Creating visually random images

How complicated does a mathematical function, pseudorandom(x), have to be to create something that seems random, and how do you check whether it seems random? Looking at a long list of numbers is not a good way, because our ability to look at long lists of numbers is very limited. Our ability to look at images is much better.

So, to inspect whether a list of numbers is seemingly random, a fun way is to create an image using the numbers for each pixel, and simply look at it.

Given a number x in the series {0, 1, 2, …, huge n}, let’s create an image that plots a random function that’s implemented using the sin() function and a 3rd degree polynomial function:

color_range = 2**8
a = -3.0
b = -5.0
c = 13.0

def pseudo_randombit(x):
    color255 = math.sin(a*x**3 + b*x**2 + c*x) % color_range
    # make black/white
    bit = color255 / 127
    return bit

I’m no signal processing expert, but his approach intuitively seems similar to distortion. Create a big number and make it fit inside a small room, throwing sin() into the mix for to make the number flip between positive and negative number. Note: I should refine the function given this realization, and simply check the sign…

The full example that uses GDAL (sinpol3_raster.py). Nevermind the geographical projection stuff, I copy-pasted some code, sorry:

import osr
import numpy
import gdal
import math
import sys

color_range = 2**8
a = -3.0
b = -5.0
c = 13.0

def pseudo_randombit(x):
    
    color255 = math.sin(a*x**3 + b*x**2 + c*x) % color_range
    # make black/white
    bit = color255 / 127
    return bit
    
def main(argv):
    
    width = int(argv[1])
    height = int(argv[2])

    format = "GTiff"
    driver = gdal.GetDriverByName( format )

    dst_ds = driver.Create( "sinpol3.tif", width, height, 1, gdal.GDT_Byte )

    dst_ds.SetGeoTransform( [ 444720, 30, 0, 3751320, 0, -30 ] )

    srs = osr.SpatialReference()
    srs.ImportFromEPSG(25832)
    dst_ds.SetProjection( srs.ExportToWkt() )

    raster = numpy.zeros( (height, width), dtype=numpy.uint32 )

    for x in range(0, height*width-1):

        bit = pseudo_randombit(x)
        # make black/white
        color = bit * 255
        i = (x / width)
        j = x % width
        raster[i][j] = color
    
    dst_ds.GetRasterBand(1).WriteArray( raster )

if __name__ == '__main__':
    main(sys.argv)

Calling sinpol3_raster.py:

$ python sinpol3_raster.py 500 500
$ ls *.tif
sinpol3.tif

sinpol3.tif:

Close-up of sinpol3.tif:

An idea that takes this idea further is cryptographic QR-codes… Is the pseudo-random function given above directly reversible? I have to think about this. But now I have other matters to attend to, like a phd in high-availability geospatial services.