1. ICONIC Training School, Freiburg, May 25th to 29th, 2010 Image processing for ion/electron imaging Dr. Lionel POISSON Laboratoire Francis PERRIN Saclay, France

2. Introduction Experimental device Images display Image processing From 2D to 1D From 3D to 2D Image projection : Abel transform Image inversion : Inverse Abel transform The 3 algorithms : - Hankel - BASEX - pBASEX From image to the information True life… Limitation for the use of the inversion ICONIC Training School, Freiburg, 2010 Guideline

3. Historical aspects J. Chem. Phys. 87, 1445 (1987) 419 citations (April 2010) CH 3 I + 266 nm (CH 3 + )

4. Historical aspects Rev. Sci. Instrum. 68, 3477 (1997) 689 citations (April 2010) O + fine mashO + VMI optics

5. ICONIC Training School, Freiburg, 2010 Experimental Setup O 2 molecule (IP=12.07 eV) + 273 nm (3 photons) Max electron energy =1.52 eV4000 V2870 V 0 V What to do with this images ? z x

6. ICONIC Training School, Freiburg, 2010 Image representation 3D presentation 2D presentation What do we see? It is also the projection of a 4D signal!

7. ICONIC Training School, Freiburg, 2010 Image Colormap

8. ICONIC Training School, Freiburg, 2010 Angular Mapping angle  0°360° radius Angular mappingImage  radius Where is the image center? n n+1 m+1 m Z= (y-n) ( (x-m) Img[n+1,m+1] +(1+m-x) Img[n+1,m] )+ (1+n-y)( (x-m) Img[n,m+1] +(1+m-x) Img[n,m] )

9. ICONIC Training School, Freiburg, 2010 Radial Average radius No scientific meaning…. yet Average image intensity for the corresponding radius

10. Introduction Experimental device Images display Image processing From 2D to 1D From 3D to 2D Image projection : Abel transform Image inversion : Inverse Abel transform The 3 algorithms : - Hankel - BASEX - pBASEX From image to the information True life… Limitation for the use of the inversion ICONIC Training School, Freiburg, 2010 Guideline

11. ICONIC Training School, Freiburg, 2010 Sphere Projection

12. ICONIC Training School, Freiburg, 2010 Sphere Projection

13. ICONIC Training School, Freiburg, 2010 Slicing

14. ICONIC Training School, Freiburg, 2010 Slicing

15. ICONIC Training School, Freiburg, 2010 Mathematics… Projection Projection

16. ICONIC Training School, Freiburg, 2010 Abel Transform August 5, 1802 - April 6, 1829 Niels Henrik Abel

17. ICONIC Training School, Freiburg, 2010 Mathematics… Projection Projection Hypothesis: Angular Symmetry

18. ICONIC Training School, Freiburg, 2010 Projection… exemple Projection Full disk

19. ICONIC Training School, Freiburg, 2010 Mathematics… Projection Projection Forward Abel Transform Note that this function is linear

20. ICONIC Training School, Freiburg, 2010 Projection Exemple Projection « Sharp Ring »

21. ICONIC Training School, Freiburg, 2010 Projection Exemple Projection « Smooth Ring »

22. ICONIC Training School, Freiburg, 2010 Projection Exemple Projection Sum of each contributions

23. ICONIC Training School, Freiburg, 2010 Sphere Projection

24. ICONIC Training School, Freiburg, 2010 Sphere Projection Each line is the projection of a ring

25. ICONIC Training School, Freiburg, 2010 Sphere Projection - anisotropy Symmetry axis

26. ICONIC Training School, Freiburg, 2010 Sphere Projection Each line is the projection of a ring Laser polarization

27. ICONIC Training School, Freiburg, 2010 Inverstion Inverse Abel Transform Slice of an Experimental Result Scientifically important information

28. ICONIC Training School, Freiburg, 2010 Inverse Abel Transform

29. ICONIC Training School, Freiburg, 2010 Inverse Abel Transform t r x x

30. ICONIC Training School, Freiburg, 2010 Inverse Abel Transform

31. ICONIC Training School, Freiburg, 2010 Inverse Abel Transform http://integrals.wolfram.com Note that this function is linear Note that the function f should be derivable

32. ICONIC Training School, Freiburg, 2010 How to proceed…? Inverse Abel Transform The whole signal distribution should be on the image

33. Introduction Experimental device Images display Image processing From 2D to 1D From 3D to 2D Image projection : Abel transform Image inversion : Inverse Abel transform The 3 algorithms : - Hankel - BASEX - pBASEX From image to the information True life… Limitation for the use of the inversion ICONIC Training School, Freiburg, 2010 Guideline

34. ICONIC Training School, Freiburg, 2010 How to proceed…? Direct numerical calculation ? Singular point for t=x  Result strongly dependant on the numerical sampling used for the calculation of the integral… f should be smooth enough to be derivate…

35. ICONIC Training School, Freiburg, 2010 How to proceed…? Play with analytic functions… H is the zero order Hankel Transform J 0 is the zero order Bessel Function

36. ICONIC Training School, Freiburg, 2010 Hankel Transform FFT algorithm  Fast calculation  Signal can be smoothed  Information on the location of the center since the FT should be real because f(y)=f(-y)  No more singularity  No derivation of the image  For small x, high t values have more importance => high frequencies => noise…

37. ICONIC Training School, Freiburg, 2010 Hankel Transform Z zoom x 50

38. ICONIC Training School, Freiburg, 2010 Hankel Transform Laser Polarization

39. ICONIC Training School, Freiburg, 2010 BASEX Play with algebra: use a fitting procedure A is a basis-set matrix B is the object to fit X is the solution Can be calculated and saved prior to inversion Build a basis-set Make the Forward Abel Fit Apply the coefficients to the initial basis-set

40. ICONIC Training School, Freiburg, 2010 BASEX BASEX Inversion method: Fit line by line Basis set = Gaussian functions Matrix A is inverted by the Tikhonov regularization method Preparation for the inversion 1- Calculate the basis-set functions (each column of the matrix is the function sampled) 2- Calculate the forward Abel transform of the basis-set functions 3- Invert the matrix 4- Save the result (total currently takes less than 5 minutes) Inversion (0-Load inverted matrix) 1- Multiply each line of the image by the inverted matrix 2- Apply the solution to the basis-set functions  Fast method  Depends on the image size and image center

41. ICONIC Training School, Freiburg, 2010 BASEX BASEX fitting

42. ICONIC Training School, Freiburg, 2010 pBASEX A is a basis-set matrix B is the object to fit X is the solution Singular value decomposition (SVD) S Diagonal matrix Can be calculated and saved prior to inversion pBASEX Inversion method: Fit of the whole image

43. ICONIC Training School, Freiburg, 2010 pBASEX pBASEX Inversion method: Fit of the whole image radiusThe image is read as a list of points : N radial points and M angular points for each radius (NxM points) The basis set is build and read as well How to choose the Basis-set?

44. ICONIC Training School, Freiburg, 2010 How to proceed…?

45. ICONIC Training School, Freiburg, 2010 pBASEX One photon transition, vertically polarized of a randomly oriented distribution: Light polarization (symmetry axis)

46. ICONIC Training School, Freiburg, 2010 pBASEX P0P0 P2P2 P1P1 P6P6 P4P4

47. ICONIC Training School, Freiburg, 2010 pBASEX P0P0 P2P2 P4P4 P6P6 Forward Abel transform

48. ICONIC Training School, Freiburg, 2010 pBASEX pBASEX fitting Image reconstruction

49. ICONIC Training School, Freiburg, 2010 pBASEX  Fast method  Can take into account the apparatus function  Independent on the image center  Gives directly the contributions of each P i  Basis calculation can take time (>4 hours…)

50. Introduction Experimental device Images display Image processing From 2D to 1D From 3D to 2D Image projection : abel transform Image inversion : Inverse abel transform The 3 algorythms : - Hankel - BASEX - pBASEX From image to the information True life… Limitation for the use of the inversion ICONIC Training School, Freiburg, 2010 Guideline

51. ICONIC Training School, Freiburg, 2010 From Image to data Jacobian  

52. ICONIC Training School, Freiburg, 2010 From Image to data Small cone = small contribution to total signal large cone = large contribution to total signal

53. ICONIC Training School, Freiburg, 2010 From Image to data Good news! No more central noise Number of pixels between r and r+dr :

54. ICONIC Training School, Freiburg, 2010 From Image to data Radius is usually velocity, what about the energy?...

55. ICONIC Training School, Freiburg, 2010 From Image to data Same aera

56. ICONIC Training School, Freiburg, 2010 From Image to data Angular informations? pBASEX …or fit of the images

57. Introduction Experimental device Images display Image processing From 2D to 1D From 3D to 2D Image projection : abel transform Image inversion : Inverse abel transform The 3 algorythms : - Hankel - BASEX - pBASEX From image to the information True life… Limitation for the use of the inversion ICONIC Training School, Freiburg, 2010 Guideline

58. ICONIC Training School, Freiburg, 2010 Looking for the center 4 pixels off vertical 4 pixels off horizontal

59. ICONIC Training School, Freiburg, 2010 Image Distortion 6 % stretch of the image

60. ICONIC Training School, Freiburg, 2010 Image Distortion Strong distortion…. O 2 molecule (IP=12.07 eV) + 273 nm (3 photons) Max electron energy =1.52 eV

61. ICONIC Training School, Freiburg, 2010 Image Distortion

62. ICONIC Training School, Freiburg, 2010 Image Distortion

63. ICONIC Training School, Freiburg, 2010 Slow electrons Photoelectron microscopy

64. ICONIC Training School, Freiburg, 2010 Slow electrons Photoelectron microscopy

65. ICONIC Training School, Freiburg, 2010 Slow electrons

66. ICONIC Training School, Freiburg, 2010 Slow electrons

67. ICONIC Training School, Freiburg, 2010 Multiphoton ionization / Orientation O+O+ O-O- CH 3 from photodissociated CH 3 I Probed by 403 nm. Both lasers polarization are vertical.

68. ICONIC Training School, Freiburg, 2010 Multiphoton ionization O+O+ O-O-

69. ICONIC Training School, Freiburg, 2010 Reference books

70. ICONIC Training School, Freiburg, 2010 Conclusion O+O+ O-O-