I'm guessing that here we assume the surface is smooth so that in particular, there is a plane tangent to the surface at the point of interest and that is why we only integrate over a hemisphere. What if our surface was not smooth?
Or we could have a clear surface? Which means light can come from bottom half plane?
How big is H?
Is this something that is easily approximated with low resource overhead? For instance, if we want to approximate an integral like this, it may require storing a very large explicit representation of a surface, supposing that we would like to use approximate methods and not precise symbolic integration.