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I was referring to this paper here related to permutohedral lattice. I didn't get what this remainder $k$ point means when defining the permutohedral lattice.

I am beginner. Can anyone provide me any pointers?

I am confused at this line

$$ A_{d^*}= \bigcup_{k=0}^{d}\{\vec{x}\in H_d\ |\ \vec{x}\text{ is a remainder-$k$ point}\} $$

we call $\vec{x}\in H_d$ a remainder-$k$ point for some $k \in \{0,\ldots,d\}$ iff all coordinates are congruent to $k$ modulo $d +1$.

I am having difficulty in visualizing it.

enter image description here

Well I got this example of the lattice for $d=2$. However, I didn't get how they plotted the points. The points are in the $Z^3$ space. I cannot actually get how they plotted the points $(2,-1,-1)$ and $(1,1,-2)$ in this space. I suppose they are wrongly placed or something. I am not sure though. I mean suppose the normal 3D space. point $(2,-1,-1)$ should be somewhere else isn't it?

enter image description here

Here is something that I drew for 2D space, where $H_d$ is a line, and I have drawn the remainder-$k$ points with their remainder.

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    Could you clarify what part of the definition (given on the very next line of the paper) you have trouble with?2012-09-08
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    @Erick I have modified the question to show where I am confused2012-09-08
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    You should add the definition in paper to your question; here, copy between the quotes: `we call $\vec{x}\in H_d$ a remainder-$k$ point for some $k \in \{0,\ldots,d\}$ iff all coordinates are congruent to $k$ modulo $d +1$`.2012-09-08
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    Is it the word, "congruent", that you don't understand? It just means every coordinate is $k$ more than a multiple of $d+1$.2012-09-09
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    @Gerry I am finding it difficult to visualize what it is going to look like. Also what is meant by all coordinates. If (x,y) is a coordinate, it means both x and y is k more than multiple of d+12012-09-09
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    @user34790 Perhaps you are confused about "coordinate": $(x,y)$ is a _point_, not a coordinate. $x$ and $y$ are the coordinates of $(x,y)$.2012-09-09
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    @user34790 Would it help to know that $(1,1,-2)$ is a remainder-$1$ point for $d=2$?2012-09-09
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    It might help you to visualize it if you do an example in 2-d. Draw yourself a grid, and circle all the remainder-$1$ points for $d=2$. What's it look like?2012-09-09
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    @Eric. No, I didn't get how (1,1,-2) is a remainder-1 point for d=2. Sorry for my ignorance. I am weak and just a beginner. Is it because 1 is greater than 0*(2+1) by 1 and -2 is greater than -1(2+1) by 12012-09-09
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    Yes, each of the numbers 1, 1, and $-2$ is 1 greater than a multiple of $2+1$. I see you have found a diagram illustrating the situation in three dimensions, but you haven't taken up my suggestion of drawing yourself a diagram to see what happens in two dimensions. Why not learn to walk before you try to run?2012-09-09
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    @Gerry Thanks for the info. Actually, I have tried that in the 2D space. Please provide some feedback, so that I can know better in 3D as well2012-09-09
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    @all I think I need to get the basic right to understand this paper. Can anyone please provide some pointers. I want to understand this paper, related to Gaussian filering on permutohedral lattice.2012-09-09
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    Sorry, what I meant was circle **all** the remainder 1 points for $d=2$, not just the ones on some line.2012-09-10
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    @Gerry, I circled only the ones on the line because my $H_d$ is the straight line and as given in the definition it is all the remainder-k points belonging to the plane $H_d$ isn't it?2012-09-10
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    Sure. I just thought that your visualization might be improved if you saw all the points, not just the ones on that one line. But if you want to stick to a line, then pick another one and see what you get. And another one. And another one, until you've got your visualization down.2012-09-10
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    @Gerry. Thanks. Actually, I am not concerned about the points given in the 3D space shown in the figure. Since it is $R^3$ space, my $H_d$ is a 2D plane. The thing I am confused about is how they plotted the points (2,-1,-1), &(1,1,-2) in that picture. The figure looks like a 3D space, with the x,y and z axis. I got how to know which k-remainder point a point belongs to based upon its coordinates. But the issue is in the figure, how come that point (2,-1,-1) lies there. I mean taking 2 steps in + x axis, - 1 steps in -ve y and z axis, the point should lie somewhere else isn't it?2012-09-10
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    I don't understand the labelling of the points in that picture, either. There ought to be some context, explaining how the diagram is to be interpreted, but you haven't provided it, nor have you provided a link so someone else could look for it.2012-09-11
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    @Gerry. Here is the link for the paper http://www.google.com/url?sa=t&rct=j&q=permutohedral%20lattice%20for%20gaussian%20filter&source=web&cd=3&cad=rja&ved=0CC8QFjAC&url=http%3A%2F%2Fgraphics.stanford.edu%2Fpapers%2Fpermutohedral%2Fpermutohedral.pdf&ei=o2NPUMqPCK-20AGUw4HwCg&usg=AFQjCNHbBY3PwLlD1DzdpylkgBQxc6KSzg2012-09-11
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    @Gerry. Also I didn't get this construction of $A_{d^*}$ for d=2 in the Fig 2 in this previous paper http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CB8QFjAA&url=http%3A%2F%2Fgraphics.stanford.edu%2Fpapers%2Fpermutohedral%2Fpermutohedral_techreport.pdf&ei=6ZJPUN__HYr10gHvkYDgBQ&usg=AFQjCNGBdVr27F8eROwdrfiSrskQWULtJg. As it characterizes $A_{d^*}$ as the projection of $Z^{d+1}$ along the diagonal vector $\vec{1}$ with a scale factor of d+1 to keep the coordinates in Z2012-09-11

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