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RGB has the advantage over schemes like YPBPR in that it's more intuitive to understand, so why use the latter? The only advantage I can think of right now is that it allows you to encode greyscale images more easily. Is this the reason, or are there other reasons?

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From a historical context, the YUV color encoding was originally designed to support the existing black-and-white infrastructure. In addition, as jooja mentioned, humans are generally less perceptive to changes in chroma and more perceptive to changes in luma. Both NTSC and PAL use less bandwidth for the chroma channel than the luma channel. In a digital context, we can use chroma subsampling to achieve the same effect.

YUV/YIQ/YDbDr is used for color PAL/NTSC/SECAM television broadcasts.
YPbPr is used for component video cables.
YCbCr is used for everything else.
YCgCo is like YCbCr but is easier to compute.

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Human eye is less sensitive to chrominance than luminance, which means you have a nifty way to compress the data by not measuring chrominance for each pixel, while you do measure luminance of all pixels. Although the black and white thing is part of the reason.

On the subject RGB might be more intuitive, but its not actually a very good intuition when you start going deeper into how the human senses actually work. As it turns out RGB is a gross simplification. RGB is not good for measuring how far away different colors are from each other visually for example.

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