An algorithm for associating the features of two images.

In this paper we describe an algorithm that operates on the distances between features in the two related images and delivers a set of correspondences between them. The algorithm maximizes the inner product of two matrices, one of which is the desired 'pairing matrix' and the other a 'proximity matrix' with elements exp (-rij2/2 sigma 2), where rij is the distance between two features, one in each image, and sigma is an adjustable scale parameter. The output of the algorithm may be compared with the movements that people perceive when viewing two images in quick succession, and it is found that an increase in sigma affects the computed correspondences in much the same way as an increase in interstimulus interval alters the perceived displacements. Provided that sigma is not too small the algorithm will recover the feature mappings that result from image translation, expansion or shear deformation--transformations of common occurrence in image sequences--even when the displacements of individual features depart slightly from the general trend.

Visual motion ambiguity.

This paper discusses a number of points that arise in the visual interpretation of optic flow fields. Particular attention is given to the case in which a flow field is visually ambiguous, and the paper concludes with a detailed description of the case of the moving plane.

The visual ambiguity of a moving plane.

It is shown that the optic flow field arising from motion relative to a visually textured plane may be characterized by eight parameters that depend on the observer's linear and angular velocity and the coordinate vector of the plane. These three vectors are not, however, uniquely determined by the values of the eight parameters. First, the optic flow field does not supply independent values for the observer's speed and distance from the plane; it only gives the ratio of these two quantities. But more unexpectedly, the equations relating the observer's linear velocity and the plane's coordinate vector to the eight parameters are still satisfied if the two vectors are interchanged or reversed in direction, or both. So in addition to the veridical interpretation of the optic flow field there exist three spurious interpretations to be considered and if possible excluded. This purpose is served by the condition that an interpretation can be seriously entertained only if it attributes every image element to a light source in the observer's field of view. This condition immediately eliminates one of the spurious interpretations, and exhibits the other two as mutually inconsistent: one of them is tenable only if all the visible sources lie on the forward half of the plane (relative to the observer's linear velocity); the other only if they all lie on the backward half-plane. If the sources are distributed over both halves of the plane, only the veridical interpretation survives. Its computation involves solving a 3 X 3 eigenvalue problem derived from the flow field. If the upper two eigenvalues coincide, the observer must be moving directly towards the plane; if the lower two eigenvalues coincide, his motion must be directly away from it; in both cases the spurious interpretation merges with the veridical one. If all three eigenvalues are equal, it may be inferred that either the observer's linear velocity vanishes or the plane is infinitely distant.

A computational model of binocular depth perception.

We point out that the horizontal disparities between a pair of retinal images are inadequate for computing the three-dimensional structure of a scene unless supplemented by independent information about the distance and direction of the fixation point. We suggest that this supplementary information is derived not from non-visual sources, but from the vertical disparities of a few non-meridional image points. This hypothesis is shown to account quantitatively for Ogle's induced effect--the marked distortion of a scene by a vertically magnifying lens placed in front of one eye.

The role of the vertical dimension in stereoscopic vision.

It is shown how the full three-dimensional structure of a scene may in principle be computed from a correlated pair of retinal images, when all that is known about the orientations of the two eyes is that the planes of their horizontal meridians accurately coincide. The vertical dimension plays a crucial role in the theory; visible points which lie on the horizontal meridian supply no information about the angle of convergence or the direction of gaze. (i) If the scene contains three or more nonmeridional points, not all lying in a vertical plane, then their positions in space are fully determined by the horizontal and vertical coordinates of their images on the two retinas. (ii) If just two nonmeridional points are visible, or more than two, lying in a vertical plane, then their retinal images admit, in general, just two distinct three-dimensional interpretations. One of these is usually unrealistic; but a choice between them may be perceptually difficult if the vertical plane containing the points is nearly perpendicular to the interocular axis. These results suggest that vertical disparities may play an important role in the binocular perception of absolute depth. Elsewhere (Mayhew, this issue) this suggestion is found to provide a quantitative explanation of Ogle's 'induced effect'.

The interpretation of a moving retinal image.

It is shown that from a monocular view of a rigid, textured, curved surface it is possible, in principle, to determine the gradient of the surface at any point, and the motion of the eye relative to it, from the velocity field of the changing retinal image, and its first and second spatial derivatives. The relevant equations are redundant, thus providing a test of the rigidity assumption. They involve, among other observable quantities, the components of shear of the retinal velocity field, suggesting that the visual system may possess specialized channels for computing these components.

The perception of music.

A common but none the less remarkable human faculty is the ability to recognize and reproduce familiar pieces of music. No two performances of a given piece will ever be acoustically identical, but a listener can perceive, in both, the same rhythmic and tonal relationships, and can judge whether a particular note or phrase was played out of time or out of tune. The problem considered in this lecture is that of describing the conceptual structures by which we represent Western classical music and the processes by which these structures are created. Some new hypotheses about the perception of rhythm and tonality have been cast in the form of a computer program which will transcribe a live keyboard performance of a classical melody into the equivalent of standard musical notation.

The kinetics of osmotic transport through pores of molecular dimensions.

This paper presents a theoretical analysis of the kinetics of osmotic transport across a semipermeable membrane. There is a thermodynamic connection between the rate of flow under a hydrostatic pressure difference and the rate of flow due to a difference in solute concentration on the two sides. One might therefore attempt to calculate the osmotic transport coefficient by applying Poiseuille's equation to the flow produced by a difference in hydrostatic pressure. Such a procedure is, however, inappropriate if the pores in the membrane are too small to allow molecules to "overtake." It then becomes necessary to perform a statistical calculation of the transport coefficient, and such a calculation is described in this paper. The resulting expression for the number of solvent molecules passing through a pore per second is J = m D(1) deltan(1)/l(2) where m is the number of solvent molecules in the pore, l is the length of the pore, D(1) is the self-diffusion coefficient of the solute, and deltan(1) the difference in solvent mole fraction on the two sides of the membrane. This equation is used for estimating the number of pores per unit area of the squid axon membrane; the result is 6 x 10(9) pores/cm(2).