It is unwise to pretend that gamma corrected signals can successfully be multiplied, added and subtracted in a matrix as if they represented linear light. Yet in television it is done all the time.
In the specific case of analog broadcasts received only by black and white CRTs and with minimal signal manipulation between camera and display, gamma applied at source seems to be a good thing, because it applies the rendering intent for a display of known brightness, it cancels the gamma of the CRT, simplifies the receiver and improves the apparent signal-to-noise ratio by applying perceptual uniformity to the transmitted signal. Win, win, win.
Today we have almost the opposite situation. We don’t have analog broadcasts, we don’t have black and white signals, we don’t have CRTs; in fact we don’t know what type of display will be used or how bright it is. Complex production steps are used all the time. All but one of the original conditions under which gamma correction at source was chosen have vanished, in particular the requirement for minimal signal manipulation. The only remaining benefit is perceptual uniformity. When the larger picture is considered, the changed circumstances mean that the drawbacks of gamma start to become serious.
One of the characteristics of human vision that we would like to exploit in television is that the ability of the HVS to discern detail in color changes is much less than in brightness. That is the whole reason for the adoption of color difference signals going right back to I and Q of NTSC, U and V of PAL and later the Cr and Cb of digital systems. The theory holds that if color difference signals contain no luminance information, their bandwidth can dramatically be reduced and no human viewer can tell.
There is nothing wrong with the theory. When true luminance is calculated from R, G, and B and that true luminance is subtracted from the R and B, there really is no luminance in the color differences. The problem is that in television we simply don't adhere to the theory and the full bandwidth saving possible is not obtained.
Fig.1 - Addition with non-linear signals simply doesn't give the expected result. Here, 5 + 5 after gamma becomes 6.7.
Instead, television adds R', G' and B' to create luma or Y'. All of those are gamma corrected by a power function and so the Y' matrix function does nor perform additions, Y’ is not luminance and R' - Y' and B' - Y' are not subtractions. The non-linear curve of gamma is not so different from a logarithmic curve. Yet when we add logarithms, we multiply and when we subtract logarithms we divide.
Fig.1 shows the problem. In school, we learn that 5 + 5 is 10. But in the presence of gamma correction, a linear light, or luminance, value of 5 has a value of about 2. Adding 2 to 2 gives 4, and in the linear domain that corresponds to about 6.7. With the help of gamma, 5 + 5 equals 6.7. In other words you can’t do analog computing when your signal is not an analog of the quantity you want to compute.
Attempting addition and subtraction on non-linear signals simply doesn't work and the resulting inadvertent multiplication produces instead a mess of intermodulation. As a result, the color difference signals are not free of luma information and cannot be reduced in bandwidth without information loss.
Fig.2 shows the difference. With linear light, a) the full luminance bandwidth is calculated from R, G and B and can be sent to the receiver along with band limited R –Y and B – Y. Full luminance bandwidth is available.
If we kid ourselves that a matrix in the gamma domain works, as in Fig.2b), we apply what we think is red, green and blue to a matrix and calculate what we think is luminance, when it isn’t. Using R’, G’ and B’, the matrix produces a signal Y’ that is not true luminance, but is instead lacking high frequencies. When Y’ is “subtracted” from R’ and B’, the HF luminance missing from Y’ is not subtracted so there is negative high frequency luminance in R’ – Y’ and in B’ – Y’.
Fig.2 - At a) in a linear light system, full luminance bandwidth is available even though color difference bandwidth has been reduced. At b) in a system using gamma, the inverse matrix only works if the color difference signals are not band limited. At c) the color difference signals contain negative luma. If they are band limited, the luma signal is impaired at the receiver.
All is not lost at this stage, because if we send those three signals to a further matrix that re-calculates R’, G’ and B’ and put them into a color CRT, everything comes out right, because whatever was initially subtracted is added back in the same domain.
But the whole point of matrixing to color differences was so that we could reduce their bandwidth. In linear light, that would have worked, but in the gamma domain it simply doesn’t. In Fig.2c), we apply what we think is a color difference signal to a low-pass filter and the high frequency negative luma information that is still in there is removed. The R', G' and B' calculated in the receiver is not what was input to the color difference calculation because high frequencies are missing, causing a loss of resolution in Y’ that is a function of saturation.
Fig.3 - The loss of luminance resolution as a function of saturation due to the use of gamma. Courtesy BBC.
Widening the color gamut makes it worse……
The BBC published a paper back in 1972 that explained the loss of resolution caused by matrixing gamma corrected primaries. The relevant figure is reproduced here as Fig.3. One small caution is required, which is that the authors assumed a display gamma of 2.8, whereas CRTs actually have a gamma of 2.5. However, that does not in any way alter the conclusion, which is that the use of gamma in color difference working causes a significant reduction in the modulation transfer function of luma in the presence of saturated colors.
Anyone who has ever studied color bars on a monitor fed by a color difference format such as NTSC, PAL, SECAM or 601 will have seen that the green-magenta transition in the center of the screen isn't correct. There is a dark bar clearly visible there, because the brightness has not correctly been calculated.
Clearly a green color bar requires no other components, whereas a magenta bar requires red and blue but no green. At the transition, the green component goes down and the red and blue components come up. The slope of the transition is finite because there is a bandwidth limit. Fig.4a) shows that in a linear system the luminance derived from G crossfades to the luminance derived from R and B, and it remains correct throughout.
However, in the presence of gamma, the crossfade is still linear but the result will be subject to display gamma. Fig. 4b) shows that in the middle of the crossfade the light from the screen is temporarily lower in level than it should be. That obvious dark bar is the answer to those who claim that failure of constant luminance isn't a problem.
Failure of constant luminance, is simply the failure to accept a compressed signal for what it is. The other failures don’t have names, but as will be seen, there are so many that they can be lumped together and called failure of everything.
NTSC, PAL and 601 were impaired by retaining the gamma that worked for black and white television but caused a loss of quality when any image processing, such as color difference matrixing was used. Gamma has always been problematic in color television and was retained primarily for compatibility with black and white TV sets having CRTs. Today there are no such sets and that reason no longer stands.
Fig.4 - At a) in a linear system the cross fade between green and magenta proceeds correctly. At b) when gamma is employed, the non-linearity causes the mid-point of the cross fade to be too dark.
The existence of the dark bar at the green/magenta transition is due to the nonlinearity of gamma, but the width of the dark bar is increased by the resolution loss shown in Fig.3. Color difference working with gamma damages the resolution most in the presence of saturated colors, making the dark bar wider.
The dark bar is still there in R’, G’, B’ images, but it is narrower because the resolution loss of color difference working has been avoided. Many graphics systems today work in linear light to avoid the dark lines where color changes. Chroma keying works better when performed in the linear light domain and is increasingly done that way. In rendering, the complex calculations performed make the color difference equations look simple, and accurate rendering requires the use of linear light, especially to get shading to look correct.
In practice, if gamma is used, the bandwidth of any color difference signals transmitted has to be higher than necessary to allow for the losses shown in Fig.3. There is a contradiction here, because following the demise of the CRT we are using gamma as a compression system that saves between 2 and 3 bits per luma pixel. Yet the use of gamma forces the data sent in the color difference signals to be greater, erasing the advantage of gamma.
In a linear system, the luma pixels would need another two or three bits, but the amount of color difference data to be sent would compensate for that because greater color subsampling factors could be used.
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