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As far as I know, the RGB to YCbCr conversion is just a linear transformation, yet the Y channel successfully captures the intensity value of each pixel. How?

In other words, where do the following magic numbers come from?

$$ \begin{align} E'_Y &= 0.299E'_R + 0.587E'_G + 0.114E'_B \\ E'_R - E'_Y &= 0.701E'_R - 0.587E'_G - 0.114E'_B \\ E'_B - E'_Y &= -0.299E'_R - 0.587E'_G + 0.886E'_B \end{align} $$

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Focus on the first equation for EY. Back in the day when color television was being developed, the color signal had to be compatible with black and white TVs and vice versa. So the compatible brightness signal (luma Y) has to be calculated from the three primary color signals (R, G B) for transmission. Human visual system does not perceive brightnesses of different wavelenghts equally. So the actual coefficients depend on which specific RGB phosphors are used on the CRT, the spectrum that each phosphor emits, their brightness differences and how human visual system perceives them. See Color FAQ by Charles Poynton for better explanation.

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