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One of the ways we perceive shape is through seeing motion. Visual motion may be actively generated (for example, in locomotion), or passively observed. In the study of how we perceive 3D structure from motion (SfM), the non-moving, passive observer in an environment of moving rigid objects has been used as a substitute for an active observer moving in an environment of stationary objects; the 'rigidity hypothesis' has played a central role in computational and experimental studies of SfM. Here we demonstrate that this substitution is not fully adequate, because active observers perceive 3D structure differently from passive observers, despite experiencing the same visual stimulus: active observers' perception of 3D structure depends on extra-visual self-motion information. Moreover, the visual system, making use of the self-motion information treats objects that are stationary (in an allocentric, earth-fixed reference frame) differently from objects that are merely rigid. These results show that action plays a central role in depth perception, and argue for a revision of the rigidity hypothesis to incorporate the special case of stationary objects.

The prediction of future positions of moving objects occurs in cases of actively produced and passively observed movement. We study the difference between active and passive movement prediction by asking subjects to estimate displacements of an occluded moving target, where the movement is produced by the subject or passively observed; in the passive condition, the target trajectory is either a replay of a preceding active trajectory, or a constant-speed approximation. In the active condition estimates are more anticipatory than in the passive conditions, but in all conditions, estimates become less anticipatory as the prediction distance increases, or the prediction time decreases. Decreasing the congruence between motor action and visual feedback diminishes but does not eliminate the anticipatory effect of action; introducing eye tracking, however, does eliminate it. Our results are compatible with common mechanisms underlying both active and passive movement prediction, with additional movement-related information in the active case making predictions more anticipatory.

Much indirect evidence supports the hypothesis that transformations of mental images are at least in part guided by motor processes, even in the case of images of abstract objects rather than of body parts. For example, rotation may be guided by processes that also prime one to see results of a specific motor action. We directly test the hypothesis by means of a dual-task paradigm in which subjects perform the Cooper-Shepard mental rotation task while executing an unseen motor rotation in a given direction and at a previously learned speed. Four results support the inference that mental rotation relies on motor processes. First, motor rotation that is compatible with mental rotation results in faster times and fewer errors in the imagery task than when the two rotations are incompatible. Second, the angle through which subjects rotate their mental images, and the angle through which they rotate a joystick handle are correlated, but only if the directions of the two rotations are compatible. Third, motor rotation modifies the classical inverted V-shaped mental rotation response time function, favoring the direction of the motor rotation; indeed, in some cases motor rotation even shifts the location of the minimum of this curve in the direction of the motor rotation. Fourth, the preceding effect is sensitive not only to the direction of the motor rotation, but also to the motor speed. A change in the speed of motor rotation can correspondingly slow down or speed up the mental rotation.

A perturbative technique, the low-temperature expansion, is developed for matrix models of random surfaces. It can be applied to models with arbitrary target spaces, including ones with c>1. As a simple illustration, the series is worked out to 10th order for the surface coupled to a q-state Potts model. Accurate estimates for, e.g., $\gamma_{str}$ are obtained both in the low q (c<1) and high q (branched polymer) regimes, including the logarithmic corrections to scaling., Comment: 31 pages, uuencoded Postscript (also available at http://parthe.lpthe.jussieu.fr/~wexler/ )

We study $q=10$ and $q=200$ state Potts models on dynamical triangulated lattices and demonstrate that these models exhibit continuous phase transitions, contrary to the first order transition present on regular lattices. For $q=10$ the transition seems to be of 2nd order, while it seems to be of 3rd order for $q=200$. For $q=200$ the phase transition also induces a transition between typical fractal structures of the piecewise linear surfaces corresponding to the triangulations. The typical surface changes from having a tree-like structure to a fractal structure characterizing pure gravity when the temperature drops below the critical temperature. An investigation of the alignment of spin clusters shows that they are strongly correlated to the underlying fractal structure of the triangulated surfaces., Comment: 22 pages, uuencoded compressed ps-file. Use csh file.uu to get ps-file

The matrix model of random surfaces with c = inf. has recently been solved and found to be identical to a random surface coupled to a q-states Potts model with q = inf. The mean field-like solution exhibits a novel type of tree structure. The natural question is, down to which--if any--finite values of c and q does this behavior persist? In this work we develop, for the Potts model, an expansion in the fluctuations about the q = inf. mean field solution. In the lowest--cubic--non-trivial order in this expansion the corrections to mean field theory can be given a nice interpretation in terms of structures (trees and ``galaxies'') of spin clusters. When q drops below a finite q_c, the galaxies overwhelm the trees at all temperatures, thus suppressing mean field behavior. Thereafter the phase diagram resembles that of the Ising model, q=2., Comment: 25 pp. (voodoo PostScript replaced by original LaTeX), NBI-HE-94-28

Recently, the free energy of the target space mean field (TSMF) matrix model has been calculated in the low temperature phase, order-by-order in a low temperature expansion. The TSMF model is a matrix model whose discrete target space has an infinite coordination number, and whose free energy assumes a universal form, corresponding to baby universes joined into a tree. Here the free energy is summed to all orders, and expressed through a transcendental algebraic equation, using which we analyze the critical phenomena that occur in the TSMF model. There are two critical curves, at which the matter and the geometry become critical, and for which the critical exponents are alpha = 1/2 and gamma_str = -1/2, respectively. There is a bicritical point where the curves meet, at which gamma_str = +1/3., Comment: 12 pp., TeX + harvmac + epsf, PUPT-1412

We consider the spherical limit of multi-matrix models on regular target graphs, for instance single or multiple Potts models, or lattices of arbitrary dimension. We show, to all orders in the low temperature expansion, that when the degree of the target graph $\Delta\to\infty$, the free energy becomes independent of the target graph, up to simple transformations of the matter coupling constant. Furthermore, this universal free energy contains contributions only from those surfaces which are made up of ``baby universes'' glued together into trees, all non-universal and non-tree contributions being suppressed by inverse powers of $\Delta$. Each order of the free energy is put into a simple, algebraic form., Comment: 19pp. (uses harvmac and epsf), PUPT-1398

We show how to expand the free energy of a matrix model coupled to arbitrary matter in powers of the matter coupling constant. Concentrating on $\nu$ uncoupled Ising models---which have central charge $\nu/2$---we work out the expansion to sixth order for $\nu$ = 1, 2, and 3. Analyzing the series by the ratio method, we exhibit the spin-ordering phase transition. We discuss the limit $\nu \rightarrow \infty$, which is especially clear in the low temperature expansion; we prove that in this limit the dependence of the model on $\nu$ becomes trivial., Comment: 10 pp. and 5 figures (included), TeX + harvmac, PUPT-1384