When does a color channel such as RGB carry more info than a grayscale channel or image?

Typically a gray-scale image gives us more information about an image in terms of a signal processing problem. But when does a color channel carry more info than a gray-scale channel?

I understand the use of FFT applied to gray-scale channels for this usual phenomena, but I want to know what is an instance of the opposite being true.

• Let's say you generate a greyscale image from an RGB by defining it as average of the three color channels: $Y= \frac13(R + G+B)$. Now, lets have an image where the green channel always decreases in intensity when the red channel gains intensity (for example: picture of cherries getting ripe. They're either green or red.) Then, the sum will contain less information on the ripeness than the red channel alone. I found that example easy to come up with. Can you find another one? – Marcus Müller Dec 2 '18 at 9:57
• RGB has three color channels. Do you mean just one of them (R, G, or B)? – Olli Niemitalo Dec 2 '18 at 17:05