variations in the characteristic curve

 
The appearance of the standard stimulus/response curves or characteristic curves will change radically depending on whether log or linear values are plotted on the stimulus and/or response axes. This page is a brief overview.

 

alternate forms of a stimulus/response curve
A: linear stimulus luminance plotted on linear response; B: log stimulus luminance plotted on linear response; C: log stimulus luminance plotted on log response

 
In a linear/linear plot (diagram A, above) the characteristic curve rises rapidly toward the response maximum, then flattens out near the response ceiling. This is the common format for a luminance/lightness diagram. This (or diagram C) is the form commonly used in perceptual psychophysics.

In a log/linear plot (diagram B), the characteristic curve assumes a familiar "S" shape; the lower part or toe of the curve is displayed as a rising exponential function, becoming approximately straight through the middle (highest contrast) portion of the curve. Eventually the response ceiling curbs this slope and the curve bends downward at the knee. This is the form commonly used in many imaging applications, including photography and printing.

In a log/log plot (diagram C), the characteristic curve assumes a "hockey stick" shape, straight through all the lower portions of the curve but flattening out near the response ceiling. This is the form commonly used in receptor psychophysics and studies of luminance adaptation.

The linear/linear curve above appears to have a much steeper slope and a much sharper bend near the response maximum than the typical luminance/lightness diagram. But this is because the lightness (response) axis is truncated at the "white" response, and the luminance curve is truncated at low values. When the luminance axis is extended to show the curve in more detail, the luminance/lightness plot appears as a small portion of the curve at luminances below the luminance of a "white" surface.

 

complete and truncated views of linear/linear curve

 
As shown above, the range of lightnesses that characterize surface or object colors represents only a small part of the total range of luminance sensitivity. In fact, under moderate illuminance levels, vision can still discriminate between the brightness of lights whose luminance is up to 100 times greater than the luminance of a white surface, although brightness discrimination is relatively coarse (the response curve is flat) when compared to the discrimination possible for the relative luminance (lightness) of surfaces.

The characteristic curve is cited in a variety of topic areas, and as a result the measure of gamma or contrast ratio is not always consistent. In photography or printing applications, it is more common to take the slope of the straight portion of the curve in linear/log units. However this must be done when the total response range and total luminance range are formatted to have equal length on their respective dimensions (the diagram fits into a square). In diagram B, this slope is approximately 1.51.

It is not convenient to use the slope method to estimate gamma from a linear/linear plot, as in diagram A. Instead the curve must be fitted as exactly as possible by a power function; then gamma is the reciprocal of the exponent. Most lightness scales, in the truncated portion of the curve, correspond to a power function of 0.43, equal to a gamma of about 2.3.
 

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