Closed-form solutions to image flow equations for 3D structure and motion

Abstract

A major source of three-dimensional (3D) information about objects in the world is available to the observer in the form of time-varying imagery. Relative motion between textured objects and the observer generates a time-varying optic array at the image, from which image motion of contours, edge fragments, and feature points can be extracted. These dynamic features serve to sample the underlying "image flow" field. New, closed-form solutions are given for the structure and motion of planar and curved surface patches from monocular image flow and its derivatives through second order. Both planar and curved surface solutions require at most, the solution of a cubic equation. The analytic solution for curved surface patches combines the transformation of Longuet-Higgins and Prazdny [25] with the planar surface solution of Subbarao and Waxman [43]. New insights regarding uniqueness of solutions also emerge. Thus, the "structure-motion coincidence" of Waxman and Ullman [541 is interpreted as the "duality of tangent plane solutions." The multiplicity of transformation angles (up to three) is related to the sign of the Gaussian curvature of the surface patch. Ovoid patches (i.e., bowls) are shown to possess a unique transform angle, though they are subject to the local structure-motion coincidence. Thus, ovoid patches almost always yield a unique 3D interpretation. In general, ambiguous solutions can be resolved by requiting continuity of the solution over time.

Extraction of three-dimensional shape from optic flow: a geometric approach

“…9 Then we introduce some notation to describe a surface patch up to second order. Many descriptors of shape for the purpose of vision and image processing have been used, but the descriptors that have been used most often are the principal curvatures Kmax and Kmm and the Gaussian and mean curvatures K and H.' 3 The principal curvatures are the maximum and minimum of the normal curvature Kn. The normal curvature is obtained as the curvature of the curve that one gets when one cuts the surface with a plane through the normal of the surface.…”

“…From the point of view of human vision, the use of the group of rotations of the plane (corresponding to torsional eye movements or torsional movements of the object) is not so easily defended. It would be more natural to study the invariants of the velocity field under the full rotation group in three dimensions, SO (3). Since this is computationally much more complex and since SO(2) is a subgroup of S0 (3), one can view the current approach as a first step.…”

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

“…In effect, their motion and structure algorithm divides the computation into two steps: use a normal velocity distribution to compute image velocity and its first-and second-order spatial derivatives at an image point and then use these as input to an algorithm that solves the nonlinear equations relating motion and structure to the image velocity and its first-and second-order derivatives. More recently, Subbarao [1986] and Waxman et al [1987] have proposed closed-form solutions for motion and structure. This basically involves solving a cubic equation and a set of decoupled nonlinear equations.…”

“…In this article the discrepancy between a motion field and the measured optic flow field is considered as another source of input error. Recentl); Waxman et al [1987] and Subbarao [1986] have presented closed-form solutions to motion and structure (at one time) even if the surface is nonplanar. As Subbarao [1986] notes, any such algorithm's error behavior can be predicted analytically.…”

Motion recovery from image sequences: Discrete viewpoint vs. differential viewpoint

“…This problem has also been explored by many researchers: an algorithm was proposed in 1984 by Zhuang et al [20] with a simplified version given in 1986 [21]; and a first order algorithm was given by Waxman et al [8] in 1987. Most of the algorithms start from the basic bilinear constraint relating optical flow to the linear and angular velocities and solve for rotation and translation separately using either numerical optimization techniques (Bruss and Horn [2]) or linear subspace methods (Heeger and Jepson [3,4]).…”