1. CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware

2. Image RegistrationLecture 26 2 Deformable Registration

3. Image RegistrationLecture 26 3 Deformable Transforms BSplines

4. Image RegistrationLecture 26 4 Deformable Transformation y Fixed Image Transform x y Moving Image x

5. Image RegistrationLecture 26 5 Deformable Transformation y Fixed Image Transform x y Moving Image x

6. Image RegistrationLecture 26 6 Deformable Transformation y x

7. Image RegistrationLecture 26 7 Deformable Transformation y x

8. Image RegistrationLecture 26 8 Image Resampling Fixed Image Moving Image Transform Interpolator Resample Image Filter Deformed Image BSpline Transform

9. Image RegistrationLecture 26 9 Deformable Transforms BSplines in Geometrical Terms

10. Image RegistrationLecture 26 10 BSplines -2120 Order Zero

11. Image RegistrationLecture 26 11 BSplines -2120 Order Zero Piece-Wise Y = 1

12. Image RegistrationLecture 26 12 BSplines Convolution -2120

13. Image RegistrationLecture 26 13 BSplines Area under the product curve -2120

14. Image RegistrationLecture 26 14 BSplines -2120

15. Image RegistrationLecture 26 15 BSplines Order One -2120

16. Image RegistrationLecture 26 16 BSplines Order One -2120 Piece-Wise Y = ( 1 - X )Y = ( X + 1 )

17. Image RegistrationLecture 26 17 BSplines Convolution -2120

18. Image RegistrationLecture 26 18 BSplines Convolution -2120

19. Image RegistrationLecture 26 19 BSplines Order Two -2120

20. Image RegistrationLecture 26 20 BSplines Order Two -2120 Piece-Wise Y = ( X – 3/2 ) 2 / 2 Y = ( 1 – 2 X 2 ) Y = ( X + 3/2 ) 2 / 2

21. Image RegistrationLecture 26 21 BSplines Convolution -2120

22. Image RegistrationLecture 26 22 BSplines Convolution -2120

23. Image RegistrationLecture 26 23 BSplines Order Three -2120

24. Image RegistrationLecture 26 24 BSplines Order Three -2120 Piece-Wise Y = ( 3X 3 - 6X 2 + 4 )/6 Y = (2-X) 3 / 6 Y = (2+X) 3 / 6 Y = ( - 3X 3 - 6X 2 + 4 )/6

25. Image RegistrationLecture 26 25 BSplines Interpolating with BSplines

26. Image RegistrationLecture 26 26 BSplines Interpolation -2120 Zero OrderNearest Neighbor

27. Image RegistrationLecture 26 27 BSplines Interpolation -2120 Zero OrderNearest Neighbor

28. Image RegistrationLecture 26 28 BSplines Interpolation -2120 Zero OrderNearest Neighbor

29. Image RegistrationLecture 26 29 BSplines Interpolation -2120 Zero OrderNearest Neighbor

30. Image RegistrationLecture 26 30 BSplines Interpolation -2120 First OrderLinear Interpolation

31. Image RegistrationLecture 26 31 BSplines Interpolation -2120 First OrderLinear Interpolation

32. Image RegistrationLecture 26 32 BSplines Interpolation -2120 First OrderLinear Interpolation

33. Image RegistrationLecture 26 33 BSplines Interpolation -2120 First OrderLinear Interpolator

34. Image RegistrationLecture 26 34 BSplines Interpolation -2120 Second OrderQuadratic Interpolation

35. Image RegistrationLecture 26 35 BSplines Interpolation -2120 Second OrderQuadratic Interpolation

36. Image RegistrationLecture 26 36 BSplines Interpolation -2120 Second OrderQuadratic Interpolation

37. Image RegistrationLecture 26 37 BSplines Interpolation -2120 Second OrderQuadratic Interpolator

38. Image RegistrationLecture 26 38 BSplines Interpolation -2120 Third OrderCubic Interpolation

39. Image RegistrationLecture 26 39 BSplines Interpolation -2120 Third OrderCubic Interpolation

40. Image RegistrationLecture 26 40 BSplines Interpolation -2120 Third OrderCubic Interpolation

41. Image RegistrationLecture 26 41 BSplines Interpolation -2120 Third OrderCubic Interpolator

42. Image RegistrationLecture 26 42 BSplines Cubic BSplines in Two Dimensions

43. Image RegistrationLecture 26 43 BSplines in 2D Tensor Product

44. Image RegistrationLecture 26 44 BSplines in 2D Tensor Product

45. Image RegistrationLecture 26 45 BSplines in 2D Tensor Product Separable

46. Image RegistrationLecture 26 46 BSplines in 2D Support Nodes in Region of Influence #Nodes = 4 N

47. Image RegistrationLecture 26 47 BSplines in 2D Interpolation BSpline Grid Resampling Grid

48. Image RegistrationLecture 26 48 BSplines in 2D Interpolation BSpline Grid Resampling Grid

49. Image RegistrationLecture 26 49 Sampling Grid y x

50. Image RegistrationLecture 26 50 BSpline Grid y x BSpline Grid Resampling Grid

51. Image RegistrationLecture 26 51 BSplines Grid & Image Grid

52. Image RegistrationLecture 26 52 BSplines Grid & Image Grid BSpline Grid

53. Image RegistrationLecture 26 53 BSplines Grid & Image Grid BSpline Grid Valid Region

54. Image RegistrationLecture 26 54 BSplines Grid & Image Grid BSpline Grid

55. Image RegistrationLecture 26 55 BSplines Grid & Image Grid BSpline Grid Valid Region

56. Image RegistrationLecture 26 56 BSplines Interpolation Interpolating Vectors

57. Image RegistrationLecture 26 57 Interpolating Vectors y x BSpline Grid Resampling Grid

58. Image RegistrationLecture 26 58 Interpolating Vectors y x BSpline Grid Resampling Grid X Compo- nents

59. Image RegistrationLecture 26 59 Interpolating Vectors y x BSpline Grid Resampling Grid X Compo- nents

60. Image RegistrationLecture 26 60 Interpolating Vectors y x BSpline Grid Resampling Grid Y Compo- nents

61. Image RegistrationLecture 26 61 Interpolating Vectors y x BSpline Grid Resampling Grid X Compo- nents

62. Image RegistrationLecture 26 62 X Grid Feeding the BSpline Transform x BSpline Transform Dimension N N Grid Y Grid ….. Z Grid Input Point Output Point

63. Image RegistrationLecture 26 63 Array of Parameters Feeding the BSpline Transform x BSpline Transform Input Point Output Point

64. Image RegistrationLecture 26 64 Array of Parameters Feeding the BSpline Transform x BSpline Transform Input Point Output Point x 1 x 2 x 3 x 4 x 5... y 1 y 2 y 3 y 4 y 5 ….…. z 1 z 2 z 3 z 4….…

65. Image RegistrationLecture 26 65 Feeding the BSpline Transform x BSpline Transform Input Point Output Point x 1 x 2 x 3 x 4 x 5... y 1 y 2 y 3 y 4 y 5 ….…. z 1 z 2 z 3 z 4….… x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 x 4 y 4 z 4 …. x N y N z N Input File Parameters Array

66. Image RegistrationLecture 26 66 BSplines Interpolation Image Resampling BSpline Warping

67. Image RegistrationLecture 26 67 Image Resampling Fixed Image Moving Image Transform Interpolator Resample Image Filter Deformed Image

68. Image RegistrationLecture 26 68 BSplines Interpolation Original Lena

69. Image RegistrationLecture 26 69 BSplines Interpolation Deformed with BSpline Transform

70. Image RegistrationLecture 26 70 BSplines Interpolation Original LenaDeformed Lena

71. Image RegistrationLecture 26 71 Deformable Registration Image Registration BSpline Transform

72. Image RegistrationLecture 26 72 Image Registration Framework Fixed Image Moving Image Metric Transform Interpolator Optimizer Parameters Array

73. Image RegistrationLecture 26 73 BSplines Deformable Registration Deformed with BSpline Transform

74. Image RegistrationLecture 26 74 BSplines Deformable Registration Registered with BSpline Transform

75. Image RegistrationLecture 26 75 BSplines Deformable Registration Original Lena

76. Image RegistrationLecture 26 76 BSplines Deformable Registration Difference After Registration Difference Before Registration

77. Image RegistrationLecture 26 77 End Enjoy ITK !