How do you convert a color image to grayscale? If each color pixel is described by a triple (R, G, B) of intensities for red, green, and blue, how do you map that to a single number giving a grayscale value? The GIMP image software has three algorithms.
The lightness method averages the most prominent and least prominent colors: (max(R, G, B) + min(R, G, B)) / 2.
The average method simply averages the values: (R + G + B) / 3.
The luminosity method is a more sophisticated version of the average method. It also averages the values, but it forms a weighted average to account for human perception. We’re more sensitive to green than other colors, so green is weighted most heavily. The formula for luminosity is 0.21 R + 0.71 G + 0.07 B.
The example sunflower images below come from the GIMP documentation.
The lightness method tends to reduce contrast. The luminosity method works best overall and is the default method used if you ask GIMP to change an image from RGB to grayscale from the Image -> Mode menu. However, some images look better using one of the other algorithms. And sometimes the three methods produce very similar results.
Update: See More on colors and grayscale for more details and more examples.