• transformation
• initialization files
• default initialization
• Programs that incorporate this model:
  • align_warp

Transformation

Given a coordinate (x,y) in the standard file, the coordinates of the corresponding voxel in the reslice file (x',y') are given by the equations:

First order polynomial (3 coefficients per coordinate)

• x'=k_x1+k_x2x+k_x3y
• y'=k_y1+k_y2x+k_y3y

Second order polynomial (6 coefficients per coordinate)

• x'=k_x1+k_x2x+k_x3y +k_x4x²+k_x5xy+k_x6y²
• y'=k_y1+k_y2x+k_y3y +k_y4x²+k_y5xy+k_y6y²

Third order polynomial (10 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 10 terms from the ordered list of polynomial terms below

Fourth order polynomial (15 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 15 terms from the ordered list of polynomial terms below

Fifth order polynomial (21 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 21 terms from the ordered list of polynomial terms below

Sixth order polynomial (28 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 28 terms from the ordered list of polynomial terms below

Seventh order polynomial (36 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 36 terms from the ordered list of polynomial terms below

Eighth order polynomial (45 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 45 terms from the ordered list of polynomial terms below

Ninth order polynomial (55 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 55 terms from the ordered list of polynomial terms below

Tenth order polynomial (66 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 66 terms from the ordered list of polynomial terms below

Eleventh order polynomial (78 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 78 terms from the ordered list of polynomial terms below

Twelfth order polynomial (91 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 91 terms from the ordered list of polynomial terms below

Ordered List of Polynomial Terms

1. 1
2. x
3. y (end of 1st order)
4. x²
5. xy
6. y² (end of 2nd order)
7. x³
8. x²y
9. xy²
10. y³ (end of 3rd order)
11. x⁴
12. x³y
13. x²y²
14. xy³
15. y⁴ (end of 4th order)
16. x⁵
17. x⁴y
18. x³y²
19. x²y³
20. xy⁴
21. y⁵ (end of 5th order)
22. x⁶
23. x⁵y
24. x⁴y²
25. x³y³
26. x²y⁴
27. xy⁵
28. y⁶ (end of 6th order)
29. x⁷
30. x⁶y
31. x⁵y²
32. x⁴y³
33. x³y⁴
34. x²y⁵
35. xy⁶
36. y⁷ (end of 7th order)
37. x⁸
38. x⁷y
39. x⁶y²
40. x⁵y³
41. x⁴y⁴
42. x³y⁵
43. x²y⁶
44. xy⁷
45. y⁸ (end of 8th order)
46. x⁹
47. x⁸y
48. x⁷y²
49. x⁶y³
50. x⁵y⁴
51. x⁴y⁵
52. x³y⁶
53. x²y⁷
54. xy⁸
55. y⁹ (end of 9th order)
56. x¹⁰
57. x⁹y
58. x⁸y²
59. x⁷y³
60. x⁶y⁴
61. x⁵y⁵
62. x⁴y⁶
63. x³y⁷
64. x²y⁸
65. xy⁹
66. y¹⁰ (end of 10th order)
67. x¹¹
68. x¹⁰y
69. x⁹y²
70. x⁸y³
71. x⁷y⁴
72. x⁶y⁵
73. x⁵y⁶
74. x⁴y⁷
75. x³y⁸
76. x²y⁹
77. xy¹⁰
78. y¹¹ (end of 11th order)
79. x¹²
80. x¹¹y
81. x¹⁰y²
82. x⁹y³
83. x⁸y⁴
84. x⁷y⁵
85. x⁶y⁶
86. x⁵y⁷
87. x⁴y⁸
88. x³y⁹
89. x²y¹⁰
90. xy¹¹
91. y¹² (end of 12th order)

Representation in initialization files

AIR no longer uses ASCII files to initialize polynomial warps. Instead, a .warp file can be used to initialize polynomials having an order one greater than the order of the .warp file

Default initialization

The default initialization for this model was modified in AIR 5.1 to make it identical to the default for alignlinear even when voxel sizes in the two files are different

If no initialization .warp file is specified, the default initialization is:

• k_x1 =((rx_dim-1) - (sx_dim-1)*(sx_size / rx_size)) / 2
• k_x2 =sx_size / rx_size
• k_y1 =((ry_dim-1) - (sy_dim-1)*(sy_size / ry_size)) / 2
• k_y2 =sy_size / ry_size
• All other parameters=0

where:

• sx_size is the voxel x size of the standard file
• sy_size is the voxel y size of the standard file
• rx_size is the voxel x size of the reslice file
• ry_size is the voxel y size of the reslice file
• sx_dim is the x dimension of the standard file
• sy_dim is the y dimension of the standard file
• rx_dim is the x dimension of the reslice file
• ry_dim is the y dimension of the reslice file

This results in the exact centers of the two files being aligned to one another.