Ordinal hashing of multidimensonal data and geographic coordinates via Morton coding / Z-ordering.

In mathematical analysis and computer science, Z-order, Morton-order, or a Morton-code is a function which maps multidimensional data to one dimension while preserving locality of the data points. It was introduced in 1966 by IBM researcher, G. M. Morton. The z-value of a point in multidimensions is calculated by interleaving the binary representations of its coordinate values. Once the data are sorted into this ordering, any one-dimensional data structure can be used, such as binary search trees, B-trees, skip lists, or hash tables. The resulting ordering can equivalently be described as the order one would achieve from a depth-first traversal of a quadtree, where {x, y, ..., K} are combined into a single ordinal value that is easily compared, searched, and indexed against other Morton numbers.

At the highest level, pymorton is split into two logical functions:

• (de)interleave: encodes/decodes hashes representing two or three dimensionsal integer sets. {x, y, z ∈ Z} or {x, y ∈ Z}, where Z represents all integer values.

• (de)interleave_latlng: encodes and decodes hashes representing latitude and longitude.

• Given a directory of images, sort the images by color (average RGB):

  from statistics import mean
  from glob import glob
  from PIL import Image
  import pymorton
  
  imgs = [(fname, Image.open(fname)) for fname in glob('imgpath/*.jpg')[:100]]
  
  # for each image, generate a tuple of len==3, representing the image's average RGB value
  avg_rgb_values = [
      [int(mean(img.getdata(band))) for band in range(3)] for _, img in imgs]
  
  # using the average RGB values, compute the Z-order of each image
  hashed_imgs = list(zip([fname for fname, _ in imgs],
                     [pymorton.interleave(*avg_rgb) for avg_rgb in avg_rgb_values]))
  
  # returns a sorted-by-color list of photos found within the directory
  return sorted(hashed_imgs, key=lambda img_tuple: img_tuple[1])

While the above use-case is fairly uncommon in the context of Morton-coding, I believe it illustrates the utility of the algorithm quite well. Morton-coding is most commonly used within the realm of geospatial indexing, but its potential applications are infinite!

via pip:

pip install pymorton

via source:

git clone https://github.com/trevorprater/pymorton.git
cd pymorton
python setup.py install

• 3D-hashing

import pymorton as pm

mortoncode = pm.interleave(100, 200, 50)  # 5162080
mortoncode = pm.interleave3(100, 200, 50) # 5162080

pm.deinterleave3(mortoncode)              # (100, 200, 50)

• 2D-hashing

import pymorton as pm

mortoncode = pm.interleave(100, 200)     # 46224
mortoncode = pm.interleave2(100, 200)    # 46224

pm.deinterleave2(mortoncode)             # (100, 200)

• geo-hashing

import pymorton as pm

geohash = pm.interleave_latlng(40.723471, -73.985361)     # '03023211233202130332202203002303'

pm.deinterleave_latlng(geohash)                           # (40.723470943048596, -73.98536103777587)

• pymorton.interleave(*args)

  • Hashes x, y or x, y, z into a single value. This function wraps interleave2() and interleave3() by supporting variable-length args.

• pymorton.interleave2(x, y)

  • Returns a hash (int) representing x, y.

• pymorton.interleave3(x, y, z)

  • Returns a hash (int) representing x, y, z.

• pymorton.interleave_latlng(lat, lng)

  • Returns a hash (string base-4) representing lat, lng.

• pymorton.deinterleave2(hash)

  • Returns a tuple representing the arguments to the corresponding interleave2() call.

• pymorton.deinterleave3(hash)

  • Returns a tuple representing the arguments to the corresponding interleave3() call.

• pymorton.deinterleave_latlng(hash)

  • Returns a tuple representing the arguments to the corresponding interleave_latlng() call.

From the project's root directory, execute nosetests.

Please feel free to contact trevor.prater@gmail.com regarding any questions/comments/issues.

• Z-order curve
• Implementation for the algorithm (1)
• Implementation for the algorithm (2)
• Extended explanation with different algorithms

MIT