How does perspective transformation work in PIL?

To apply a perspective transformation you first have to know four points in a plane A that will be mapped to four points in a plane B. With those points, you can derive the homographic transform. By doing this, you obtain your 8 coefficients and the transformation can take place.

The site http://xenia.media.mit.edu/~cwren/interpolator/ (mirror: WebArchive), as well as many other texts, describes how those coefficients can be determined. To make things easy, here is a direct implementation according from the mentioned link:

import numpy

def find_coeffs(pa, pb):
    matrix = []
    for p1, p2 in zip(pa, pb):
        matrix.append([p1[0], p1[1], 1, 0, 0, 0, -p2[0]*p1[0], -p2[0]*p1[1]])
        matrix.append([0, 0, 0, p1[0], p1[1], 1, -p2[1]*p1[0], -p2[1]*p1[1]])

    A = numpy.matrix(matrix, dtype=numpy.float)
    B = numpy.array(pb).reshape(8)

    res = numpy.dot(numpy.linalg.inv(A.T * A) * A.T, B)
    return numpy.array(res).reshape(8)

where pb is the four vertices in the current plane, and pa contains four vertices in the resulting plane.

So, suppose we transform an image as in:

import sys
from PIL import Image

img = Image.open(sys.argv[1])
width, height = img.size
m = -0.5
xshift = abs(m) * width
new_width = width + int(round(xshift))
img = img.transform((new_width, height), Image.AFFINE,
        (1, m, -xshift if m > 0 else 0, 0, 1, 0), Image.BICUBIC)
img.save(sys.argv[2])

Here is a sample input and output with the code above:

We can continue on the last code and perform a perspective transformation to revert the shear:

coeffs = find_coeffs(
        [(0, 0), (256, 0), (256, 256), (0, 256)],
        [(0, 0), (256, 0), (new_width, height), (xshift, height)])

img.transform((width, height), Image.PERSPECTIVE, coeffs,
        Image.BICUBIC).save(sys.argv[3])

Resulting in:

You can also have some fun with the destination points: