Abstract
In an earlier paper [J. Opt. Soc. Am. A 2, 1769 (1985)] a class of nonlinear image processing operators was introduced in which each photoreceptor creates a nonnegative point-spread function whose center height is proportional to its quantum catch and whose volume is constant, so that the local spatial-summation area varies inversely with the local quantum catch. These constant-volume (CV) operators are designed to maximize spatial resolution in the presence of photon noise. In the previous paper it was shown that when CV operators are applied to deterministic images, they produce a surprising range of effects that are reminiscent of human vision, including Mach bands and Weber's-law behavior. In this paper the consequences of applying CV operators to images containing Poisson noise are analyzed. It is shown that a fixed-parameter CV operator can duplicate the global qualitative properties of spatial vision for retinal illuminances ranging from absolute threshold to 1000 Td. Although there are fundamental obstacles to modeling the exact quantitative properties of human spatial vision by CV operators, these operators seem likely to be useful in machine vision.
Highlights
This paper is a sequel to a recent paper by Cornsweet and Yellott,l which introduced a class of nonlinear image-processing operators based on the following idea: each point in the input image gives rise to a nonnegative point-spread function whose center height is proportional to the light intensity at that point and whose volume is constant, so that the area covered by the point spread, the local spatial-summation area, varies inversely with the input intensity at each point
Their paper began with a motivation of intensity-dependent summation (IDS) operators in terms oftheir ability to deal efficientlywith photon noise, in particular, to adjust automatically the size of the spatial-summation area according to the prevailing quantum catch, so as to maximize spatial resolution at every light level while maintaining a fixed reliability for spatialcontrast detection
For input contrasts of the order of 10% or less, CV operators act as linear operators at all light levels, returning sinusoidal outputs for sinusoidal inputs. - (In the noisy-input case, this means that when the expected input image is a sinusoid, the expected output image is a sinusoid.) a modulation transfer function (MTF) can be defined, and, combined with the variances derived in Section 6, these MTF's can be used to predict spatial contrast-sensitivity functions (CSF's)
Summary
This paper is a sequel to a recent paper by Cornsweet and Yellott,l which introduced a class of nonlinear image-processing operators based on the following idea: each point in the input image gives rise to a nonnegative point-spread function whose center height is proportional to the light intensity at that point and whose volume is constant, so that the area covered by the point spread, the local spatial-summation area, varies inversely with the input intensity at each point. In the model considered here, photon-noisy retinal images (the quantum catches ofthe photoreceptors over a 250-msectime interval) are transformed by a fixed-parameter CV operator (the Gaussian operator G defined below, with its scale parameter a held constant across all illuminance levels), and it is assumed that any test image becomes discriminable from a uniform field when the peak value of d' across its output image reaches some fixed threshold level This model correctly predicts that as the retinal illuminance rises from 10-4 to 103 Td, (1) Ricco's area shrinks 5 ; (2) visual acuity rises (for gratings, by an overall factor on the order of 100, which matches the increase in human acuity 6 ); (3) peak spatial-contrast sensitivity rises: the sensitivity at the best spatial frequency growsas the square root of the mean retinal illuminance up to 0.1 Td (Ref. 4) and reaches an asymptote of the order of 100 (threshold contrast -1%) at 10 Td (Ref. 4); (4) the shape of the spatial CSF changes from low pass to bandpass as the illuminance rises above 1 Td (Ref. 4); (5) increment thresholds obey the deVries-Rose law at low background illuminances and Weber's law at high ones; (6) the background illuminance at which Weber's law begins to hold is higher for small test spots than for large ones.[2] The model correctly predicts that two sinusoidal gratings whose frequencies are both above the resolution limit at a given mean luminance level (and are invisible when viewed individually) can give rise to visible beats when superimposed.
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