Mercurial > hg > orthanc-stone
annotate Resources/Computations/ComputeWarp.py @ 886:5cfb479fdbde am-dev
fix BasicScene WASM sample
author | Alain Mazy <alain@mazy.be> |
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date | Tue, 09 Jul 2019 17:36:47 +0200 |
parents | 4abddd083374 |
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rev | line source |
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ShearWarpProjectiveTransform::ApplyAxial()
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1 #!/usr/bin/python |
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2 |
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3 from sympy import * |
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4 from sympy.solvers import solve |
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5 import pprint |
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6 import sys |
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7 |
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8 init_printing(use_unicode=True) |
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9 |
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10 |
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11 # Create a test 3D vector using homogeneous coordinates |
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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 |
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15 |
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16 # Create a shear matrix, and a scale/shift "T * S" transform as in |
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17 # Lacroute's thesis (Equation A.16, page 209) |
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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') |
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20 |
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21 TS = Matrix([[ sx, 0, 0, tx ], |
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22 [ 0, sy, 0, ty ], |
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23 [ 0, 0, 1, 0 ], |
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24 [ 0, 0, 0, 1 ]]) |
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25 |
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26 pureShear = Matrix([[ 1, 0, ex, 0 ], |
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27 [ 0, 1, ey, 0 ], |
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28 [ 0, 0, 1, 0 ], |
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29 [ 0, 0, ew, 1 ]]) |
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30 |
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31 |
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32 # Create a general warp matrix, that corresponds to "M_warp" in |
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33 # Equation (A.17) of Lacroute's thesis: |
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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') |
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35 |
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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 ]]) |
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40 |
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41 |
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42 # Create the matrix of intrinsic parameters of the camera |
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43 k11, k22, k14, k24 = symbols('k11 k22 k14 k24') |
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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 |
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48 |
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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 |
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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 |
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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 |
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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 |
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84 |
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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 |
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100 |
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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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ShearWarpProjectiveTransform::ApplyAxial()
Sebastien Jodogne <s.jodogne@gmail.com>
parents:
diff
changeset
|
110 |
4abddd083374
ShearWarpProjectiveTransform::ApplyAxial()
Sebastien Jodogne <s.jodogne@gmail.com>
parents:
diff
changeset
|
111 print("\nThis gives the warp transform.") |
4abddd083374
ShearWarpProjectiveTransform::ApplyAxial()
Sebastien Jodogne <s.jodogne@gmail.com>
parents:
diff
changeset
|
112 print("QED: line after Equation (A.17) on page 209.\n") |