J. J. Koenderink

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.

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.

Generic neighborhood operators

A method that treats linear neighborhood operators within a unified framework that enables linear combinations, concatenations, resolution changes, or rotations of operators to be treated in a canonical manner is presented. Various families of operators with special kinds of symmetries (such as translation, rotation, magnification) are explicitly constructed in 1-D, 2-D, and 3-D. A concept of order is defined, and finite orthonormal bases of functions closely connected with the operators of various orders are constructed. Linear transformations between the various representations are considered. The method is based on two fundamental assumptions: a decrease of resolution should not introduce spurious detail, and the local operators should be self-similar under changes of resolution. These assumptions merely sum up the even more general need for homogeneity isotropy, scale invariance, and separability of independent dimensions of front-end processing in the absence of a priori information. >

Dynamic shape

Many useful notions of partial order and/or similarity and relatedness of different geometrical features of smooth shapes that occur in psychologically valid descriptions of shape have no equivalents in the usual geometrical shape theories. This is especially true where similarities are noted between objects of different connectivity: in almost all of the present theories the topological type generates the primary categorization. It is argued that such relations find a logical place only in shape theories that involve morphogenesis. Any object can be embedded uniquely in a morphogenetic sequence if one takes resolution as the parameter of the sequence. A theory of measurement is presented that allows one to define surfaces and (boundary-) curves on multiple levels of resolution. The embedding is essentially unique and is generated via a partial differential equation that governs the evolution. A canonical projection connects any high resolution specimen to lower resolution versions. The bifurcation set of the projection generates natural part boundaries. Singularities of the evolution are completely characterized as emergence, accretion and versification processes (involving topological change) and singularities by which inflections (inflection points for curves, parabolic curves for surfaces) are generated. The latter singularities involve a single process for the generation of inflections and three other processes by which the existing inflection structure may be changed. Relations with existing theories in vogue in robotics and AI, as well as in psychophysics are discussed.

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.

The distribution of human motion detector properties in the monocular visual field

The detection of coherent movement in stroboscopically (100 Hz) displayed moving random checkerboard (Julesz-) patterns is studied psychophysically for eccentricities up to 48 degrees in the temporal visual field. Starting from the assumption that the studied visual subsystem consists of ensembles of bilocal movement detectors (Reichardt-detectors), the parameters of these elementary detectors are deduced from the experimental results. This leads to the following interesting insights into the functional architecture of the system. At any eccentricity there is a critical velocity value Vc (near the center of the range of detectable velocities) at which both the spans and the delays reach their minimum value. Thus Vc can be defined as the ratio of the minimum span to the minimum delay values. At velocities below Vc the spans are constant and the delays are inversely proportional to V. At velocities above Vc the delays are constant and the spans increase proportional to V. The critical velocity Vc at any given eccentricity equals N times Vco, where Vco, is the critical velocity for foveal vision and N an eccentricity scaling factor. (N is the inverse normalized cortical magnification factor). Thus there is a complete structural invariance in terms of eccentricity-scaled units. Given the eccentricity scaling factor, the determination of two subject dependent constants of foveal vision, the minimum span and minimum delay, suffices to predict the main properties of the motion detection system at any eccentricity.

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).

The influence of the retinal inhomogeneity on the perception of spatial patterns.

From the fact that the retina is rather inhomogeneous, it can be inferred that the perception of spatial patterns of appreciable extent will be dependent on the retinal location. Anatomical, electrophysiological and psychophysical findings substantiate the claim that the retina is very inhomogeneous of composition. In order to investigate the influence of this inhomogeneity on the perception of patterns, a model of spatiotemporal signal processing in the retina was developed on the basis of a paradigm for the Weber type adaptation. Such “scaling-ensembles” proved successful in the prediction of spatiotemporal modulation transfer in the human fovea (Koenderink et al., 1971). One prediction of the present model is that certain spatial patterns are optimally detected at well defined retinal locations, dependent on the spatial frequency content of the stimulus. A confrontation of the models predictions with measurements published by Bryngdahl (1966) enabled us to estimate some of the relevant parameters of the retinal receptive fields as a function of the eccentricity. We obtained estimates that compare reasonably well with previously known values; for instance with values of acuity and anatomical measurements. The present discussion bears relevance on the question of whether the retina is composed of independently tuned spatial frequency filters at any retinal location, or whether the tuning is with respect to the eccentricity.

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.

Influence of contrast on foveal and peripheral detection of coherent motion in moving random-dot patterns.

The detection of coherent motion was studied in stroboscopically displayed moving random-dot patterns disturbed by incoherent noise. We determined the threshold signal-to-noise ratio S as a function of velocity V at eccentricities of 0 degrees, 3 degrees, 6 degrees, 12 degrees, 24 degrees, 48 degrees in the temporal visual field of the right eye. At each eccentricity the measurements of S = f(V) were repeated for a range of rms contrast values from 60% (0 dB) in steps of 3 dB down to 1.9% (-30 dB). All stimuli were scaled with eccentricity to keep the ratio of pixel size to acuity constant (about 2). It is shown that the S values in our paradigm are never determined by contrast-threshold effects. They are true correlational thresholds. Bilocal movement detectors are assumed to underlie the detection of coherent motion. The bilocal correlation proves to be rather insensitive to rms contrast down to contrast levels of about 10%. Despite the eccentricity scaling, which is quite effective at high contrast levels, differences between the eccentricities become noticeable at lower contrast levels (below about 30-20%). The fovea is the least, and the far periphery the most, resistent to contrast degradation.