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NOTES.md

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Notes

(Tight-ish?) Upperbound on neighbor count

Upper bound for a voxel in/on the

  • Center of the image: (2d + 1)^3 - 1
    • Count: (x - 2) * (y - 2) * (z - 2)
  • Side of the image: (d + 1)(2d + 1)^2 - 1
    • Count: 2 * ((x - 2) * (y - 2) + (x - 2) * (z - 2) + (y - 2) * (z - 2))
  • Edge of the image: (2d + 1)(d + 1)^2 - 1
    • Count (ignore corners): 4 * (x + y) + 4 * z - 8
  • Corner of the image: (d + 1)^3 - 1
    • Count: 8

We can make this bound tighter (when d > 1 -- tight as possible for d = 1), but I'm not sure that it's worth it (yet). Bound can be improved by considering the voxels v that are less than d away from the side of the image.