BijectiveHilbert

Data in two or more dimensions is stored linearly, so places nearby in the data can be far in memory or on disk. This package offers functions that make multi-dimensional data storage and retrieval more efficient by sorting them so that nearby data is nearby in memory.

julia> using Pkg; Pkg.add("BijectiveHilbert")
julia> using BijectiveHilbert
julia> xy = zeros(Int, 8, 8)
julia> for y in 1:size(xy, 2)
           for x in 1:size(xy, 1)
               z = encode_hilbert(Simple2D(Int), [x, y])
               xy[x, y] = z
           end
       end
julia> X = zeros(Int, 2)
julia> decode_hilbert!(Simple2D(Int), X, xy[5, 7])
julia> X == [5, 7]
julia> xy
8×8 Array{Int64,2}:
  1   2  15  16  17  20  21  22
  4   3  14  13  18  19  24  23
  5   8   9  12  31  30  25  26
  6   7  10  11  32  29  28  27
 59  58  55  54  33  36  37  38
 60  57  56  53  34  35  40  39
 61  62  51  52  47  46  41  42
 64  63  50  49  48  45  44  43

This function, called a Hilbert curve, is used most often for geospatial work or database implementation but is equally appropriate for dealing with large TIFF files. It belongs to the class of space-filling, self-avoiding, simple, and self-similar (FASS) curves, which includes Peano curves, and Morton z-curves.

Included are several variations of the Hilbert curve. They are type-stable and thoroughly tested, including bug fixes on three of the implementations.

Simple2D, shown above, two-dimensional. Doesn't need to know axis dimensions, from Chen, Wang, and Shi.
GlobalGray, an n-dimensional curve where all axis dimensions must be equal, from Skilling.
SpaceGray, an n-dimensional curve with a different path. All axis dimensions must be equal, from Hamilton and Rau-Chaplin.
Compact, an n-dimensional curve the permits each axis to be a different size, from Hamilton and Rau-Chaplin.
FaceContinuous, an n-dimensional curve, the oldest version from Butz and Lawder.

Hilbert curves for computation

A video about using Hilbert curves: