv, 1989).…”

The feasibility of motion and structure from noisy time-varying image velocity information

“…The latter is by far the most important issue if a motion and structure algorithm is to be judged a success. As Waxman and Ullman [1983; and others have noted, motion and structure algorithms that use image velocities of neighboring image points require accurate differences of these similar velocities. That is, solving systems of equations effectively requires subtraction of very similar quantities: the error in the quantities themselves may be quite small but since the magnitudes of their differences are quite small, the relative error in them can be quite large.…”

“…Table 2 shows the 5 motions (labeled W6 to Wl0 to correspond to examples 6 to 10 in Waxman and Ullman [1983]) and their duals, DW6 to DWl0. These are not realistic, everyday, motions one might expect an autonomous vehicle to undergo.…”