Motion Vector - how to calculate it properly?

Question

I'm trying to wrap my head around calculating motion vectors (also called velocity buffer). I found this tutorial, but I'm not satisfied with explanations of how motion vector are calculated. Here is the code:

vec2 a = (vPosition.xy / vPosition.w) * 0.5 + 0.5;
vec2 b = (vPrevPosition.xy / vPrevPosition.w) * 0.5 + 0.5;
oVelocity = a - b;

Why are we multiplying our position vectors by 0.5 and then adding 0.5? I'm guessing that we're trying to get from clip space to NDC, but why? I completly don't understand that.

Answer 1

This is a mapping from the [-1, 1] clip space onto the [0, 1] texture space. Since lookups in the blur shader have to read from a textured at a position offset by the velocity vector, it's necessary to perform this conversion.

Note, that the + 0.5 part is actually unnecessary, since it cancels out in a-b anyway. So the same result would have been achieved by using something like

vec2 a = (vPosition.xy / vPosition.w);
vec2 b = (vPrevPosition.xy / vPrevPosition.w);
oVelocity = (a - b) * 0.5;

I don't know if there is any reason to prefer the first over the second, but my guess is that this code is written in the way it is because it builds up on a previous tutorial where the calculation had been the same.