Where We Go with a Little Good Information

When observers move through an environment, they are immersed in a sea of motions that guide their further movements. The horizontal relative motions of all possible pairs of stationary objects fall into three classes: They converge, diverge and slow down, or diverge with increasing velocity. Conjoined with ordinal depth information, the first two motions reveal nominal invariants, constraining heading to one side of the visual field. When two object pairs yield invariants on opposing sides of the heading, they can constrain judgments to a narrow region. Distributional analyses of responses in an experiment involving simulated observer movement suggest that observers follow these constraints. When people walk, run, or drive through the world, how do they know where they are going? How do they know their heading so they can safely avoid obstacles in their path? Over the past 50 years, particularly because of the work of Gibson (1950, 1979), these questions have received sustained attention. Over the past 25 years, advances in neurophysiology have revealed cells in the visual systems of monkeys, pigeons, and cats that respond to relative motion pooled over wide regions of the visual field (Allman, Miezin, & McGuinness, 1985; Bradley, Maxwell, Andersen, Banks, & Shenoy, 1996; Bridgeman, 1972; Frost & Nakayama, 1983; Pasternak, Albano, & Harvitt, 1990). Thus, many researchers have sought computational and psychophysical evidence that human beings might negotiate environments on the basis of such cells (Hildreth, 1992; Nakayama & Loomis, 1974; Rieger & Lawton, 1985; Warren & Saunders, 1995). The supporting evidence is substantial, provided that test environments contain many elements. Motion pooling fails, however, when simulated navigation occurs through relatively sparse environments (Cutting, 1996). In such environments, however, considerable accuracy is achieved by human observers from the relative motions of a few pairs of stationary objects. Our data suggest that this accuracy is based on multiple constraints derived from relative motions of these object pairs. Accurate heading judgments in natural environments can also be accomplished in this way. Mathematically, the motions of stationary objects around a moving observer can be parsed in several ways (see Cutting, 1986; Cutting, Springer, Braren, & Johnson, 1992; Koenderink & van Doorn, 1975; Longuet-Higgins & Prazdny, 1980; Regan & Beverley, 1982). Here we develop a new approach, considering the relative motions of pairs of stationary objects with respect to the eye of the moving observer. The horizontal motion of all possible pairs falls into three classes: Object pairs can converge, they can diverge and slow down, or they can diverge with increasing velocity (Cutting, 1996). In the forward visual field, convergence is always acceleratory, except in certain cases when the observer is moving along a curved path. Figure 1 shows these three classes of motion with respect to an object to the right of one’s path. Two important rules about heading, also shown in Figure 1, then follow. First, when objects converge, one’s instantaneous heading is always outside of the nearest member of the pair, in this case to the left. This rule has no exceptions. Second, when objects decelerate apart, the same is true. This rule is qualified in that both objects must be within 45° of one’s heading, but this condition is not overly restrictive because about 90% of pedestrian gazes fall within this bound (Wagner, Baird, & Barbaresi, 1981). These two rules are optical invariants, or statistically certain sources of information (Gibson, 1979). There is also a third relative-motion class. When a pair of objects accelerate apart, one’s heading is unsure. To compute the efficacy of accelerating divergence as it predicts one’s heading, we assumed that a pedestrian’s gaze is within ±90° of the heading on a reference object at 30 m, near the median distance for pedestrian fixation (Wagner et al., 1981), and that the second object under consideration is between 1 and 100 m and within ±20° to either side of the first. We then sampled gaze-heading angles at 1° intervals between ±90°, computed the relative areas of the three region types shown in Figure 1, and then weighted angular gaze-heading calculations by their naturally occurring frequency (Wagner et al., 1981). The resulting values are shown at the bottom of Figure 1. (If the ratio of depths is known, further refinements can be made.) Thus, this motion class yields an optical heuristic, a probabilistic information source (Gilden & Proffitt, 1989), suggesting that heading is most often to the outside of the farther object in the pair. Notice that, taken singly, none of these potential sources of information predicts one’s absolute heading, or the exact direction in which one is moving. In particular, the two invariants nominally constrain one’s heading direction; they specify that it is left or right of a particular object, but not by how much. Yet there is ample evidence that observers can report their absolute headings with reasonable accuracy (Royden, Banks, & Crowell, 1992; van den Berg & Brenner, 1994; Warren & Hannon, 1988). How might nominal invariants yield accurate, near-metric heading judgments? Our answer is that invariants on both sides of one’s heading constrain judgments to a narrow region.