--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Resources/Computations/ComputeWarp.py	Tue Mar 20 14:05:39 2018 +0100
@@ -0,0 +1,112 @@
+#!/usr/bin/python
+
+from sympy import *
+from sympy.solvers import solve
+import pprint
+import sys
+
+init_printing(use_unicode=True)
+
+
+# Create a test 3D vector using homogeneous coordinates
+x, y, z, w = symbols('x y z w')
+p = Matrix([ x, y, z, w ])
+
+
+# Create a shear matrix, and a scale/shift "T * S" transform as in
+# Lacroute's thesis (Equation A.16, page 209)
+ex, ey, ew = symbols('ex ey ew')
+sx, sy, tx, ty = symbols('sx sy tx ty')
+
+TS = Matrix([[ sx, 0,  0, tx ],
+             [ 0,  sy, 0, ty ],
+             [ 0,  0,  1, 0  ],
+             [ 0,  0,  0, 1  ]])
+
+pureShear = Matrix([[ 1, 0, ex, 0 ],
+                    [ 0, 1, ey, 0 ],
+                    [ 0, 0, 1,  0 ],
+                    [ 0, 0, ew, 1 ]])
+
+
+# Create a general warp matrix, that corresponds to "M_warp" in
+# Equation (A.17) of Lacroute's thesis:
+ww11, ww12, ww13, ww14, ww21, ww22, ww23, ww24, ww31, ww32, ww33, ww34, ww41, ww42, ww43, ww44 = symbols('ww11 ww12 ww13 ww14 ww21 ww22 ww23 ww24 ww31 ww32 ww33 ww34 ww41 ww42 ww43 ww44')
+
+WW = Matrix([[ ww11, ww12, ww13, ww14 ],
+             [ ww21, ww22, ww23, ww24 ],
+             [ ww31, ww32, ww33, ww34 ],
+             [ ww41, ww43, ww43, ww44 ]])
+
+
+# Create the matrix of intrinsic parameters of the camera
+k11, k22, k14, k24 = symbols('k11 k22 k14 k24')
+K = Matrix([[ k11, 0,   0, k14 ],
+            [ 0,   k22, 0, k24 ],
+            [ 0,   0,   0, 1   ]])
+
+
+# The full decomposition is:
+M_shear = TS * pureShear
+M_warp = K * WW * TS.inv()
+AA = M_warp * M_shear
+
+# Check that the central component "M_warp == K * WW * TS.inv()" that
+# is the left part of "A" is another general warp matrix (i.e. no
+# exception is thrown about incompatible matrix sizes):
+M_warp * p
+
+if (M_warp.cols != 4 or
+    M_warp.rows != 3):
+    raise Exception('Invalid matrix size')
+
+
+# We've just shown that "M_warp" is a general 3x4 matrix. Let's call
+# it W:
+w11, w12, w13, w14, w21, w22, w23, w24, w41, w42, w43, w44 = symbols('w11 w12 w13 w14 w21 w22 w23 w24 w41 w42 w43 w44')
+
+W = Matrix([[ w11, w12, w13, w14 ],
+            [ w21, w22, w23, w24 ],
+            [ w41, w43, w43, w44 ]])
+
+# This shows that it is sufficient to study a decomposition of the
+# following form:
+A = W * M_shear
+print('\nA = W * M_shear =')
+pprint.pprint(A)
+
+sys.stdout.write('\nW = ')
+pprint.pprint(W)
+
+sys.stdout.write('\nM_shear = ')
+pprint.pprint(M_shear)
+
+
+
+# Let's consider one fixed 2D point (i,j) in the intermediate
+# image. The 3D points (x,y,z,1) that are mapped to (i,j) must satisfy
+# the equation "(i,j) == M_shear * (x,y,z,w)". As "M_shear" is
+# invertible, we solve "(x,y,z,w) == inv(M_shear) * (i,j,k,1)".
+
+i, j, k = symbols('i j k')
+l = M_shear.inv() * Matrix([ i, j, k, 1 ])
+
+print('\nLocus for points imaged to some fixed (i,j,k,l) point in the intermediate image:')
+print('x = %s' % l[0])
+print('y = %s' % l[1])
+print('z = %s' % l[2])
+print('w = %s' % l[3])
+
+
+# By inspecting the 4 equations above, we see that the locus entirely
+# depends upon the "k" value that encodes the Z-axis
+
+print('\nGlobal effect of the shear-warp transform on this locus:')
+q = expand(A * l)
+pprint.pprint(q)
+
+print("\nWe can arbitrarily fix the value of 'k', so let's choose 'k=0':")
+pprint.pprint(q.subs(k, 0))
+
+print("\nThis gives the warp transform.")
+print("QED: line after Equation (A.17) on page 209.\n")