Roger S. Pinkham

A space-variant differential operator for visual sensitivity

All the elements of a Fourier analysis can be derived from the experiments of Graham and Robson on contrast sensitivity. Once their experiment is posed as an eigenvalue problem, a complete orthonormal set of eigenfunctions results from solving the associated differential equation. Neither sine and cosine nor Gabor functions result. Instead, the Hermite functions arise as the eigenfunctions of a space-variant differential operator used to model the contrast sensitivity of human observers. These functions, up to a constant, are their own Fourier transforms, and in principle can be used to exactly represent the Fourier transform of naturally occuring visual images.

Nonmonotone backward masking functions and brightness reversals

Increasing the target-field luminance aids detection for a simultaneously presented black target disc and a black masking annulus. At an intermediate interval separating the onset of the target from the mask, increasing the target-field luminance reduces target detection. This decrease in performance occurs with both temporal and spatial forced choice tasks. With a spatial forced choice, an observer’s performance can fall below chance. We associate below-chance performance with a brightness reversal of the black target disc, such that the target disc appears brighter than its surround. The occurrence of brightness reversals follows from our model of the Broca–Sulzer effect, and nonmonotone masking functions result from a generalization of luminance summation.

Representing contrast detection as an eigenvalue problem.

Contrast detection can be formulated as an eigenvalue problem. One of the simplest resulting models has only two parameters. The model is space variant and employs the Hermite functions as eigenfunctions. Computing the response to a sinusoidal acuity grating yields the observers contrast response. The model itself, however, is developed within an abstract mathematical framework which is general enough to include Fourier analysis as a special case. Consequently, the methods of Fourier analysis are generalized to those of eigenfunction expansion and the spectral theory of linear operators.

Space-variant models of visual acuity using self-adjoint integral operators

The purpose of this paper is to outline a more general approach to visual acuity experiments than the classical methods borrowed from the optical sciences. A theory based on integral operators with symmetric kernels replaces the standard use of filters constructed from windowed sines and cosines. This more general approach allows greater latitude in the range of phenomena to be modelled. It also permits all the standard techniques used in Fourier expansion of a stimulus and response to be generalized to a space-variant system such as the human visual field.

A Class of Integration Formulas

Gregorys formula for numerically integrating a function is one of the most promising formulas for use in a computer library. This paper shows how Gregorys formula can be generalized, and examines special cases which have a number of very favorable properties for library use.

Hermite acuity gratings and models of space-variant acuity

A coherent mathematical framework for the psychophysics of contrast perception emerges when contrast sensitivity is posed as an eigenvalue problem. This more general mathematical theory is broad enough to encompass Fourier analysis as it is used in vision research. We present a model of space-variant contrast detection to illustrate the main features of the theory, and obtain a new contrast sensitivity function using acuity gratings based on the Hermite functions. The Hermite gratings have several advantages: they represent a complete orthogonal basis, are easy to manipulate, and are of finite extent. A theoretical Hermite csf results from posing contrast perception as an eigenvalue problem. Surprisingly, the theoretical Hermite csf is determined by a single empirical parameter.

Mathematics and Modern Technology

This paper is an attempt to illustrate and emphasize three points. First, that modern technology allows one to encourage students to ask and answer questions heretofore impossible to address fruitfully. Second, many (probably most) classical methods are as necessary as ever, but perhaps in a different setting, and third, simple calculus and a bit of reflection is amazingly effective. Although the examples presented are specific, I have attempted to approach each in a manner that has general applicability. Throughout I have tried to show how modern technology can provide insight and foster a spirit of inquiry. Machines do some things very well, some poorly. The same is true of humans. The two are often complementary; it seems best to search for uses of each that capitalize on their individual strengths. The topic of infinite series provides convenient examples. Given an infinite series, one asks two things. Does it converge, and if so, to what? The former question has been the concern of most calculus texts, while the latter has usually been left in abeyance. This is unfortunate for the applications-oriented student, for it is the value of the series that is most often the thing of primary concern.

A WORD-SUPERIORITY EFFECT IN THE PRESENCE OF FOVEAL LOAD

Foveal stimuli have been shown to disrupt visual information processing in the parafovea and periphery by their mere presence. In the present study, 6 subjects were presented letter triads 3.58° to the right or left of the point of fixation. At the same time, a single letter was presented at the point of fixation that was either the same as the middle letter in the triad or different from any of the triad letters. On other trials, no letter was presented at the point of fixation. Analysis indicated a word superiority effect when a foveal letter was presented that was the same as the letter in the triad. Performance between words and nonwords did not differ significantly when the foveal letter was different or absent. It was concluded that the mere presence of foveal load alone is not disruptive to performance. Depending on the visual context of the target to be reported, the presence of a foveal stimulus may improve performance.

Transposition errors in context-free languages

Abstract This article examines the problem of context-free languages that include strings containing transposition errors. The precise construction of a transposition error-correcting algorithm is given. Error detection and correction has been analyzed by others. Context-free grammar transformations to allow for the derivation of strings containing replacement, deletion, and insertion errors have been defined. This article presents an algorithm that, when given a Chomsky Normal Form context-free grammar G , constructs a Chomsky Normal Form context-free grammar, G T , to derive strings that contain transposition errors. The strings in the new language are one transposition error away from those in the original language, L ( G ). G T can be used to parse incorrect strings and to correct them. This approach also allows for the detection, classification, and correction of transposition errors occurring in the syntactic description of patterns. The number of new productions in G T is related by a small polynomial to the number of productions in G .

Production probability estimators for context-free grammars

Abstract The problem of production probability estimation is considered. We examine the estimation of production probabilities for context-free probabilistic grammars (CFPGs). Given a context-free grammar, G, and a random sample of strings from L(G), we define ratio estimators to estimate the production probabilities of G. It is shown that the statistical analysis of the estimators becomes much less complex by using the theory of random walks. The main result of the article is that the biases and variances of all ratio estimators for any Chomsky Normal Form CFPG are approximately directly proportional to 1 n , where n is the size of the random sample. Random samples consisting of strings generated from an independent source are used to validate theoretical results. We show that the estimates can be used to increase the efficiency of parsing strings in a language. The ability to parse strings efficiently is extremely useful in compiler theory and the theory of artificial intelligence.