Mathematical Psychology

Information-sharing and aggregation models for interacting minds

by Piotr Migdal

Natural groups of transformations underlying apparent motion and perceived object shape and color

by David Foster

Shepard's analysis of how shape, motion, and color are perceptually represented can be generalized. Apparent motion... more Shepard's analysis of how shape, motion, and color are perceptually represented can be generalized. Apparent motion and shape may be associated with a group of spatial transformations, accounting for rigid and plastic motion, and perceived object color may be associated with a group of illuminant transformations, accounting for the discriminability of surface-reflectance changes and illuminant changes beyond daylight. The phenomenological and mathematical parallels between these perceptual domains may indicate common organizational rules, rather than specific ecological adaptations.

Recognizing novel three-dimensional objects by summing signals from parts and views

by David Foster

Visually recognizing objects at different orientations and distances has been assumed to depend either on extracting... more Visually recognizing objects at different orientations and distances has been assumed to depend either on extracting from the retinal image a viewpoint-invariant, typically three-dimensional (3D) structure, such as object parts, or on mentally transforming two-dimensional (2D) views. To test how these processes might interact with each other, an experiment was performed in which observers discriminated images of novel, computer-generated, 3D objects, differing by rotations in 3D space and in the number of parts (in principle, a viewpoint-invariant, ‘non-accidental’ property) or in the curvature, length or angle of join of their parts (in principle, each a viewpoint-dependent, metric property), such that the discriminatory cue varied along a common physical scale. Although differences in the number of parts were more readily discriminated than differences in metric properties, they showed almost exactly the same orientation dependence. Overall, visual performance proved remarkably lawful: for both long (2 s) and short (100 ms) display durations, it could be summarized by a simple, compact equation with one term representing generalized viewpoint-invariant parts-based processing of 3D object structure, including metric structure, and another term representing structure-invariant processing of 2D views. Object discriminability was determined by summing signals from these two independent processes.

Characterization of discrete and continuous modes of visual pattern discrimination

by David Foster

Discrete and continuous modes of visual pattern discrimination performance are analyzed using a model for the... more Discrete and continuous modes of visual pattern discrimination performance are analyzed using a model for the investigation of discrete internal pattern representations described in previous papers (Foster, 1980a, b). A simple quantitative criterion is derived to characterize the two kinds of visual discrimination performance. Values predicted by this criterion are then compared with values obtained from experimental data.

Selective internal operations in the recognition of locally and globally point-inverted patterns

by David Foster

Performance in discriminating rotated 'same' patterns from 'different' patterns may decrease with rotation angle up to... more Performance in discriminating rotated 'same' patterns from 'different' patterns may decrease with rotation angle up to about 90° and then increase with angle up to 180°. This anomalously improved performance under 180° pattern rotation or point-inversion can be explained by assuming that patterns are internally represented in terms of local features and their spatial-order relations ('left of', 'above', etc.), and that, in pattern comparison, an efficient internal sense-reversal operation occurs (transforming 'left of' to 'right of', etc.). Previous experiments suggested that local features and spatial relations could not be efficiently separated in some pattern-comparison tasks. This hypothesis was tested by measuring 'same-different' discrimination performance under four transformations: point-inversion i 1 of the whole pattern, point-inversion lF of local features alone, point-inversion lp of local-feature positions alone, and identity transformation Id. The results suggested that internal sense-reversal operations could be applied selectively and efficiently, provided that local features were well separated. Under this condition performances for lF and l were about the same whereas performance for lp was significantly worse, the latter performance resulting possibly from an attempt to apply internal global and local sense-reversal operations serially.

Short-term memory scanning viewed as exemplar-based categorization

by Chris Donkin

Psychological Review, 2011

Increasing capacity: Practice effects in absolute identification

by Chris Donkin

Journal of Experimental Psychology: Learning, Memory & Cognition, 2011

Diffusion versus linear ballistic accumulation: Different models for response time, same conclusions about psychological mechanisms?

by Chris Donkin

Psychonomic Bulletin & Review, 2011

Drawing conclusions from response time models: A tutorial using the linear ballistic accumulator model

by Chris Donkin

Journal of Mathematical Psychology, 2011

Converging measures of workload capacity

by Chris Donkin

Psychonomic Bulletin & Review, 2010

The overconstraint of response time models: Rethinking the scaling problem

by Chris Donkin

Psychonomic Bulletin & Review, 2009

Getting more from accuracy and response time data: Methods for fitting the linear ballistic accumulator

by Chris Donkin

Behavior Research Methods, 2009

An Integrated Model of Choices and Response Times in Absolute Identification

by Chris Donkin

Psychological Review, 2008

A power-law model of psychological memory strength in short-term and long-term recognition

by Chris Donkin

Psychological Science 2012

Closing the Gap Between Ideal and Real Behavior: Scientific Vs. Engineering Approaches to Normativity

by Sergei Gepshtein

2009

Early normative studies of human behavior revealed a gap between the norms of practical rationality (what humans ought... more Early normative studies of human behavior revealed a gap between the norms of practical rationality (what humans ought to do) and the actual human behavior (what they do). It has been suggested that, to close the gap between the descriptive and the normative, one has to revise norms of practical rationality according to the Quinean, engineering view of normativity. On this view, the norms must be designed such that they effectively account for behavior. I review recent studies of human perception which pursued normative modeling and which found good agreement between the normative prescriptions and the actual behavior. I make the case that the goals and methods of this work have been incompatible with those of the engineering approach. I argue that norms of perception and action are observer-independent properties of biological agents; the norms are discovered using methods of natural sciences rather than the norms are designed to fit the observed behavior.

Viewpoint-invariant Weber fractions and standard contour-curvature discrimination

by David Foster

It is proposed that any cue for the visual discrimination of shape, in particular the discrimination of curved... more It is proposed that any cue for the visual discrimination of shape, in particular the discrimination of curved contours, should be such that the perceptual relationships defined by the cue are invariant under changes in observer viewpoint. Such relationships may be quantified by the Weber fraction; that is, the ratio Dc/c, where, for any particular value c of the cue, Dc is the smallest difference in c that can be detected. Eight geometric attributes of curved contours having one symmetry axis and parallel chords (a standard stimulus configuration) were examined for invariance of the Weber fraction under symmetry-preserving affine transformations of the image plane (changes in viewpoint are well approximated by affine transformations when depth is small relative to viewing distance). The attributes, each investigated in previous psychophysical studies, were equivalent-curvature, radius-of- curvature, turning-angle, arc-length-divided-by-chord-length, arc-length, maximum-deviation (sag), area, and mean-deviation. Three of the attributes, namely sag, area, and mean-deviation, satisfied the viewpoint-invariance condition; the remainder failed. These results are considered in relation to previously published empirical data on the Weber fraction for contour-curvature discrimination.

Overshoot of curvature in visual apparent motion

by David Foster

If a curved line and a straight line are presented briefly, one above the other, in sequence to the eye, then, under... more If a curved line and a straight line are presented briefly, one above the other, in sequence to the eye, then, under appropriate conditions, visual apparent motion is obtained. Subjects report that the illusory figure moving and changing from the curved line to the straight line appears to overshoot the latter, gaining a small curvature in the opposite sense. Three experiments are described. In the first, the magnitude of this apparent curvature was quantified as a function of the delay between the onsets of the curved line and straight line (the stimulus onset asynchrony, SOA). It is shown that overshoot in curvature cannot be attributed to inappropriate patterns of eye fixations. In the second experiment, the stimulus configuration was modified to reveal the contribution to apparent curvature of classical curvature-contrast effects. Curvature overshoot due to apparent motion alone was thus estimated as a function of SOA. In the third experiment, an analogous position overshoot was measured for apparent motion elicited by two brief sequentially presented parallel line segments. It is argued that a combination of such position overshoots cannot explain curvature overshoot. Two schemes of a more general kind that might be used to interpret curvature overshoot are then outlined. One scheme is based on a neural-net model of apparent motion, and the other on a functional model of apparent motion that operates by laws analogous to those governing real physical motion.

Classical and fuzzy differential methods in shape analysis

by David Foster

This study considers four means of defining differential operators for extracting local aspects of shape in... more This study considers four means of defining differential operators for extracting local aspects of shape in ill-specified environments: fuzzy differentiation as kernel smoothing; differentiation in the sense of weak or generalized derivatives; differentiation for fuzzy functions between normed spaces; and fuzzy differentiation for mappings between fuzzy manifolds. More consideration is given to the last, norm-free approach, which involves the notions of an abstract fuzzy topological vector space, fuzzy differentiation between fuzzy topological vector spaces, fuzzy atlases, and tangent vectors of fuzzy manifolds.

Elements of a fuzzy geometry for visual space

by David Foster

This study introduces the notions of fuzzy location and fuzzy proximity to capture the imprecision associated with... more This study introduces the notions of fuzzy location and fuzzy proximity to capture the imprecision associated with judgements of absolute and relative visual position. These notions are used to establish the elements of a fuzzy geometry for visual space, including the fuzzy betweenness of points, the fuzzy orientation of a pair of points, and the fuzzy collinearity of three or more points. Fuzzy orientation and fuzzy collinearity are, in turn, used to define the fuzzy straightness of a curve and the fuzzy tangency of two curves.

Visual apparent motion and the calculus of variations

by David Foster

The rapid sequential presentation of two distinct objects in the human visual field induces, under suitable... more The rapid sequential presentation of two distinct objects in the human visual field induces, under suitable conditions, the illusion of a single object undergoing a smooth continuous transformation from the first to the second form. It is suggested that in generating this illusion the visual system operates according to variational principles and chooses those impleting motions which, in some suitable space, have minimum energy. Implications of this hypothesis are discussed in relation both to experimental data on apparent motion and to the general problem of visual pattern recognition.