If instead we generate the RGB samples by simply averaging over
square cells, we
average across boundaries which yields the artifacts we can see here.
You can see that the space
above the earth now has a bluish halo around it and the Red Sea now
Thus it is important to only average in pixels that are inside the deformed cellso that discontinuities are properly preserved.
[ Note: When the image for this slide was generated, the antialiasing removed some of the silhouette map cell borders so it appears that it is a 32x32 silmap instead of a 64x64 silhouette map. ]