Figure 27.4 BT.709, sRGB, and CIE L* encoding functions are compared. They are all approximately perceptually uniform; however, they are not sufficiently close to be interchangeable.

Tristimulus value, T (relative)

Figure 27.4 sketches the sRGB encoding function, overlaid on the BT.709 encoding and CIE L* functions.

Transfer functions in SD

Historically, transfer functions for SD have been very poorly specified. The FCC NTSC standard adopted in 1953 referred to a “transfer gradient (gamma exponent) of 2.2.” It isn’t clear whether 2.2 was intended to characterize the camera’s OECF or the display’s EOCF. In any event, modern CRTs have power function laws very close to 2.4! The FCC statement is widely interpreted to suggest that encoding should approximate

a power of 1⁄2.2; the reciprocal of 1⁄2.2, 0.45, appears in modern standards such as BT.709. However, as I mentioned on page 321, BT.709’s effective overall curve is very close to a square root. The FCC specification should not be taken seriously: Use BT.709 for encoding.

Standards for 576i SD also have poorly specified transfer functions. An “assumed display power function” of 2.8 is mentioned in EBUspecifications; some people interpret this as suggesting an encoding exponent of 1⁄2.8. However, the 2.8 value is unrealistically high. In fact, European displays are comparable to displays in other parts of the world, and encoding to BT.709 is appropriate.

Surprisingly, no current standards specify viewing conditions in the studio. Only in 2011 was a standard adopted that specifies the transfer function of an ideal-

In BT.601 coding with 8 bits, the black-to-white range without footroom or headroom encompasses 220 levels. For linear-light coding of this range, 10 bits suffices:

4.5 220 = 990; 990 < 210

4.5 880 = 3960; 3960 < 212

ized studio display! In the absence of a studio display EOCF, consumer display manufacturers adopted their own (nonstandard) practices, one factor leading to unpredictable image display in consumers’ premises.

Bit depth requirements

In Figure 10.1 on page 108, in Chapter 10’s discussion of constant luminance, I indicated that conveying relative luminance directly would require about 11 bits.

That observation stems from two facts. First, studio video experience proves that 8 bits is barely sufficient to convey gamma-corrected R’G’B’ – that is, 28 (or 256) nonlinear levels are sufficient. Second, the transfer function used to derive gamma-corrected R’G’B’ has a certain maximum slope; a maximum slope of 4.5 is specified in BT.709. The number of codes necessary in a linear-light representation is the product of these two factors: 256 times 4.5 is 1152, which requires 11 bits.

In studio video, 8 bits per component barely suffice for distribution purposes. Some margin for roundoff error is required if the signals are subject to processing operations. For this reason, 10-bit studio video is now usual. To maintain 10-bit BT.709 accuracy in a linearlight system would require 12 bits per component. The BT.709 transfer function is suitable for video intended for display in the home, where contrast ratio is limited by the ambient environment. For higher-quality video, such as home theater, or for the adaptation of HD to digital cinema, we would like a higher maximum gain. When scaled to a lightness range of unity, CIE L* has a maximum gain of 9.033; sRGB has a gain limit of 12.92. For these systems, linear-light representation requires 4 bits in excess of 10 on the nonlinear scale – that is, 14 bits per component.

If RGB or XYZ tristimulus components were conveyed directly, then 16 bits in each component would suffice for any realistic image-reproduction purpose. Linearlight 16-bit coding is now practical in high-end production, for example, scene-linear workflows using OpenEXR coding. For now, such approaches don’t havve realtime hardware. In most applications, the nonlinear characteristics of perception are exploited and nonlinear image data coding is used.

PDP and DLP devices are commonly described as employing PWM. However, it is not exactly the widths of the pulses that are being modulated, but the number of unit pulses per frame.

Concerning the conversion between BT.601 levels and the full-swing levels commonly used in computing, see Figure 31.3, on page 384.

Gamma in modern display devices

Modern display devices, such as liquid crystal displays (LCDs), have transfer functions different from that of CRTs. Plasma display panels (PDPs) and Digital Light Processors (DLPs) both achieve apparent continuous tone through pulse width modulation (PWM): They are intrinsically linear-light devices, with straight-line transfer functions. Linear-light devices, such as PDPs and DLPs, potentially suffer from the “code 100” problem explained on page 31: In linear-light, more than 8 bits per component are necessary to achieve high quality.

No matter what transfer function characterizes the display, it is economically important to encode image data in a manner that is well matched to perceptual requirements. The BT.1886 EOCF is well matched to CRTs, but more importantly, it is well matched to perception! The performance advantage of perceptual coding, the wide deployment of equipment that expects BT.1886 decoding, and the huge amount of program material already encoded to this standard preclude any attempt to establish new standards optimized to particular devices.

A display device whose transfer function differs from a CRT must incorporate local correction, to adapt from its intrinsic transfer function to the transfer function that has been standardized for image interchange.

Estimating gamma

Knowing that a CRT is intrinsically nonlinear, and that its response is based on a power function, many researchers have attempted to summarize the nonlinearity of a CRT display in a single numerical parameter γ using this relationship, where V is code (or voltage) and T is luminance (or tristimulus value):

The model forces zero voltage to map to zero luminance for any value of gamma. Owing to the model being “pegged” at zero, it cannot accommodate blacklevel errors: Black-level errors that displace the transfer function upward can be “fit” only by an estimate of gamma that is much smaller than 2.4. Black-level errors that displace the curve downward – saturating at zero

Equation 27.8 below is written with logs to base 10; however, because the ratio oof logs is taken, any base would do. In this calculation, luminance values L1 through LN must be strictly greater than L0. If the video signal values are not at equal intervals, replace i/n in the denominator by Vi where each Vi is the appropriate video signal level strictly between 0 and 1.

EBU Tech. 3325 (2008), Methods for the Measurement of the performance of Studio Monitors,

Version 1.1 (Sep.).

over some portion of low voltages – can be “fit” only with an estimate of gamma that is much larger than 2.4. The only way the single gamma parameter can fit a black-level variation is to alter the curvature of the function. The apparent wide variability of gamma under this model has given gamma a bad reputation.

A much better model is obtained by fixing the exponent of the power function at 2.4, and using the single parameter to accommodate black-level error, :

This model fits the observed nonlinearity much better than the variable-gamma model.

A simple technique to estimate gamma uses luminance measurements for video signal codes 0.08 and 0.8. Take the log10 of these two luminance values. The arithmetic difference between the two logs is a decent gamma estimate. The two video signal values are one decade apart; the 0.08 video signal is high enough to avoid potential black-level issues, and the 0.8 signal is low enough to avoid CRT saturation.

Figure 27.1, on page 317, graphs several pure 2.4-power functions. Gamma is 2.4 everywhere along these curves. Consider measuring a display at n+1 video signal values at equal intervals of 1/n between 0 and 1. (Usually ten values are used, 0.1, 0.2, …, 0.9, 1.0.) Using L0 to symbolize the luminance produced by zero signal value and LN to symbolize the luminance produced by unity signal value, average gamma can be estimated as follows:

A variant of this formulation is described in EBU Tech. 3325; it is commonly used in home theatre calibration. In my view, this formulation of average gamma gives the luminance produced at reference white video level undue influence over the estimated gamma value. If reference white luminance is depressed – as will be the case for a CRT entering saturation, or an LCD mimicking that behaviour – then all of the contributing