A. J. van Doorn

Local structure of movement parallax of the plane

The movement parallax field due to the translation of an observer relative to a plane surface is studied in an infinitesimal neighborhood of a visual direction. The parallax field is decomposed into elementary transformations: a translation, a rigid rotation, a similarity, and a deformation. A topologically invariant classification based on critical-point analysis is also obtained. It is shown that the field is either that of a node or that of a saddle point. Numerical results for a general case are offered as illustration. We discuss the relevance of the local, as opposed to the global structure of the parallax field for visual perception and the visual space sense.

Photometric Invariants Related to Solid Shape

An attempt is made to identify the structural invariants of the field of isophotes on lambertian surfaces of arbitrary shape under arbitrary illumination. It is a priori clear that such invariants play an important part in the perception of solid shape by means of chiaroscuro. We have found several such invariants. The topological structure of the field of isophotes is determined by the collection of nested, closed parabolic curves on the surface. An important class of stationary points of the field clings to these parabolic lines. The isophotes cut the parabolic lines at a constant angle, irrespective of the location of the light sources. For certain canonical surface undulations the field of isophotes is explicitly given.

Temporal properties of the visual detectability of moving spatial white noise.

SummaryWe obtained movement detection thresholds for two-dimensional random speck-patterns (“Julesz” patterns) homogeneously moving over the whole target field (5.21×5.31 degrees of visual angle). We alternated between two uncorrelated but otherwise similar patterns, one moving with velocity →v1, the other with velocity →v2, such that each pattern was on for T ms. We masked this pattern (signal) with spatio-temporal white noise (“snow”). The total r.m.s. contrast was kept constant, whereas the ratio of the r.m.s. contrasts of signal and noise was varied. The square of this ratio was designated SNR.At low SNR values the pattern was not perceptually different from the snow alone. At high SNR values the subject detected spatio-temporal correlation (e.g., movement). In these experiments we determined the threshold SNR values as a measure of the detectability of spatio-temporal correlation as a function of the parameters T, →v1 and →v2.When →v1 and →v2 were sufficiently dissimilar one of three percepts occurred: for very large T the alternation could be followed, for very small T two transparent, simultaneously moving sheets of noise-pattern with different velocities could be seen. For intermediate T-values no systematic movement at all could be observed. At these T-values the threshold SNR was maximal. This “critical” T-value decreased with increasing velocity.We found that it was possible to have more than one percept of uniform smooth movement at a single location in the visual field if these movements had velocity vectors with an angular difference of at least 30 deg or if their magnitudes differed by at least a factor of 4.

Exterospecific component of the motion parallax field

For the egocentric orientation of observers moving with respect to a plane (e.g., pilots and automobile drivers), the movement parallax field provides the main cue. The parallax field is split into a lamellar and a solenoidal part, and it is shown that the solenoidal part is purely propriospecific. For instance, it can be shown that this component can be completely canceled by an appropriate eye movement. Thus all exterospecific information is contained in the lamellar part, and this part is completely determined by the divergence of the parallax field. Thus the measure of expansion of the visual field as a function of direction of gaze is sufficient to provide all information available for egocentric orientation. It is further shown that the widely used focus of expansion, as introduced by Gibson, is not invariant against eye movements and does not (in general) correspond to extrema of the divergence.

Detection of coherent movement in peripherally viewed random-dot patterns

We studied the detection of coherent motion in stroboscopically moving random-dot patterns for foveal vision and at eccentricities of 6, 12, 24, and 48 deg in the temporal visual field. Threshold signal-to-noise ratios (SNRs) were determined as a function of velocity for a range of stimulus sizes. It was found that the motion-detection performance is roughly invariant throughout the temporal visual field, provided that the stimuli are scaled according to the cortical magnification factor to obtain equivalent cortical sizes and velocities at all eccentricities. The maximum field velocity compatible with the percept of coherent motion increased about linearly with the width of the square stimuli. At this high-velocity threshold any pixel crossed the field in five to nine equal steps with a constant total crossing time of 50-90 msec, regardless of stimulus size or eccentricity. The lowest SNR values were reached at the optimal or tuning velocity V0. They approached the amazingly low values of 0.04-0.05 for large stimuli and at all eccentricities. Regardless of stimulus size, the parameter V0 increased about linearly with eccentricity from roughly 1 deg sec-1 at the fovea to some 8 deg sec-1 at 48 deg in the temporal visual field.

Depth and shape from differential perspective in the presence of bending deformations

It is shown that there exist classes of useful partial solutions to what has become known as the structure from motion problem, that is, the problem of finding the three-dimensional shape as well as the movement in space of objects on the basis of their optical projections in several distinct phases of the movement. The class of solutions treated in this paper is quite different from the usual treatment in that the rigidity condition is relaxed so as to include the important class of bending deformations. For instance, we show a case in which important aspects of shape are obtained for a configuration of seven points, no four of which move in a rigid way, and which can be obtained from only two views. (This violates all requirements of the structure from motion theorem.) We discuss an algorithm that obtains the three-dimensional configuration of a polyhedral vertex up to a relief transformation (a similar transformation to the one that maps object to image space in paraxial optics), in the presence of arbitrary flections of the edges, from the projection and the instantaneous-movement parallax. Such a solution can, for instance, be used to predict side views of the vertex, as is demonstrated for a specific example. The solution can be used in an iterative fashion on triangulations to solve the case of extended curved surfaces suffering arbitrary bendings. The class of bending deformations encompasses all deformations that conserve distances along the surface (but not necessarily through space); thus stretchings are excluded. Practical examples include, e.g., deformations of flexible but nonstretchable shells. Actual numerical examples illustrate the practicability of the method.

Geometrical modes as a general method to treat diffuse interreflections in radiometry

The problem of interreflections for Lambertian surfaces of arbitrary shape and with varying reflectance is of interest for many practical applications. We present a general method to approach this problem. We define photometric modes that are uncoupled in the sense that each mode may be assigned a (pseudo) reflectance and that interreflections among modes vanish. Then the problem is formally identical with that of a convex body, in which interreflections are of no importance. The photometric modes depend on the shape of the body. In many practical cases one or a few modes dominate, and the reflected radiance depends more on the shape of the body (the dominant mode) than on the precise irradiance distribution. A few examples are treated explicitly. The redistribution of radiation described by the modes is treated by means of the net vector flux and the space density of radiation. Knowledge of these fields for the dominant mode yields considerable intuitive insight in the physical situation and provides the means to estimate the effects of painting part of the surface or of the introduction of screens.

Detectability of velocity gradients in moving random-dot patterns

Abstract We present a subject with two moving spatial noise patterns at both sides of a common border. The patterns are masked with spatio-temporal white noise and we measure the threshold signal-to-noise ratio at which a subject can discriminate the case of two differently moving patterns from that of a single uniformly moving pattern over the whole field. At some experiments the patterns move in the same direction but the magnitude of the velocity of one of the patterns is varied. In other experiments both patterns move with the same velocity but in different directions. In no case do we find a dependence of the thresholds on the orientation of the common border with respect to the direction of movement of the patterns. In all cases we find a certain region of magnitudes of the variable velocity or of the orientation-difference between the two velocities in which the thresholds are very high, whereas for larger differences either in magnitude or in direction the thresholds are generally at values of less than one for the signal-to-noise ratio. The region of the raised thresholds for both magnitude and orientation differences can be described with a single simple expression: thresholds are high whenever the magnitude of the velocity difference of the two patterns is less than one half of the magnitude of the common component of the two velocities. This is a kind of Weber law for the velocity vectors. It is estimated that you may not pick many more than 150 differently moving patterns (both with respect to magnitude and direction) in such a way that any pair of them leads to easy discrimination (low threshold signal-to-noise ratio).

Viewing-distance invariance of movement detection.

SummarySince visual movement information is often presented in electronic displays or films it is amazing that there is a paucity of research on the influence of viewing distance on motion detection in cinematograms. We report a relatively high degree of detection constancy with changing viewing distance for coherent motion in random-pixel cinematograms. A constant performance irrespective of viewing-distance is called ‘distance-invariance’ and for motion detection it proves to hold reasonably well for a relatively wide range of viewing distances both for foveal and eccentric vision. The limits of this viewing-distance invariance are explored as a function of screen velocity. Detection performance is quantified by a theshold signal-to-noise-ratio (SNR-) value, S, which is determined as a function of velocity for a range of viewing distances from 53 to 13476 mm for foveal vision and from 60 to 1925 mm at 24° eccentricity on the nasal horizontal meridian of the right eyes retina. The data can be explained, at least qualitatively, by a model in which a spatial-resolution stack has a stack of velocity-tuned motion detectors at every resolution layer. Such a ‘stack-of-stacks’ model is in line with proposals for contrast-detection stack-models, but it suggests that the usual hypothesis that motion perception is based on the activity of two separate systems, the short-range and the long-range system, might be superfluous. This two-systems distinction was largely based on the different performance found for moving random dot patterns and moving form-defined stimuli. A moving random pixel array viewed at very close range (e.g. 6 cm) presents the subject with relatively large almost square ‘blobs’, which are less dissimilar from the phi-stimuli used in classic motion perception studies than random dot stimuli at the usual medium to large viewing distances. It leads to maximum displacement threshold (Dm-) values that are not untypical of the ‘long-range’ system, but by gradually increasing the viewing-distance and thus decreasing the pixel-size a continuous change is found from typical long-range to typical short-range values of Dm. The two-systems distinction for motion detection appears to refer to the stimulus rather than to the visual system: The motion-detection system might be forced into a local or a global ‘mode of operation’ by the choice of stimulus.

Inhomogeneity and anisotropies for motion detection in the monocular visual field of human observers.

Signal-to-noise-ratio (SNR) thresholds were measured for the detection of coherent motion in moving random pixel arrays of constant root-mean-square contrast (35%) and constant average luminance (48 cd/m2) for 8 or 16 directions of motion at 25 positions in the visual field of the right eye. Five observers took part in this perimetric study of motion detection. The 24 eccentric positions were chosen on 8 equally spaced radial lines at the eccentricities 6, 24, and 48 degrees, the 25th position was centred on the fovea. At these positions we analysed the threshold SNR-value as a function of motion direction alpha. A significant modulation of the threshold with alpha is called an anisotropy. Anisotropies were found for low to medium velocities at positions on and near the vertical meridian, where the thresholds proved to be highest for vertical motion directions (up or down). On the horizontal meridian no significant anisotropies were found. Also on the oblique radials anisotropies were found, especially at 225 degrees (lower nasal quadrant of the visual field, upper temporal quadrant of the retina), but these were milder than those on the vertical meridian. The diameter of the stimulus is an important parameter and its influence was explored, albeit incompletely. Also inhomogeneities were found. This is defined as a consistent modulation of the threshold SNR-value with position A, the position along an equi-eccentricity circle (A-inhomogeneity), or with eccentricity E (E-inhomogeneity) or both. A simple acuity-scaling optimized for the nasal retina takes care of most of the E-inhomogeneity, but an A-inhomogeneity stays rather prominent. It too is characterized by higher thresholds near the vertical meridian than near the horizontal meridian. The findings suggest that iso-threshold curves are elliptical or egg-shaped with their long axis on the horizontal meridian and shifted somewhat out of naso-temporal symmetry towards the nasal half of the retinal field. As with the anisotropies the inhomogeneity grows in amplitude for decreasing velocity below medium velocity values of 1-2 pixels/frame, but in contradistinction to the anisotropies it is present and even increases in amplitude for increasing velocities above these medium values of 1-2 pixels/frame as well. The results are discussed in the light of other perimetric studies of motion detection and acuity, in the light of a model postulating the cooperation of groups of velocity-tuned bilocal motion detectors, and in the light of recent ideas on structure and function of primate cortical areas and processing streams.