written by Jarno Elonen <elonen@iki.fi>, November 2007, released into the Public Domain
The following Python function finds, by least squares fitting, an affine transformation that (approximately) transforms given set of points/vertices/vectors (from_pts) to another (to_pts). It works with arbitrary dimensional points and requires at least dim points, when your points are dim-dimensional.
First, a usage example:
from_pt = ((1,1),(1,2),(2,2),(2,1)) # a 1x1 rectangle
to_pt = ((4,4),(6,6),(8,4),(6,2)) # scaled x 2, rotated 45 degrees and translated
trn = Affine_Fit(from_pt, to_pt)
print "Transformation is:"
print trn.To_Str()
err = 0.0
for i in range(len(from_pt)):
fp = from_pt[i]
tp = to_pt[i]
t = trn.Transform(fp)
print ("%s => %s ~= %s" % (fp, tuple(t), tp))
err += ((tp[0] - t[0])**2 + (tp[1] - t[1])**2)**0.5
print "Fitting error = %f" % err
This outputs:
Transformation is: x0' = x0 * 2.000000 + x1 * 2.000000 + -0.000000 x1' = x0 * -2.000000 + x1 * 2.000000 + 4.000000 (1, 1) => (3.9999999999999987, 3.9999999999999973) ~= (4, 4) (1, 2) => (5.9999999999999982, 5.9999999999999991) ~= (6, 6) (2, 2) => (8.0, 4.0000000000000018) ~= (8, 4) (2, 1) => (6.0000000000000018, 2.0) ~= (6, 2) Fitting error = 0.000000
The fitting function:
def Affine_Fit( from_pts, to_pts ):
"""Fit an affine transformation to given point sets.
More precisely: solve (least squares fit) matrix 'A'and 't' from
'p ~= A*q+t', given vectors 'p' and 'q'.
Works with arbitrary dimensional vectors (2d, 3d, 4d...).
Written by Jarno Elonen <elonen@iki.fi> in 2007.
Placed in Public Domain.
Based on paper "Fitting affine and orthogonal transformations
between two sets of points, by Helmuth Späth (2003)."""
q = from_pts
p = to_pts
if len(q) != len(p) or len(q)<1:
print "from_pts and to_pts must be of same size."
return false
dim = len(q[0]) # num of dimensions
if len(q) < dim:
print "Too few points => under-determined system."
return false
# Make an empty (dim) x (dim+1) matrix and fill it
c = [[0.0 for a in range(dim)] for i in range(dim+1)]
for j in range(dim):
for k in range(dim+1):
for i in range(len(q)):
qt = list(q[i]) + [1]
c[k][j] += qt[k] * p[i][j]
# Make an empty (dim+1) x (dim+1) matrix and fill it
Q = [[0.0 for a in range(dim)] + [0] for i in range(dim+1)]
for qi in q:
qt = list(qi) + [1]
for i in range(dim+1):
for j in range(dim+1):
Q[i][j] += qt[i] * qt[j]
# Ultra simple linear system solver. Replace this if you need speed.
def gauss_jordan(m, eps = 1.0/(10**10)):
"""Puts given matrix (2D array) into the Reduced Row Echelon Form.
Returns True if successful, False if 'm' is singular.
NOTE: make sure all the matrix items support fractions! Int matrix will NOT work!
Written by Jarno Elonen in April 2005, released into Public Domain"""
(h, w) = (len(m), len(m[0]))
for y in range(0,h):
maxrow = y
for y2 in range(y+1, h): # Find max pivot
if abs(m[y2][y]) > abs(m[maxrow][y]):
maxrow = y2
(m[y], m[maxrow]) = (m[maxrow], m[y])
if abs(m[y][y]) <= eps: # Singular?
return False
for y2 in range(y+1, h): # Eliminate column y
c = m[y2][y] / m[y][y]
for x in range(y, w):
m[y2][x] -= m[y][x] * c
for y in range(h-1, 0-1, -1): # Backsubstitute
c = m[y][y]
for y2 in range(0,y):
for x in range(w-1, y-1, -1):
m[y2][x] -= m[y][x] * m[y2][y] / c
m[y][y] /= c
for x in range(h, w): # Normalize row y
m[y][x] /= c
return True
# Augement Q with c and solve Q * a' = c by Gauss-Jordan
M = [ Q[i] + c[i] for i in range(dim+1)]
if not gauss_jordan(M):
print "Error: singular matrix. Points are probably coplanar."
return false
# Make a result object
class Transformation:
"""Result object that represents the transformation
from affine fitter."""
def To_Str(self):
res = ""
for j in range(dim):
str = "x%d' = " % j
for i in range(dim):
str +="x%d * %f + " % (i, M[i][j+dim+1])
str += "%f" % M[dim][j+dim+1]
res += str + "\n"
return res
def Transform(self, pt):
res = [0.0 for a in range(dim)]
for j in range(dim):
for i in range(dim):
res[j] += pt[i] * M[i][j+dim+1]
res[j] += M[dim][j+dim+1]
return res
return Transformation()