Roger N. Shepard

Mental Rotation of Three-Dimensional Objects

The time required to recognize that two perspective drawings portray objects of the same three-dimensional shape is found to be (i) a linearly increasing function of the angular difference in the portrayed orientations of the two objects and (ii) no shorter for differences corresponding simply to a rigid rotation of one of the two-dimensional drawings in its own picture plane than for differences corresponding to a rotation of the three-dimensional object in depth.

The analysis of proximities: Multidimensional scaling with an unknown distance function. II

The first in the present series of two papers described a computer program for multidimensional scaling on the basis of essentially nonmetric data. This second paper reports the results of two kinds of test applications of that program. The first application is to artificial data generated by monotonically transforming the interpoint distances in a known spatial configuration. The purpose is to show that the recovery of the original metric configuration does not depend upon the particular transformation used. The second application is to measures of interstimulus similarity and confusability obtained from some actual psychological experiments.

Mental Images and Their Transformations

This book collects some of the most exciting pioneering work in perceptual and cognitive psychology.

CHRONOMETRIC STUDIES OF THE ROTATION OF MENTAL IMAGES

Publisher Summary This chapter discusses chronometric studies of the rotation of mental images. It describes experimental paradigms for investigating the nature of mental images, selective reduction of reaction times, and preceding reaction-time studies of mental rotation. In the experiments described in the chapter, mental transformations and the selective reduction of reaction times are used, jointly, to establish that the internal representations and mental operations upon these representations are to some degree analogous or structurally isomorphic to corresponding objects and spatial transformations in the external world. In all of these experiments, each spatial transformation consists simply of single rigid rotation of a visual object about a fixed axis. However, in related work reported elsewhere, reaction times have been measured for much more complex sequences of imagined operations in space.

Attention and the metric structure of the stimulus space

Abstract Three experiments were performed in an investigation of how differences in size and inclination combine to determine the over-all similarity between otherwise identical visual stimuli. Similarity was defined both in terms of direct subjective judgments of over-all resemblance and in terms of the frequencies with which the stimuli were actually confused during identification learning. The results were incompatible with the static, Euclidean metric assumed by multidimensional scaling models. Apparently, when the stimuli vary along perceptually distinct dimensions, the psychological metric changes as subjects shift their attention more to one dimension or the other. The interstimulus similarities for any one state of attention, however, appear to conform with a Minkowski metric somewhere between the Euclidean and “city-block” varieties.

Multidimensional Scaling, Tree-Fitting, and Clustering

American mathematical psychologists have developed computer-based methods for constructing representations of the psychological structure of a set of stimuli on the basis of pairwise measures of similarity or confusability. Applications to perceptual and semantic data illustrate how complementary aspects of the underlying psychological structure are revealed by different types of representations, including multidimensional spatial configurations and nondimensional tree-structures or clusterings.

Second-order isomorphism of internal representations: Shapes of states ☆

Abstract It is argued that, while there is no structural resemblance between an individual internal representation and its corresponding external object, an approximate parallelism should nevertheless hold between the relations among different internal representations and the relations among their corresponding external objects. In support of this “second-order” type of isomorphism, subjective judgments of the similarities among the shapes of 15 states of the U. S. are found (a) to be very much the same whether the states to be compared are pictorially displayed or only imagined, and (b) to be related, in both cases to identifiable properties of their actual cartographic shapes.

Representation of structure in similarity data: Problems and prospects

ConclusionAfter struggling with the problem of representing structure in similarity data for over 20 years, I find that a number of challenging problems still remain to be overcome—even in the simplest case of the analysis of a single symmetric matrix of similarity estimates. At the same time, I am more optimistic than ever that efforts directed toward surmounting the remaining difficulties will reap both methodological and substantive benefits. The methodological benefits that I forsee include both an improved efficiency and a deeper understanding of “discovery” methods of data analysis. And the substantive benefits should follow, through the greater leverage that such methods will provide for the study of complex empirical phenomena—perhaps particularly those characteristic of the human mind.

Quantification of the hierarchy of tonal functions within a diatonic context.

Listeners rated test tones falling in the octave range from middle to high C according to how well each completed a diatonic C major scale played in an adjacent octave just before the final test tone. Ratings were well explained in terms of three factors. The factors were distance in pitch height from the context tones, octave equivalence, and the following hierarchy of tonal functions : tonic tone, other tones of the major triad chord, other tones of the diatonic scale, and the nondiatonic tones. In these ratings, pitch height was more prominent for less musical listeners or with less musical (sinusoidal) tones, whereas octave equivalence and the tonal hierarchy prevailed for musical listeners, especially with harmonically richer tones. Ratings for quarter tones interpolated halfway between the halftone steps of the standard chromatic scale were approximately the averages of ratings for adjacent chromatic tones, suggesting failure to discriminate tones at this fine level of division. The study of perceived pitch and of the perceived relations between tones differing in pitch has generally been approached from one of two quite different traditions: the psychoacoustic and the musical. The psychoacoustic approach has typically focused on simple, physically specifiable properties of tones isolated from any musical context— properties of frequency, separation in log frequency, or simplicity of integer ratios of frequencies. The results of such studies have provided some precise information about how the ear responds to isolated tones or tones in random sequences. We believe that they have been less informative with regard to how the listener perceives tones in organized musical sequences. Music theory suggests that the perception of such sequences may rely on the listeners sensitivity to different and structurally richer principles associated with tonal and diatonic organization. Such principles are useful in explaining the cognitive phenomena of reference point, motion, tension, and resolution that underlie the dynamic force of virtually all tonal music. They have, however, been subjected to relatively little systematic laboratory investigation or quantitative formulation.

Metric structures in ordinal data

Abstract Under appropriate conditions, data merely about the ordering of objects—or of the separations between objects—is sometimes sufficient to fix the positions of those objects on an essentially numerical scale. This paper uses both mathematical and “Monte Carlo” results to establish and clarify the possibility of thus extracting metric information from purely ordinal data for two multidimensional cases: (a) analysis of proximities, in which one is given a single rank order of all n(n−1) 2 pairs of n objects with respect to psychological similarity or “proximity”; and (b) nonmetric factor analysis, in which one is given a different rank order of n individual objects with respect to each of m psychological attributes. As n (and m) increase, the ordinal data are found to determine a spatial representation of the objects more and more nearly to within a general similarity transformation, in the case of analysis of proximities, or an affine transformation, in the case of nonmetric factor analysis. Extensions of these results to other cases are also considered.