The optic flow field

Optic flow is typically represented as an instantaneous velocity field in which each vector corresponds to the optical motion of a point in the environment, as in Figure 84.1A. It is immediately apparent that this flow field has a radial structure, with a focus of expansion (FOE) lying in the direction of self-motion. When one views the moving display, it is also apparent that the environment is a planar surface receding in depth. The radial pattern of vector directions depends solely on the observer's direction of translation, or heading, and is independent of the 3D structure. The magnitude of each vector, on the other hand, depends on both heading and depth and decreases quickly with distance. The radial flow pattern thus specifies one's current heading, whether or not the local FOE itself is visible. This was Gibson's fundamental hypothesis about the perception of self-motion from optic flow, and was the starting point for both psychophysical and neurophysiological experiments.

Figure 84.1..  

Retinal flow field for a ground plane with rotation about a diagonal axis. a, Translational component produced by observer translation toward the x: radial flow from the heading point. b, Rotational component produced by eye rotation downward and to the right: lamellar flow upward and to the left. c, Flow field produced by the sum of a and b due to heading toward the x while rotating to track the o on the ground plane. The singularity in the field is now at the fixation point. Note the motion parallax (differences in vector direction and length) at different distances across the surface.

Although the velocity field description is compatible with the motion selectivity of cortical areas V1 and MT, it does not represent higher-order temporal components of the optic flow such as acceleration, or the trajectories of points over time. This appears to be a reasonable approximation, for the visual system is relatively insensitive to acceleration and relies primarily on the first-order flow to determine heading (Paolini et al., 2000; Warren et al., 1991a).

Observer Rotation

However, the detection of optic flow by a moving eye is complicated by the fact that the eye can also rotate (Gibson, 1950, pp. 124–127). If the observer simply translates on a straight path, the flow pattern on the retina is radial (Fig. 84.1A). This is called the translational component of retinal flow, and recovery of heading from it is straightforward. A rotation of the observer, such as a pursuit eye or head movement, merely displaces the image on the retina, producing the rotational component of retinal flow.1 Specifically, pitch or yaw of the eye create patterns of vertical or horizontal lamellar flow (Fig. 84.1B), and roll about the line of sight creates a pattern of rotary flow. But if the eye is simultaneously translating and rotating, which commonly occurs when one fixates a point in the world during locomotion, the retinal flow is the vector sum of these two components (Fig. 84.1C). The resulting flow field is more complex, without a qualitative feature corresponding to one's heading; indeed, the singularity in the flow field is now at the fixation point.

Thus, to determine heading, the visual system must somehow analyze the translational and rotational components and recover the direction of self-motion. This has come to be known as the rotation problem. Fortunately, the retinal flow in a 3D scene contains sufficient information to solve this problem in principle. Specifically, motion parallax between points at different depths corresponds to observer translation, whereas common lamellar motion across the visual field corresponds to observer rotation.

The Path of Self-motion

Even if the rotation problem can be solved, however, it only yields one's instantaneous heading. A further problem is that the velocity field does not specify one's path over time—for instance, whether one is traveling on a straight or a curved path. In fact, the same flow field can be generated by a straight path together with eye rotation or by a circular path of self-motion. A particularly troublesome case appears in Figure 84.2, in which translation plus rotation about a vertical axis produces the same velocity field as a circular path on the ground plane. When presented with such flow displays, observers often report seeing a curved path of self-motion rather than a straight path plus rotation. How, then, can one determine whether one is traveling on a straight or a curved path? We will call this the path problem.

Figure 84.2..  

Retinal flow field for a ground plane with rotation about a vertical axis. a, Translational component produced by observer translation toward the x. b, Rotational component produced by eye rotation to the right. c, Flow field produced by the sum of a and b due to translating toward the x while rotating to track the o on the post, which is at eye level. An identical velocity field is generated by travel on a circular path to the right, with instantaneous heading along the tangent toward the x.

Before tackling these issues, let's consider the detection of optic flow patterns.