Visibility of continuous luminance gradients

doi:10.1016/0042-6989(74)90158-8

Vision Research

Volume 14, Issue 10, October 1974, Pages 917-927, I

https://doi.org/10.1016/0042-6989(74)90158-8 Get rights and content

Abstract

A plateau of illumination was modulated with various patterns of gradual change: linear slopes and small numbers of low spatial frequency sinusoidal oscillations. Over the range of parameters tested, the threshold contrast necessary for the detection of these modulations was found to be largely independent of the steepness of the gradient, the frequency of the sinusoids, and the size of the target on the retina. Visibility was found to be a function of the fractional change in luminance across the target (contrast) and the pattern of the modulation (characterized by the number of cycles of sinusoid).

Résumé

On module un plateau d'éclairement par divers types de changement graduel: pentes linéaires et oscillations sinusoïdales de basse fréquence spatiale en petit nombre. Dans le domaineétudiépour ces paramètres, le seuil de contraste nécessaire pour détecter ces modulations est largement indépendant de la raideur du gradient, de la fréquence des sinusoïdes, et de la taille du test sur la rétine. On trouve que la visibilitéest fonction de la fraction de changement de luminanceàtravers la cible (contraste) et du type de modulation (caractérisépar le nombre de cycles de la sinusoïde).

Zusammenfassung

Ein Feld homogener Leuchtdichte wurde mit verschiedenen stetigen Leuchtdichtemustern variiert: Mit linearen Gradienten sowohl wie mit Sinusgittern niedriger Ortsfrequenz. Bei allen untersuchten Parametern wurde gefunden, dass der Schwellenkontrast für die Erkennbarkeit dieser Modulationen weitgehend von der Steilheit des Gradienten, von der Ortsfrequenz des Sinus und von der Grosse des Testzeichens auf der Netzhaut unabhängig war. Die Sichtbarkeit war eine Funktion der relativen Leuchtdichteänderung (Kontrast) und des Modulationsmusters (charakterisiert durch die Zahl von Perioden im Sinusgitter).

Peзюme

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Natural environments, however, may contain more abrupt spatial variations of local illumination as a result of their complex spatial structures, which produce shading, mutual reflections, and occlusions (Chiao, Cronin, & Osorio, 2000; Endler, 1993). The spatial variations in local illumination intensity level have been well documented (Dror, Willsky, & Adelson, 2004; Morgenstern, Geisler, & Murray, 2014; Mury, Pont, & Koenderink, 2007, 2009), as has visual sensitivity to those variations (Lee & Brainard, 2011; McCann, Savoy, Hall, & Scarpett, 1974; Olkkonen & Brainard, 2011; Ruppertsberg et al., 2008). Yet unlike the characterization of natural temporal changes in illumination color, the characterization of natural spatial variations in local illumination color and their consequences for color constancy have received relatively little attention (cf. Hubel, 2000).
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We report on the extent to which human observers are able to indicate the gradient direction of a luminance ramp. In our experiment modulation depth ranged from 1 to 64% and field sizes subtended 0.5 to 47.5° of visual angle. Observers are not able to indicate the gradient direction for modulation depths below 6%. For values above this threshold, the angular standard deviation in the responses decreases proportionally with the logarithm of the modulation depth and is about 11° for a modulation depth of 64%. The angular standard deviations for supra-threshold contrast are slightly increased for the field sizes of 0.5 and 47.5° but are constant over a range of field sizes of 1 to 25°. Thus, size invariance holds well for this range of field sizes.
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The objectives of the experiment described in the paper were the determination of the effects of nonuniformities on operator performance and perception, and the validation of current recommended standards. Subjects performed two tasks in the presence of nonuniformities: an objective visual search task, and a subjective magnitude-estimation task for the determination of perceived uniformity. The results indicated that the nonuniformities did not appreciably affect search performance, and that current recommendations are appropriate, although the magnitude-estimation task results indicated that the subjects were sensitive to the nonuniformities. The subjective impressions of perceived uniformity results follow the contrast threshold function of the visual system.
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The harmonic input method of nonlinear system identification is modified to allow the Volterra series approach to be used for psychophysical investigation of various aspects of human pattern vision in the spatial frequency domain. While it is well known that only one modulation transfer function provides a complete characterization of a linear system, a number of multidimensional transfer functions are needed to identify a nonlinear system. We have shown, that so far as the contrast sensitivity to sine-wave gratings may be used for an empirical estimate of the first-order modulation transfer function of the human visual system, the contrast sensitivity to difference harmonics may be used as an empirical estimate of the second-order modulation transfer function.
A difference harmonic arises from a mixture of two sine-wave gratings resulting from the nonlinearity of the visual system. Difference harmonic, experienced as some periodic beatlike structure, may still be observed if frequencies of the component gratings are higher than the maximum visual acuity.
The visibility of the low-frequency beatlike pattern produced by pairs of sine-wave gratings, which themselves are of spatial frequencies too high to be resolved, could be accounted for either by a difference frequency distortion product (Burton, 1973) or by a special beat detector (Derrington & Badcock, 1985). We found that increasing the contrast of one component grating may be compensated for by reducing the contrast of the other component grating, the beatlike pattern being at threshold. This is exactly what would be expected if the beatlike pattern is detected because of the difference harmonics produced by nonlinearity of the visual system.
We have determined contrast thresholds for the difference harmonics which occur between two unresolved different spatial frequencies. The contrast sensitivity function for difference harmonics was found to have a marked similarity both in the shape and position of peak sensitivity to the contrast sensitivity function for single sine-wave gratings. Another important characteristic of the contrast sensitivity function for difference harmonics is that it depends only on the frequency difference, Δf = f₁ − f₂, rather than on the value of either f₁ or f₂.
All this indicates that a difference harmonic arises from local nonlinearities in the visual system. More specifically, the visual system may be represented as a cascade system, composed of a linear system with transfer function O(f) followed by a nonlinear element, r(·), without spatial spread in cascade with another linear system with transfer function P(f). The nth order transfer function of this cascade system, H_n(f₁,...,f_n) can be expressed in the following way: H_n(f₁,...,f_n) = a_nO(f₁)P(f₁ + ... + f_n) where a_n is the nth coefficient in the Taylor series expansion for the nonlinear function r(·). It follows from this that the measurement of the first- and second-order transfer functions is sufficient to determine O(f) and P(f). We have derived the estimates of cO(f) and P(f) from contrast sensitivity functions for single sine-wave gratings and difference harmonics by the least squares method.
The hybrid constraint equation for motion extraction
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A new derivation of the constraint equation for motion extraction in the image flow is presented. The derivation was obtained by applying the constraint equation to the Fourier transform of the 2D images. It resulted in a hybrid equation joining time-space and frequency notations. The hybrid equation was numerically solved for the 1D case and tested with many image sequences representing coherent, rigid and incoherent motion patterns. Outlines of the 2D case are also presented with some solved examples. The algorithm seems superior to former derivations of the constraint equation for its ability to extract the movement parameters (magnitude and direction) in non-homogeneous movement fields for displacements greater than pixel size. The relevance of the present procedures to machine and organic motion detection is discussed.
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The measurements show (i) that the visual system has a very large bandwidth and (ii) that the presence of edges or luminance transitions in the visual field can diminish the sensitivity to a stimulus considerably, even when they are rather distant.
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View full text

Visibility of continuous luminance gradients

Abstract

Résumé

Zusammenfassung

Peзюme

Contrast thresholds of the human eye

J. opt. Soc. Am.

On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images

J. Physiol.

Optical and retinal factors affecting visual resolutions

J. Physiol.

Application of Fourier analysis to the visibility of gratings

J. Physiol.

The detection of gratings in narrow-band visual noise

J. Physiol.

Visual Perception

Perturbation approach to spatial brightness interaction in human vision

J. opt. Soc. Am.

Human brightness perception near sharp contours

J. opt. Soc. Am.

Sine-wave response of the visual system—II. Sine-wave and square-wave contrast sensitivity

J. opt. Soc. Am.

Threshold contrast as a function of target complexity

Am. J. Optom.