Watson
IS&T Member.
The Field of View (FoV), the Field of Resolution, and the Field of Contrast Sensitivity describe three progressively more detailed descriptions of human spatial sensitivity at angles relative to fixation. The FoV is the range of visual angles that can be sensed by an eye. The Field of Resolution describes the highest spatial frequency that can be sensed at each angle. The Field of Contrast Sensitivity describes contrast sensitivity at each spatial frequency, at each visual angle. These three concepts can be unified with the aid of the Pyramid of Visibility, a simplified model of contrast sensitivity as a function of spatial frequency, temporal frequency, and luminance or retinal illuminance. This unified model provides simple yet powerful observations about the Field of Contrast Sensitivity. I have fit this model to a number of published measurements of peripheral contrast sensitivity. This allows us to test the validity of the model, and to estimate its parameters. Although the model is a simplification, I believe it provides an invaluable guide in a range of applications in visual technology.
The Field of View (FoV) describes the range of visual angles that can be sensed by an eye. But within that range, some angles are sensed better than others. A more complete description would describe how the contrast sensitivity function varies with angle. I call that the Field of Contrast Sensitivity. The spatial frequency at which contrast sensitivity declines to zero is the resolution of the visual system. By way of the Field of Contrast Sensitivity, one can derive the resolution at each angle, which I call the Field of Resolution.
These general ideas can be given mathematical form with the aid of the Pyramid of Visibility (PoV). This is a recent simplified model of contrast sensitivity as a function of spatial frequency, temporal frequency, and luminance or retinal illuminance. From this model, one can derive some simple yet powerful observations about the Field of Contrast Sensitivity.
Finally, I have fit this model to a number of published measurements of peripheral contrast sensitivity. This allows us to test the validity of the model, and to estimate its parameters. Although the model is a simplification, I believe it provides an invaluable guide in a range of applications in visual technology.
The FoV describes the range of angles that can be seen by the human eye. The range that can be seen by one eye while fixating straight ahead (the Fixated Field of View, or FFoV) is bounded by the occluding structures of the head (eye orbit, nose, brow) and by the cessation of photoreceptors at the margins of the retina. The literature suggests this extends for one eye from
I adopt the shorter and more intuitive terms “in,” “out,” “up,” and “down” in place of the traditional terms “nasal,” “temporal,” superior,” and “inferior.” Further, these terms are referenced to locations in the visual field, and are reversed relative to locations on the retina.
[The contrast sensitivity function (CSF) is defined as the inverse of the smallest contrast of a sinusoidal grating that can be detected at each spatial frequency [
In Fig.
Left: a sinusoidal luminance grating. Center: a contrast sensitivity function plotted on log–log coordinates. Right: the same function on log–linear coordinates.
Recently we observed that log contrast sensitivity, at moderate photopic luminances, and at high spatial or temporal frequencies, could be approximated by a linear function of spatial frequency, temporal frequency, and adapting luminance [
Pyramid of Visibility. The green diamond at the base defines the Window of Visibility. This example is for a mean luminance of 100 nits and parameters derived from data of Robson [
In this report we are concerned primarily with static spatial patterns, and the data sets we consider were all collected at fixed adapting luminances. Thus in the remainder of this paper I will fix temporal frequency at zero, and define the sum of constant and luminance term as
To measure the CSF in a local region of the visual field it is necessary to use small patches of grating. These are often produced as the product of a 2D Gaussian window and a 1D sinusoid (a Gabor function) [
Gabor functions. The top row shows four CDG with frequencies of 2, 4, 8, and 16 cycles/deg (at a viewing distance of about 28 picture heights). Each Gabor function has a Gaussian standard deviation of
In general, we would like to use as small a patch as possible, so that the measurement is as local as possible. But as a patch becomes smaller, its bandwidth increases. As we shall see, the rate of change in sensitivity with eccentricity is proportional to frequency. This means that we need to use a small patch for high frequencies, and a large patch for low. Consequently the use of CCG is preferred. CCG also have a fixed logarithmic frequency bandwidth.
Contrast sensitivity functions measured with CDG and CCG are shown in Figure
CSF measured with constant-degree (Fig.
The linear fits in Fig.
PoV parameters for luminance and retinal illuminance and for constant degrees and constant cycles conditions. Parameters are derived from ModelFest data. Luminance refers to the luminance of the display, illuminance refers to the retinal illuminance, that is the product of luminance and area of the pupil.
CDG | 0.915 | −0.060 | 0.500 | |
CCG | 0.556 | −0.091 | 0.500 | |
CDG | 1.739 | −0.060 | 0.391 | |
CCG | 1.380 | −0.091 | 0.391 |
The parameters estimated above are for one particular local region: the visual center, or point of fixation. How might the PoV model incorporate changes in position in the visual field? One simplified theory of the effect of eccentricity on visual sensitivity is that it changes the spatial scale [
Substituting Eq. (
PoV CCG CSF at several eccentricities.
Equation (
PoV CCG CS versus
Finally, we can convert eccentricity
Sensitivity versus eccentricity in grating cycles
The PoV model also provides a formula for resolution (grating acuity)
I define the local spatial scale as the inverse of resolution 1∕
Local resolution and local spatial scale as a function of eccentricity for the PoV model. Parameters are as in Figure
Here I consider existing data on contrast sensitivity measured with CCG stimuli, to evaluate the PoV model, and if valid to estimate parameters. Measurements of contrast sensitivity as a function of eccentricity have typically taken one of three formats. The first (data mode 1) is the measurement of the full CSF, for a range of frequencies, at several eccentricities, as in Fig.
Robson & Graham [
Fit of PoV to data from Robson and Graham [
PoV parameter estimates from Robson and Graham (1981).
Inf | 1.82 | −0.0536 | 0.343 | −0.0184 |
Sup | 1.87 | −0.0514 | 0.446 | −0.0229 |
Mean | 1.85 | −0.0525 | 0.393 | −0.0206 |
Watson measured contrast sensitivities for 0.25 to 16 cycles/deg at eccentricities of 0 and
Fit of PoV to data from Watson (1987).
PoV parameter estimates from Watson (1987).
1.94 | −0.0694 | 0.221 | −0.0154 |
Pointer and Hess [
Fit of PoV to data from Pointer and Hess [
PoV parameter estimates from Pointer and Hess (1989).
Nasal | 2.17 | −0.0828 | 0.174 | −0.0144 |
Temporal | 2.21 | −0.0903 | 0.171 | −0.0154 |
Mean | 2.19 | −0.0865 | 0.172 | −0.0149 |
Arnow & Geisler [
Fit of PoV to data from Arnow & Geisler (1996). Left: observer JS, right: observer AK.
PoV parameter estimates from Arnow & Geisler (1996).
JS | 1.77 | −0.0419 | 0.352 | −0.0148 |
AK | 1.79 | −0.0774 | 0.215 | −0.0166 |
Mean | 1.78 | −0.0597 | 0.263 | −0.0157 |
The authors measured contrast sensitivities for a Gabor target over horizontal eccentricities of
Fit of the PoV to data of Foley et al. (2007).
PoV parameter estimates from Foley et al. (2007).
JMF-left | 1.84 | −0.0229 |
JMF-right | 1.84 | −0.0163 |
KRH-left | 1.82 | −0.0192 |
KTH-right | 1.80 | −0.0183 |
Mean | 1.82 | −0.0192 |
Baldwin, Meese & Baker [
Fit of PoV to data from experiment 3 of Baldwin, Meese & Baker (2012).
PoV parameter estimates from experiment 3 of Baldwin et al. (2012).
ASB | 0.02 | −0.1050 | −0.333 | 0.0350 |
DHB | 0.15 | −0.1340 | −0.309 | 0.0414 |
SAW | −0.05 | −0.1360 | −0.357 | 0.0484 |
Mean | 0.04 | −0.1250 | −0.333 | 0.0416 |
In their experiments 1 and 2, sensitivity was measured for 4 cycle/deg for four meridians. Since only one frequency was used, I cannot estimate
Fit of the PoV to experiments 1 and 2 of Baldwin et al. (2012).
PoV parameter estimates from experiments 1 and 2 of Baldwin et al. (2012).
Right | 1.45 | −0.0382 |
Up | 1.39 | −0.0386 |
Left | 1.41 | −0.0327 |
Down | 1.39 | −0.0375 |
Mean | 1.41 | −0.0367 |
In general, the Field of Contrast Sensitivity model fits the ensemble of data quite well. Table
The reasons for variation among these studies (bandwidth, range of eccentricities studied, and range of spatial frequencies studied) warrant further study.
The mean values over all studies, observers, and conditions are about
PoV parameter estimates from several studies.
Robson | Inf | 1.82 | −0.0536 | 0.343 | −0.0184 |
Sup | 1.87 | −0.0514 | 0.446 | −0.0229 | |
Mean | 1.85 | −0.0525 | 0.393 | −0.0206 | |
Watson | Mean | 1.94 | −0.0694 | 0.221 | −0.0154 |
Pointer | Nasal | 2.17 | −0.0828 | 0.174 | −0.0144 |
Temporal | 2.21 | −0.0903 | 0.171 | −0.0154 | |
Mean | 2.19 | −0.0865 | 0.172 | −0.0149 | |
Arnow | JS | 1.77 | −0.0419 | 0.352 | −0.0148 |
AK | 1.79 | −0.0774 | 0.215 | −0.0166 | |
Mean | 1.78 | −0.0597 | 0.263 | −0.0157 | |
Foley | JMF-left | 1.84 | −0.0229 | ||
JMF-right | 1.84 | −0.0163 | |||
KRH-left | 1.82 | −0.0192 | |||
KTH-right | 1.8 | −0.0183 | |||
Mean | 1.82 | −0.0192 | |||
Baldwin 1&2 | Right | 1.45 | −0.0382 | ||
Up | 1.39 | −0.0386 | |||
Left | 1.41 | −0.0327 | |||
Down | 1.39 | −0.0375 | |||
Mean | 1.41 | −0.0367 | |||
Baldwin 3 | ASB | −0.105 | 0.333 | −0.035 | |
DHB | −0.134 | 0.309 | −0.0414 | ||
SAW | −0.136 | 0.357 | −0.0484 | ||
Mean | −0.125 | 0.333 | −0.0416 | ||
Grand mean |
The data analyzed above show that the Field of Contrast Sensitivity provides a simple comprehensive description of human spatial sensitivity throughout the visual field. It is of course an approximation, but one that is reasonably accurate over a wide range of conditions. It provides a principled
basis for engineering decisions in imaging systems that are concerned with the complete visual field.
In this report, we have been little concerned with the parameter
The PoV, and the associated Field of Contrast Sensitivity, describe contrast sensitivity only at mid to high spatial frequencies. We have not yet determined precisely at what low frequency the model ceases to be valid, and this value is likely to vary with adapting luminance and temporal frequency. But in the design of displays and imaging systems it is usually the sensitivities at high spatial frequencies that are critical.
One proposal for the source of the local scale is that it is based on the spacing of the mRGCs, which can be approximated as a linear function of eccentricity [
In the fovea, the visual optics prevents aliasing that might occur for frequencies above the sampling limit of the mRGC. However in peripheral vision, where the sampling is much coarser, aliasing can occur [
Our original description of the PoV included both spatial and temporal sensitivity; in this report I discuss only spatial sensitivity. The manner in which temporal sensitivity varies from fixation to periphery is also of great interest, and a topic that has attracted considerable research [
The PoV asserts that the log of contrast sensitivity declines linearly with spatial frequency with some slope (Eq. (
The corresponding radially symmetric impulse response is given by the Hankel Transform of the exponential, which is the Lorentzian function
We can implement this as a two-dimensional filter, and apply it in a space-variant manner [
An image filtered by the Field of Contrast Sensitivity. I also show the original image for comparison.
Vision is made possible by variations over space and time in the luminance of the retinal image. Consequently human sensitivity to those variations is a fundamental topic in vision science. Equally important is the manner in which that sensitivity varies with adapting luminance, and with position in the visual field. In previous work we showed how visibility as a function of contrast, spatial frequency, temporal frequency, and luminance could be described by the PoV model. In this report we have extended the PoV model into the periphery, by assuming a linear change in local scale with eccentricity. Now with one linear equation we are able to encompass over a century of research. And while the PoV and the Field of Contrast sensitivity are approximations, with acknowledged omissions, they nonetheless provide a powerful description of the bounds on the universe of visible signals.
The following are some methodological details for the six studies analyzed in this report. Unless stated otherwise, the stimulus was a CCG, viewing was binocular, and time course was Gaussian. Detection thresholds were measured with a 2IFC QUEST staircase [
Experimental details for the six studies cited in this paper.
Study | Mean | Duration | Standard deviation | Bandwidth | |||
(nit) | (cycle/deg) | (deg) | (cycles) | (ms) | (cycles) | (octave) | |
Robson & Graham (1981) | 500 | 1.5–24 | 20 | 32 | 71 | – | 0.44 |
Watson (1987) | 100 | 0.25–16 | 3 | 48 | 167 | 1.13 | 0.48 |
Pointer and Hess (1989) | 100 | 1.6–12.8 | 40 | 96 | 177 | 2.26 | 0.36 |
Arnow & Geisler (1996) | 130 | 200 | 1.09 | 0.5 | |||
Foley, Varadharajan, Koh & Farias (2007) | 110 | 4 | 5 | 20 | 90 or 240 | 1 | 0.55 |
Baldwin, Meese & Baker (2012) | 60–85 | 0.7–4 | 18 | 18 | 100 | 0.37 | 1.6 |
Robson & Graham [
Pointer and Hess (1989): The time course included a cosine modulation at 1 Hz, but with a time Gaussian with standard deviation of 177 ms this is effectively just a narrower Gaussian.
Arnow & Geisler [
Foley, Varadharajan, Koh & Farias [