Subspace methods for recovering rigid motion I: Algorithm and implementation

Abstract

As an observer moves and explores the environment, the visual stimulation in his/her eye is constantly changing. Somehow he/she is able to perceive the spatial layout of the scene, and to discern his/her movement through space. Computational vision researchers have been trying to solve this problem for a number of years with only limited success. It is a difficult problem to solve because the optical flow field is nonlinearly related to the 3D motion and depth parameters.Here, we show that the nonlinear equation describing the optical flow field can be split by an exact algebraic manipulation to form three sets of equations. The first set relates the flow field to only the translational component of 3D motion. Thus, depth and rotation need not be known or estimated prior to solving for translation. Once the translation has been recovered, the second set of equations can be used to solve for rotation. Finally, depth can be estimated with the third set of equations, given the recovered translation and rotation.The algorithm applies to the general case of arbitrary motion with respect to an arbitrary scene. It is simple to compute, and it is plausible biologically. The results reported in this article demonstrate the potential of our new approach, and show that it performs favorably when compared with two other well-known algorithms.

A neural model of visually guided steering, obstacle avoidance, and route selection.

“…Several of these alternative approaches are compared to the computational mechanisms in the STARS model. Heeger and Jepson (1992) produced an influential mathematical analysis on the recovery of the components of optic flow, namely translation, rotation, and depth, from a set of discrete samples. Their subspace algorithm sets up a least-squares optimization problem that solves for the components of the flow field, without the use of explicit extraretinal information, although psychophysical studies have shown that extraretinal signals are required for accurate heading perception during sufficiently fast eye rotation.…”

A review on egomotion by means of differential epipolar geometry applied to the movement of a mobile robot

“…Hegger and Jepson estimator: Hegger and Jepson proposed the so-called linear subspace method [24,25]. Given the optical ow of a set of n image points, the following relationship can be formulated:…”

Egomotion Estimation Using Quadruples of Collinear Image Points

“…Due to its importance, many algorithms dealing with the problem of estimating egomotion have appeared in the literature. The following paragraphs provide a short review of a few representative methods; more detailed discussions can be found in [7,8,10]. Most of the methods reviewed here rely on the availability of a dense optical flow field to describe 2D motion.…”

“…Heeger and Jepson [7] also make use of the residual function introduced in [1] and propose an efficient search technique for locating its minimum. Hummel and Sundareswaran [8] present an algorithm for finding the rotational motion and one for locating the FOE.…”

“…The first algorithm is based on the observation that the curl of the optical flow field is approximately a linear function whose coefficients are proportional to the desired rotational parameters of motion. The algorithm for locating the FOE extends the work of Heeger and Jepson [7] by considering for each candidate FOE the projection of the optical flow along vectors emanating from the former. Da Vitoria Lobo and Tsotsos [10] develop a constraint (the Collinear Point Constraint -CPC) involving flow projections at three collinear image points, which provides a means for canceling rotation and at the same time constraining the FOE to lie on the line defined by the collinear points.…”