# 7.1 General Formulas

The Grayscale Standard Display Function is defined by a mathematical interpolation of the 1023 Luminance levels derived from Barten’s model. The Grayscale Standard Display Function allows us to calculate luminance, L, in candelas per square meter, as a function of the Just-Noticeable Difference (JND) Index, j:

[pic]

with Ln referring to the natural logarithm, j the index (1 to 1023) of the Luminance levels L j of the JNDs, and a = -1.3011877, b = -2.5840191E-2, c = 8.0242636E-2, d = -1.0320229E-1, e = 1.3646699E-1, f = 2.8745620E-2, g = -2.5468404E-2, h = -3.1978977E-3, k = 1.2992634E-4, m = 1.3635334E-3.

The logarithms to the base 10 of the Luminance L j are very well interpolated by this function over the entire Luminance Range. The relative deviation of any log(Luminance)-value from the function is at most 0.3%, and the root-mean-square-error is 0.0003. The continuous representation of the Grayscale Standard Display Function permits a user to compute discrete JNDs for arbitrary start levels and over any desired Luminance Range.

Notes: 1. To apply the above formula to a device with a specific range of L values, it is convenient to also have the inverse of this relationship, which is given by:[pic]where Log10 represents logarithm to the base 10, and A = 71.498068, B = 94.593053, C = 41.912053, D = 9.8247004, E = 0.28175407, F = -1.1878455, G = -0.18014349, H = 0.14710899, I = - 0.017046845.2. When incorporating the formulas for L(j) and j(L) into a computer program, the use of double precision is recommended.

3. Alternative methods may be used to calculate the JND Index values. One method is use a numerical algorithm such as the Van Vijngaarden-Dekker-Brent method described in Numerical Recipes in C (Cambridge University press, 1991). The value j may be calculated from L iteratively given the Grayscale Standard Display Function’s formula for L(j). Another method would be to use the Grayscale Standard Display Function’s tabulated values of j and L to calculate the j corresponding to an arbitrary L by linearly interpolating between the two nearest tabulated L,j pairs.4. No specification is intended as to how these formulas are implemented. These could be implemented dynamically, by executing the equation directly, or through discrete values, such as a LUT, etc.

Annex B lists the Luminance levels computed with this equation for the 1023 integer JND Indices and Fig. 7-1 shows a plot of the Grayscale Standard Display Function. The exact value of the Luminance levels, of course, depends on the start level of 0.05 cd/m 2 .

The Characteristic Curve of a Display System represents the Luminance produced by a Display System as a function of DDL and the effect of ambient Illuminance. The Characteristic Curve is measured with Standard Test Patterns (see Annex D). In general, the Display Function describes, for example,

a) the Luminance (including ambient Illuminance) measured as a function of DDL for emissive displays such as a CRT-monitor/digital display controller system,

b) the Luminance (including ambient Illuminance) as a function of DDL measured for a transmissive medium hung in front of a light-box after a printer produced an optical density, depending on DDL, on the medium,

c) the Luminance (including ambient light) as a function of DDL measured for a diffusely reflective medium illuminated by a office lights after a printer produced a reflective density, depending on DDL, on the medium.

By internal or external means, the system may have been configured (or calibrated) such that the Characteristic Curve is consistent with the Grayscale Standard Display Function.

Some Display Systems adapt themselves to ambient light conditions. Such a system may conform to the Grayscale Standard Display Function for one level of ambient Illuminance only, unless it had the capability of adjusting its Display Function without user-intervention so that it remains in conformance with the Grayscale Standard Display Function.