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comparison OrthancStone/Resources/Computations/ComputeWarp.py @ 1512:244ad1e4e76a

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reorganization of folders
author Sebastien Jodogne <s.jodogne@gmail.com>
date Tue, 07 Jul 2020 16:21:02 +0200
parents Resources/Computations/ComputeWarp.py@4abddd083374
children 8c5f9864545f
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1511:9dfeee74c1e6 1512:244ad1e4e76a
1 #!/usr/bin/python
2
3 from sympy import *
4 from sympy.solvers import solve
5 import pprint
6 import sys
7
8 init_printing(use_unicode=True)
9
10
11 # Create a test 3D vector using homogeneous coordinates
12 x, y, z, w = symbols('x y z w')
13 p = Matrix([ x, y, z, w ])
14
15
16 # Create a shear matrix, and a scale/shift "T * S" transform as in
17 # Lacroute's thesis (Equation A.16, page 209)
18 ex, ey, ew = symbols('ex ey ew')
19 sx, sy, tx, ty = symbols('sx sy tx ty')
20
21 TS = Matrix([[ sx, 0, 0, tx ],
22 [ 0, sy, 0, ty ],
23 [ 0, 0, 1, 0 ],
24 [ 0, 0, 0, 1 ]])
25
26 pureShear = Matrix([[ 1, 0, ex, 0 ],
27 [ 0, 1, ey, 0 ],
28 [ 0, 0, 1, 0 ],
29 [ 0, 0, ew, 1 ]])
30
31
32 # Create a general warp matrix, that corresponds to "M_warp" in
33 # Equation (A.17) of Lacroute's thesis:
34 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')
35
36 WW = Matrix([[ ww11, ww12, ww13, ww14 ],
37 [ ww21, ww22, ww23, ww24 ],
38 [ ww31, ww32, ww33, ww34 ],
39 [ ww41, ww43, ww43, ww44 ]])
40
41
42 # Create the matrix of intrinsic parameters of the camera
43 k11, k22, k14, k24 = symbols('k11 k22 k14 k24')
44 K = Matrix([[ k11, 0, 0, k14 ],
45 [ 0, k22, 0, k24 ],
46 [ 0, 0, 0, 1 ]])
47
48
49 # The full decomposition is:
50 M_shear = TS * pureShear
51 M_warp = K * WW * TS.inv()
52 AA = M_warp * M_shear
53
54 # Check that the central component "M_warp == K * WW * TS.inv()" that
55 # is the left part of "A" is another general warp matrix (i.e. no
56 # exception is thrown about incompatible matrix sizes):
57 M_warp * p
58
59 if (M_warp.cols != 4 or
60 M_warp.rows != 3):
61 raise Exception('Invalid matrix size')
62
63
64 # We've just shown that "M_warp" is a general 3x4 matrix. Let's call
65 # it W:
66 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')
67
68 W = Matrix([[ w11, w12, w13, w14 ],
69 [ w21, w22, w23, w24 ],
70 [ w41, w43, w43, w44 ]])
71
72 # This shows that it is sufficient to study a decomposition of the
73 # following form:
74 A = W * M_shear
75 print('\nA = W * M_shear =')
76 pprint.pprint(A)
77
78 sys.stdout.write('\nW = ')
79 pprint.pprint(W)
80
81 sys.stdout.write('\nM_shear = ')
82 pprint.pprint(M_shear)
83
84
85
86 # Let's consider one fixed 2D point (i,j) in the intermediate
87 # image. The 3D points (x,y,z,1) that are mapped to (i,j) must satisfy
88 # the equation "(i,j) == M_shear * (x,y,z,w)". As "M_shear" is
89 # invertible, we solve "(x,y,z,w) == inv(M_shear) * (i,j,k,1)".
90
91 i, j, k = symbols('i j k')
92 l = M_shear.inv() * Matrix([ i, j, k, 1 ])
93
94 print('\nLocus for points imaged to some fixed (i,j,k,l) point in the intermediate image:')
95 print('x = %s' % l[0])
96 print('y = %s' % l[1])
97 print('z = %s' % l[2])
98 print('w = %s' % l[3])
99
100
101 # By inspecting the 4 equations above, we see that the locus entirely
102 # depends upon the "k" value that encodes the Z-axis
103
104 print('\nGlobal effect of the shear-warp transform on this locus:')
105 q = expand(A * l)
106 pprint.pprint(q)
107
108 print("\nWe can arbitrarily fix the value of 'k', so let's choose 'k=0':")
109 pprint.pprint(q.subs(k, 0))
110
111 print("\nThis gives the warp transform.")
112 print("QED: line after Equation (A.17) on page 209.\n")