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Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1. A wide variety of sigmoid functions including the logistic and hyperbolic tangent ...
There are two luminous efficiency functions in common use. For everyday light levels, the photopic luminosity function best approximates the response of the human eye. For low light levels, the response of the human eye changes, and the scotopic curve applies. The photopic curve is the CIE standard curve used in the CIE 1931 color space.
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
The curve which represents the response of the human eye to light is a defined standard function y (λ) or V(λ) established by the International Commission on Illumination (CIE, for Commission Internationale de l'Éclairage) and standardized in collaboration with the ISO.
A CRT, for example, converts a video signal to light in a nonlinear way, because the electron gun's intensity (brightness) as a function of applied video voltage is nonlinear. The light intensity I is related to the source voltage V s according to , where γ is the Greek letter gamma.
A comparison between a typical normalized M cone's spectral sensitivity and the CIE 1931 luminosity function for a standard observer in photopic vision. In the CIE 1931 model, Y is the luminance, Z is quasi-equal to blue (of CIE RGB), and X is a mix of the three CIE RGB curves chosen to be nonnegative (see § Definition of the CIE XYZ color space).
Evidence indicates that what Newton meant by "indigo" and "blue" does not correspond to the modern meanings of those color words. Comparing Newton's observation of prismatic colors with a color image of the visible light spectrum shows that "indigo" corresponds to what is today called blue, whereas his "blue" corresponds to cyan. [14] [15] [16]
The generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is sometimes named Richards's curve after F. J. Richards, who proposed the general form for the family of models in 1959.