Alleys in visual space

Geometry of frameless visual space is dealt with. First, parallel and equidistant alleys, horopters in the horizontal plane of eyes' level are discussed within the framework of the Luneburg's model that the frameless visual space is a Riemannian space of constant curvature. That basic postulate and the specific mapping functions assumed by Luneburg between the euclidean map of visual space and the physical space are kept separate, and efforts are directed to make the model applicable to more natural conditions of our visual space. A possibility is pointed out to remove the constraint “frameless” in the sense that perceptual geometrical properties are primarily determined by the convergence of optic axes. So far, only alleys in the horizontal plane extending from us toward infinity have been studied, but more often we perceive parallel lines, horizontal or vertical, in front of us like shelves of a bookcase. Hence, equations are derived for horopter plane appearing fronto-parallel in the three-dimensional visual space and alleys running horizontally or vertically on the horopter plane. It is shown that parallel and equidistant alleys are not the same in the horopter plane as in the horizontal plane, if the visual space is not euclidean. A method to evaluate the discrepancy between the two alleys without using any mapping functions is stated with some numerical examples.     

Multidimensional mapping of visual space with real and simulated stars

In each of five experiments Ss made direct assessments of interpoint distances within a stimulus configuration. A multidimensional scaling method was applied to make explicit the form of the subjective configuration. The stimulus configurations consisted of small light points arranged in a two- or three-dimensional array in a dark room, and of real stars in the sky. All the data were adequately accounted for by a configuration constructed in Euclidean space of the appropriate dimensionality. That was true even in the situation where alley experiments with the same Ss gave the result that is usually regarded as evidence for a hyperbolic binocular space. The Euclidean interpretation entails a more complicated form of correspondence between physical and visual spaces than the hyperbolic interpretation.     

Parallel—and distance—alleys with moving points in the horizontal plane

Stimulus points were presented on the horizontal plane of eye level under both dark and illuminated homogeneous spaces. When two apparent movements towards the subject were generated and positions of points were so adjusted that the two movements appeared straight and parallel (P alley) or with a constant lateral distance (D alley), the D alley lay outside the P alley, as traditionally shown with stationary sets of points. The two alleys were constructed with various patterns and velocities of movement, and the Lüneburg formulas were used as experimental equations to describe the results. The equations have two parameters: K (curvature) and o (sensitivity in depth perception). Values of K and o obtained with stationary points in previous experiments are shown too. Predominantly, K < 0 (hyperbolic), and the same is true in the present study. No first-order effect of patterns and velocities of the movement upon K and o was found.     

An approach to geometry of visual space with no a priori mapping functions: Multidimensional mapping according to Riemannian metrics

A method of multidimensional mapping is described which constructs a configuration of points {P_i} in a Euclidean map of Riemannian space of constant curvature (hyperbolic, Euclidean, and elliptic) from the dissimilarity matrix (d_ij). The method was applied to the distance matrix in visual space where stimulus points Q_i were either small light points in the dark or small black points in the illuminated field surrounded by white curtains and d_ij represent scaled values of perceptual distances. Configuration of points {Q_i} were at intersections of parallel or distance alleys and horopters for the subject in the horizontal plane of the eye level. In contrast to the theoretical equations for {Q_i} by Luneburg and Blank, no a priori assumption on mapping functions between {Q_i} and {P_i} is necessary in this procedure to fit theoretical curves to {P_i} in the Euclidean map. The data were accounted for better by equations in the hyperbolic plane than by ones in the Euclidean plane. Discussions are made on robustness of Euclidean representation and on how to approach geometry of visual space as a dynamic entity under more natural conditions than the traditional frameless condition for alley and horopter experiments.     

Recognition memory of retrieved sequences of words

The subject was asked to retrieve words belonging to a given category, natural or artificially created. After n words were retrieved, a recognition task was given either immediately after or after a delay (4 hours, 7, 14, or 28 days). Several probe words were presented and the subject had to decide whether each had been retrieved or not. The length of retrieval sequence, n, was systematically varied. Results of three experiments showed that, in general, mean decision time was almost independent of n. Namely, recognition regarding self-generated lists of words seems to be due to a different information processing than recognition regarding lists of words presented by the experimenter to memorize. Implications of the findings for retrieval from long term memory were discussed.     

A critical review of Luneburg's model with regard to global structure of visual space

Visual space (VS) is a coherent self-organized dynamic complex that is structured into objects, backgrounds, and the self. As a concrete example of geometrical properties in VS, experimental results on parallel and (equi) distance alleys in a frameless VS were reviewed, and Luneburg's interpretation on the discrepancy between these 2 alleys was sketched with emphasis on the 2 hypotheses involved: VS is a Riemannian space of constant curvature (RCC) and the a priori assumed correspondence between VS and the physical space in which stimulus points are presented. Dissociating these 2 assumptions, the author tried to see to what extent the global structure of VS under natural conditions is in accordance with the hypothesis of RCC and to make explicit the logic underlying RCC. Several open questions about the geometry of VS per se have been enumerated.     

Usefulness and limitation of spatial representation

What discussed herein is not an “open problem” in the sense of mathematics. It is a problem that psychologists should keep in mind when presenting a formal model. A model will be useful for phenomena on which the model has been formulated. However, the model may contain a number of remaining properties that not necessarily represent related psychological phenomena adequately. The situation is analogous to that the particle model of light does not represent diffraction whereas the wave model of light is not adequate for the Compton effect. When presenting a model, mathematical psychologists should be especially keen about this point. The problem is discussed with a concrete example.     

Scaling of dot numerosity

Subjective impression of numerosity of dots randomly scattered in an area of 19.5 × 24.5 cm, 36≈1,010 in number (?), was scaled by four different methods with five subjects. The absolute identification showed that the number of dots is underestimated if ? > 10. Each of two scales based uponratio judgments is a power function of $ with the exponent of 0.87, and each of two scales based upondifference judgments is also a power function of ? when the origin of the scale is appropriately defined. The relationship between the four scales and comparison with scales in other sense modalities were discussed from the viewpoint of scaling methodology.