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.