Mark Wexler
> supplementary information
Supplementary information for:
M. Wexler (2003). Voluntary head movement and allocentric perception of space. Psychological Science, 14, 340-346.

In Experiments 2 and 3, subjects performed backwards-and-forwards head movements (voluntary and involuntary). These movements we measured by a headtracker; the information from the headtracker was used, in real time, to create optic flow in a cloud of dots displayed on a computer screen, as shown in Fig. 1.

  Fig. 1

Note that only the dots (and not the edges of the screen, or anything else for that matter) were visible to the subject.

To explain why the dots moved the way they did, I will first have to explain a particular invariance of optic flow. As far as I know, this invariance was first noticed by Jacques Droulez, was first described in Wexler, Lamouret & Droulez (2001), and is related to the induced effect in binocular vision.

  Fig. 2a

Consider the situation depicted in Fig. 2a. An observer approaches a rotating surface. The approach component of the relative motion results in a radially expanding optic flow, shown on the left. The rotation of the surface gives rise to a kind of linear contraction optic flow, shown on the right.

  Fig. 2b

Now, consider the configuration in Fig. 2b. The observer is approaching a (differently) rotating surface, but this surface recedes, at the same speed as the observer. Since there’s no relative translation between the observer and the surface, there isn’t any radially expanding or contracting optic flow. On the other hand, the surface, which is now tilted in depth about a vertical axis and which also rotates about a vertical axis (compare to surface in Fig. 2a), produces a linear expansion optic flow, perpendicular to that of the surface in Fig. 2a.

  Fig. 2c

Here’s the interesting bit. If the lengths of the arrows are adjusted just right—in other words, if the relative speeds of the translation and rotation satisfy a certain relation—the optic flows depicted in Figs. 2a and 2b are approximately the same. This is illustrated in Fig. 2c. Put differently: the rotation in Fig. 2a can approximately cancel the translation, resulting in the purely rotational optic flow of Fig. 2b.

The reason why I’ve been saying “approximately” and using wiggly equal signs is that, strictly speaking, this invariance is only true in the limit of small objects (or, technically, for first-order optic flow). But, for the size of the stimuli used in these experiments, the components of optic flow that violate the above invariance are practically invisible. A mathematical derivation of all this can be found in Wexler et al. (2001).

Now we can go back to Fig. 1. The moving observers, if they are presented with just the right optic flow (and they were), can perceive two different 3D scenes.

  Fig. 3

The observer may perceive the red object: a plane whose center is fixed in space (but which moves relative to the observer, who is himself moving), and which undergoes a rotation about a horizontal axis (the object is seen sideways). This is just the configuration described in Fig. 2a. Or, the subject may perceive the black object, whose center moves in space so as to remain at a fixed distance from the observer, and which rotates about a vertical axis: the configuration of Fig. 2b.

The brain often interprets ambiguous moving stimuli in such a way as to minimize perceived speed (cf. apparent motion and the aperture problem). In the situation shown in Fig. 3, however, there is a dilemma: should we minimize relative or egocentric speed (in which case the black solution should be preferred), or absolute or allocenteric speed (which should bias towards the red solution)? In Wexler et al. (2001), we showed that there is a very strong allocentric bias. This is surprising because the retinal image, which is often considered to be the main or even sole input to vision, is egocentric.

In this article (Wexler, 2003), I show that this allocentric bias is much stronger when observers move actively than when they are displaced passively.