I was reading this paper related to Permutohedral Lattice for Gaussian Filtering - http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CCAQFjAA&url=http%3A%2F%2Fgraphics.stanford.edu%2Fpapers%2Fpermutohedral%2Fpermutohedral_techreport.pdf&ei=yaw3UJHUCOrx0gGwhoDgDw&usg=AFQjCNGBdVr27F8eROwdrfiSrskQWULtJg&sig2=9Er7ofJrQ8kCsabKzo1DDA.
However, I have a confusion understanding the lattice itself. It says that $ H_d= \{{\vec{x} | \vec{x}.\vec{1} = 0}\} \subset R^{d+1} $
The root lattice
$ A_d=\{\vec{x}=(x_0,...x_d) \epsilon Z^{d+1}| \vec{x}.\vec{1}=0\} $
$ A_d=Z^{d+1}\cap H_d, A_d\subset H_d $
Well my confusion is how come $ A_d\subset H_d $ I mean
$ Z^{d+1}\cap H_d= H_d $
itself isn't it? since $H_d$ itself is in $R^{d+1}$. Any insights?
Also it mentions $ A_d^{*} = \{{\vec{x}\epsilon H_d | \forall \vec{y} \epsilon A_d, \vec{x}.\vec{y} \epsilon Z}\} $
Also I didn't understand what the above definition of permutohedral lattice means. I am beginner so I am finding it difficult to visualize. Any help? $$
Also I didn't understand what they mean by
x is a remainder-k point while defining the lattice