My notation is outlined in Figure 28.6, on page 343. The coefficients are derived in Colour science for video, on page 287.

To compute luminance using (R+G+B)/3 is at odds with the characteristics of vision.

If the luminance of a scene element is to be sensed by a scanner or camera having a single spectral filter, then the spectral response of the scanner’s filter must – in theory, at least – correspond to the luminous efficiency function of Figure 24.1. However, luminance can also be computed as a weighted sum of suitably chosen red, green, and blue tristimulus components. The coefficients are functions of vision, of the white reference, and of the particular red, green, and blue spectral weighting functions employed. For realistic choices of white point and primaries, the green coefficient is quite large, the blue coefficient is the smallest of the three, and the red coefficient has an intermediate value.

The primaries of contemporary video displays are standardized in BT.709. Weights computed from these primaries are appropriate to compute relative luminance from red, green, and blue tristimulus values for computer graphics, and for modern video cameras and modern displays in both SD and HD:

For BT.709 primaries, luminance comprises roughly 21% power from the red (longwave) region of the spectrum, 72% from green (mediumwave), and 7% from blue (shortwave).

Blue has a small contribution to luminance. However, vision has excellent colour discrimination among blue hues. Equation 24.1 does not give you licence to assign fewer bits to blue than to red or green – in fact, it tells you nothing whatsoever about how many bits to assign to each channel.

Lightness (CIE L*)

Lightness is defined by the CIE as the brightness of an area judged relative to the brightness of a similarly illuminated area that appears to be white or highly transmitting. Lightness is most succinctly described as apparent reflectance. Vision is attuned to estimating surface reflectance factors; lightness relates to that aspect of vision. The CIE’s phrase “similarly illuminated area that appears white” involves the absolute luminance by which relative luminance is normalized. In digital imaging, the reference white luminance is ordinarily

The L* symbol is pronounced

EL-star.

closely related to the luminance of a perfectly diffusing reflector (PDR) in the scene, or the luminance at which such a scene element is presented (or will ultimately be presented) at a display.

In Contrast sensitivity, on page 249, I explained that vision has a nonlinear perceptual response to luminance. Vision scientists have proposed many functions that relate relative luminance to perceived lightness; several of these functions are graphed in Figure 24.2.

The computational version of lightness, denoted L*, is defined by the CIE as a certain nonlinear function of relative luminance. In 1976, the CIE standardized lightness, L* as an approximation of the lightness response of human vision. Other functions – such as Munsell value – specify alternate lightness scales, but the CIE L* function is widely used and internationally standardized.

The L* function has two segments: a linear segment near black, and a scaled and offset cube root (1/3-power) function everywhere else.

Eq 24.2

Eq 24.3

To compute L* from optical density D in the range 0 to 2, use this relation:

L* = 116 ·10-D/3-16

My best-fit pure power function estimate is based upon numerical (Nelder-Mead) minimization of least-squares error on L* values 0 through 100 in steps of 10. The same result is obtained by fitting 100 samples in linear-light space.

The 1976 version of the CIE standard expresses this definition of L*:

In the 2004 version of the standard, the decimals were replaced by exact rational fractions. Today’s definition is equivalent to this:

The argument Y is relative luminance, proportional to intensity. This quantity is already relative to some absolute white reference, typically the absolute luminance associated with a perfect (or imperfect, say 90%) diffuse reflector. The argument Y is tacitly assumed to lie on

a scale whose maximum value (Yn) is related to the viewer’s adaptation state. The division by Yn does not form relative luminance; rather, the normalization accommodates the tradition dating back to 1931 and earlier that tristimulus values lie on a 0 to 100 scale. For tristimulus reference range of 0 to 1, as I prefer, the division by Yn can be omitted.

The linear segment of L* is convenient for mathematical reasons, but is not justified by visual perception: The utility of L* is limited to a luminance ratio of about 100:1, and L* values below 8 don’t represent meaningful visual stimuli. (In graphics arts, luminance ratios up to about 300:1 are used with L*.)

The exponent of the power function segment of the L* function is 1⁄3, but the scale factor of 116 and the offset of -16 modify the pure power function such that the best-fit pure power function has an exponent of 0.42, not 1/3! L* is based upon a cube root, but it is not best approximated by a cube root! The best purepower function approximation to lightness is 100 times the 0.42-power of relative luminance.

In a display system having contrast ratio of 100:1, L* takes values between 9 and 100.

∆L* is pronounced delta EL-star.

For television viewing, we typically set Yn to reference white at the display. In television viewing, the viewer’s adaptation is controlled both by the image itself and by elements in the field of view that are outside the image. In cinema, the viewer’s adaptation is controlled mainly by the image itself. In cinema, setting Yn to reference white is not necessarily appropriate; it may be more appropriate to set Yn to the luminance of the representation of a perfect diffuse reflector in the displayed scene.

Relative luminance of 0.01 maps to L* of almost exactly 9. You may find it convenient to keep in mind two exact mappings of L*: Relative luminance of 1/64 (0.015625) corresponds to L* of exactly 13, and relative luminance of 1/8 (0.125) corresponds to L* of exactly 42 (which, as Douglas Adams would tell you, is the answer to Life, the Universe, and Everything).

The difference between two L* values, denoted ∆L*, is a measure of perceptual “distance.” In graphics arts, a difference of less than unity between two L* values is generally considered to be imperceptible – that is, ∆L* of unity is taken to lie at the threshold of discrimination. L* is meaningless beyond about 200 – that is, beyond about 6.5 Y/Yn.

In Contrast sensitivity, on page 249, I gave the example of logarithmic coding with a Weber contrast of 1.01. For reconstructing images for human viewing, it is never necessary to quantize relative luminance more finely than that. However, L* suggests that a ratio of 1.01 is unnecessarily fine. The inverse L* of 100 is unity; dividing that by the inverse L* of 99 yields a Weber contrast of 1.025. The luminance ratio between adjacent L* values increases as L* falls, reaching 1.13 at L* of 8 (at relative luminance of about 1%, corresponding to a contrast ratio of 100:1). L* was standardized based upon estimation of lightness of diffusely reflecting surfaces; the linear segment below L* of 8 was inserted for mathematical convenience. I consider estimating the visibility of lightness differences at luminance values less than 1% of white to be a research topic, and I recommend against using delta-L* at such low luminances.

L* provides one component of a uniform colour space; it can be described as perceptually uniform. Since we cannot directly measure the quantity in question, we

Figure 25.1 Example coordinate system

The CIE system

The Commission Internationale de L’Éclairage (CIE) has defined a system that maps a spectral power distribution

(SPD) of physics into a triple of numerical values – CIE XYZ tristimulus values – that form the mathematical coordinates of colour space. In this chapter, I describe the CIE system. In the following chapter, Colour science for video, I will explain how these XYZ tristimulus values are related to linear-light RGB values.

Colour coordinates are analogous to coordinates on a map (see Figure 25.1). Cartographers have different map projections for different functions: Some projections preserve areas, others show latitudes and longitudes as straight lines. No single map projection fills all the needs of all map users. Analogously, there are many “colour spaces,” and as in maps, no single coordinate system fills all of the needs of users.

In Chapter 24, Luminance and lightness, I introduced the linear-light quantity luminance. To reiterate, I use the term luminance and the symbol Y to refer to CIE luminance. I use the term luma and the symbol Y’ to refer to the video component that conveys an approximation to lightness. Most of the quantities in this chapter, and in the following chapter Colour science for video, involve “linear-light” values that are proportional to intensity. In Chapter 10, Constant luminance, I related the theory of colour science to the practice of video. To approximate perceptual uniformity, video uses quantities such as R’, G’, B’, and Y’ that are not proportional to intensity.