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annotate Resources/Computations/ComputeWarp.py @ 525:2549363fadc3 am-touch-events

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author Alain Mazy <am@osimis.io>
date Tue, 12 Mar 2019 17:29:12 +0000
parents 4abddd083374
children
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193
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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1 #!/usr/bin/python
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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2
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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3 from sympy import *
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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4 from sympy.solvers import solve
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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5 import pprint
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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6 import sys
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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7
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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8 init_printing(use_unicode=True)
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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9
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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10
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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11 # Create a test 3D vector using homogeneous coordinates
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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12 x, y, z, w = symbols('x y z w')
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13 p = Matrix([ x, y, z, w ])
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14
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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15
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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16 # Create a shear matrix, and a scale/shift "T * S" transform as in
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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17 # Lacroute's thesis (Equation A.16, page 209)
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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18 ex, ey, ew = symbols('ex ey ew')
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19 sx, sy, tx, ty = symbols('sx sy tx ty')
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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20
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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21 TS = Matrix([[ sx, 0, 0, tx ],
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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22 [ 0, sy, 0, ty ],
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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23 [ 0, 0, 1, 0 ],
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24 [ 0, 0, 0, 1 ]])
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25
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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26 pureShear = Matrix([[ 1, 0, ex, 0 ],
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27 [ 0, 1, ey, 0 ],
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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28 [ 0, 0, 1, 0 ],
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29 [ 0, 0, ew, 1 ]])
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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30
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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31
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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32 # Create a general warp matrix, that corresponds to "M_warp" in
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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33 # Equation (A.17) of Lacroute's thesis:
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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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')
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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35
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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36 WW = Matrix([[ ww11, ww12, ww13, ww14 ],
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37 [ ww21, ww22, ww23, ww24 ],
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38 [ ww31, ww32, ww33, ww34 ],
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39 [ ww41, ww43, ww43, ww44 ]])
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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40
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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41
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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42 # Create the matrix of intrinsic parameters of the camera
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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43 k11, k22, k14, k24 = symbols('k11 k22 k14 k24')
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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44 K = Matrix([[ k11, 0, 0, k14 ],
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45 [ 0, k22, 0, k24 ],
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46 [ 0, 0, 0, 1 ]])
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47
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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48
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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49 # The full decomposition is:
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50 M_shear = TS * pureShear
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51 M_warp = K * WW * TS.inv()
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52 AA = M_warp * M_shear
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53
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54 # Check that the central component "M_warp == K * WW * TS.inv()" that
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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55 # is the left part of "A" is another general warp matrix (i.e. no
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56 # exception is thrown about incompatible matrix sizes):
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57 M_warp * p
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58
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59 if (M_warp.cols != 4 or
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60 M_warp.rows != 3):
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61 raise Exception('Invalid matrix size')
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62
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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63
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64 # We've just shown that "M_warp" is a general 3x4 matrix. Let's call
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65 # it W:
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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')
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67
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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68 W = Matrix([[ w11, w12, w13, w14 ],
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69 [ w21, w22, w23, w24 ],
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70 [ w41, w43, w43, w44 ]])
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71
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72 # This shows that it is sufficient to study a decomposition of the
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73 # following form:
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74 A = W * M_shear
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75 print('\nA = W * M_shear =')
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76 pprint.pprint(A)
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77
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78 sys.stdout.write('\nW = ')
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79 pprint.pprint(W)
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80
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81 sys.stdout.write('\nM_shear = ')
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82 pprint.pprint(M_shear)
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83
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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84
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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85
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86 # Let's consider one fixed 2D point (i,j) in the intermediate
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87 # image. The 3D points (x,y,z,1) that are mapped to (i,j) must satisfy
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88 # the equation "(i,j) == M_shear * (x,y,z,w)". As "M_shear" is
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89 # invertible, we solve "(x,y,z,w) == inv(M_shear) * (i,j,k,1)".
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90
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91 i, j, k = symbols('i j k')
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92 l = M_shear.inv() * Matrix([ i, j, k, 1 ])
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93
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94 print('\nLocus for points imaged to some fixed (i,j,k,l) point in the intermediate image:')
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95 print('x = %s' % l[0])
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96 print('y = %s' % l[1])
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97 print('z = %s' % l[2])
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98 print('w = %s' % l[3])
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99
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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100
4abddd083374 ShearWarpProjectiveTransform::ApplyAxial()
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101 # By inspecting the 4 equations above, we see that the locus entirely
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102 # depends upon the "k" value that encodes the Z-axis
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103
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104 print('\nGlobal effect of the shear-warp transform on this locus:')
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105 q = expand(A * l)
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106 pprint.pprint(q)
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107
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108 print("\nWe can arbitrarily fix the value of 'k', so let's choose 'k=0':")
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109 pprint.pprint(q.subs(k, 0))
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110
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111 print("\nThis gives the warp transform.")
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112 print("QED: line after Equation (A.17) on page 209.\n")