Alternative Moebius Transformation

moebius2.py

""" An alternative way to do the moebius transformation,
 using scipy.ndimage.geometric_transform,
 that interpolates the points for a smoother transformation """

from pylab import *
from numpy import *

zp=[157+148j, 78+149j, 54+143j]; # (zs) the complex point zp[i]
wa=[147+143j, 78+140j, 54+143j]; # (ws) will be in wa[i]

# transformation parameters
a = linalg.det([[zp[0]*wa[0], wa[0], 1], 
                [zp[1]*wa[1], wa[1], 1], 
                [zp[2]*wa[2], wa[2], 1]]);

b = linalg.det([[zp[0]*wa[0], wa[0], wa[0]], 
                [zp[1]*wa[1], wa[1], wa[1]], 
                [zp[2]*wa[2], wa[2], wa[2]]]);         

c = linalg.det([[zp[0], wa[0], 1], 
                [zp[1], wa[1], 1], 
                [zp[2], wa[2], 1]]);

d = linalg.det([[zp[0]*wa[0], zp[0], 1], 
                [zp[1]*wa[1], zp[1], 1], 
                [zp[2]*wa[2], zp[2], 1]]);

img = fliplr(imread('mond.jpg')) # load an image

from scipy.ndimage import geometric_transform

def shift_func(coords):
    """ Define the moebius transformation, though backwards """
    #turn the first two coordinates into an imaginary number
    z = coords[0] + 1j*coords[1]
    w = (d*z-b)/(-c*z+a) #the inverse mobius transform
    return real(w),imag(w),coords[2] #take the color along for the ride
    
newimg = geometric_transform(img,shift_func,cval=255) #make the outside white

figure(2)
subplot(121)
title('Original Mondrian')
imshow(img)
subplot(122)
title('Mondrian after the transformation')
imshow(newimg)
show()

Inspired by the Glowing Python post here: http://glowingpython.blogspot.com/2011/08/applying-moebius-transformation-to.html

This version uses scipy.ndimage.geometric_transform to interpolate the in-between points.

Result can be seen here: http://i.imgur.com/qz7P0.png