Affine transformation of coordinate system using least squares to reduce error

affine_transform_coordinates_w_least_squares.py

import numpy as np

primary = np.array([[40., 1160., 0.],
                    [40., 40., 0.],
                    [260., 40., 0.],
                    [260., 1160., 0.]])

secondary = np.array([[610., 560., 0.],
                      [610., -560., 0.],
                      [390., -560., 0.],
                      [390., 560., 0.]])

def least_squares_transform():

    # Pad the data with ones, so that our transformation can do translations too
    n = primary.shape[0]
    pad = lambda x: np.hstack([x, np.ones((x.shape[0], 1))])
    unpad = lambda x: x[:,:-1]
    X = pad(primary)
    Y = pad(secondary)

    # Solve the least squares problem X * A = Y
    # to find our transformation matrix A
    A, res, rank, s = np.linalg.lstsq(X, Y)

    transform = lambda x: unpad(np.dot(pad(x), A))

    print "Target:"
    print secondary
    print "Result:"
    print transform(primary)
    print "Max error:", np.abs(secondary - transform(primary)).max()

    A[np.abs(A) < 1e-10] = 0 # set really small values to zero
    print A

Target:

[[ 610.  560.    0.]
 [ 610. -560.    0.]
 [ 390. -560.    0.]
 [ 390.  560.    0.]]

Result:

[[ 610.  560.    0.]
 [ 610. -560.    0.]
 [ 390. -560.    0.]
 [ 390.  560.    0.]]

Max error:

1.13686837722e-12

A:

[[  -1.    0.    0.    0.]
 [   0.    1.    0.    0.]
 [   0.    0.    0.    0.]
 [ 650. -600.    0.    1.]]

Reference:

https://stackoverflow.com/questions/20546182/how-to-perform-coordinates-affine-transformation-using-python-part-2