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Given an NxNxN cube (image), how can I find all the 2x2x2 boxes within the NxNxN cube? of course if N is even, we can find 2x2x2 boxes without overlapping, but when the N is odd, there is overlapping between some of the 2x2x2 boxes found in the bigger cube.

So,

1- How can I find all the non-overlapped 2x2x2 boxes in a bigger NxNxN cube where N is even?

input: NxNxN cube output: all the possible non-overlapped 2x2x2 cubes.

2- How can I find all the overlapped 2x2x2 boxes in a bigger NxNxN cube where N is odd? This time, the overlapped areas in the 2x2x2 boxes should be zero in second (or more) visits; i.e. each overlapped area should be visited (counted) once not more.

input: NxNxN cube output: all the possible overlapped 2x2x2 cubes with zero values for the overlapped voxels in 2nd or more visits.

Xio
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  • Are you always doing double posts? https://de.mathworks.com/matlabcentral/answers/281798-how-to-find-all-possible-small-boxes-e-g-2x2x2-and-their-overlaps-within-a-bigger-nxnxn-cube – tim Apr 30 '16 at 07:50

1 Answers1

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I will give you an answer for the part where N is even. The rest can be easily adapted, I hope you can do this yourself :-) Or at least try it - if you've got problems, just come back to us.

I don't have MATLAB installed anymore, so I hope this is free of typo errors. But the idea should be clear:

% 
N = 10;

% First create all possible starting coordinates of 2x2x2 cubes within the big cube:
coords = 1:2:(N-1); % could easily be adapted to other small-cube sizes like 3x3x3 if you want to...

% Now create all possible combinations of starting-coordinates in every direction (as it is a cube, the starting points in x, y, z directions are the same):
sets = {coords, coords, coords};
[x y z] = ndgrid(sets{:});
cartProd = [x(:) y(:) z(:)]; % taken from here: http://stackoverflow.com/a/4169488/701049 --> you could instead also use this function: https://www.mathworks.com/matlabcentral/fileexchange/10064-allcomb-varargin- which generates all possible combinations

% Now cartProd contains all possible start-points of small cubes as row-vectors. If you want, you can easily calculate the corresponding vectors of end-points by simply adding +1 to every entry which will effectively yield a small-cube size of 2. If you want to further modify it to support e.g. 3x3x3 cubes, simply add +2 to every dimension.
endPoints = cartProd+1;

% E.g.: the first small cube starts at [x,y,z] = cartProd(1,:) and ends at [x_e, y_e, z_e] = endPoints(1,:).

Have fun :-)

Hint: for the odd big cube -> Simply treat it as evenly-sized cube, e.g. treat a 9x9x9 cube as 10x10x10, take my algorithm from above, and then move the most outer small-cubes one step to the center. That means, take the small cubes with the biggest x, y or z coordinate and substract 1 in that direction. So that the starting coordinate for all small cubes with x_max=9 will be changed to x=8. Then the same for y_max=9 and z_max=9.

tim
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