1. 1 Tomography Reconstruction : Introduction and new results on Region of Interest reconstruction -Catherine Mennessier - Rolf Clackdoyle -Moctar Ould Mohamed Laboratoire Hubert Curien, St Etienne Bucharest, May 2008

2. 2 Table of contents 1.Introduction 2.Reconstruction in 2D tomography : standard algorithms 3.Reconstruction of a Region Of Interest from truncated data : new results.

3. 3 1. Introduction Computer Tomography : a non-destructive imaging technique for interior inspection. Waste inspectionCT scanner Some applications…

4. 4 1. Introduction Domains of application: Medical image processing : –Anatomic imaging (CT, Image Guided Surgery, Diagnostic..)  density –Functional imaging (SPECT, PET…search for tumour, heart muscle viable…)  radioactive tracer Industrial : –Non destructive techniques for characterization (drum nuclear waste..), defect detection (on production lines)… Archaeology : –Interior reconstruction (of amphora…) Astronomy : –Doppler imaging Geology : –Seismic studies (wave tomography) …

5. 5 1. Introduction In transmission tomography, the X ray (or gamma ray…) are attenuated. The degree of attenuation depends on the density of the object. The absorption of the X-ray is measured, from different positions of the source/detector system.

6. 6 1. Introduction xx N0N0 f N Beer-Lambert law: X-ray and matter interaction: Photoelectric absorption Compton scattering Rayleigh scattering X-ray attenuation Macroscopic scale Microscopic scale The absorption coefficient f depends on the material. For instance, at 60KeV, water(0,203/cm), white matter(0,210/cm), gray matter(0,212/cm) …

7. 7 1. Introduction x out x in L X-ray source X-ray sensor Patient X-ray and matter interaction :

8. 8 1. Introduction ? s 

9. 9 2. Reconstruction in 2D tomography : standard algorithms Notations  s t   p( ,s) f(x)

10. 10 2. Reconstruction in 2D tomography : standard algorithms The Radon transform :  s t   p( ,s) f(x) We note :

11. 11 2. Reconstruction in 2D tomography : the Fourier slice theorem p( ,s) Fourier domainDirect domain  f(x) F( ) 1D Fourier transform 2D Fourier transform = F(  ) P( ,  ) 

12. 12 2. Reconstruction in 2D tomography : the BackProjection x x p(  1,s) p(  2,s) p(  3,s) We note :

13. 13 2. Reconstruction in 2D tomography : the BackProjection Backprojection of the Radon transform of a centred disk of constant intensity : N  =1N  =2 N  =4 N  =180

14. 14 2. Reconstruction in 2D tomography : the FBP algorithm 1. Projection filtering For k=1:N  p f ( ,s)=(p  r ) ( ,s) where R(  )=|  | End 2. Backprojection f=R * p f Ramp filter

15. 15 2. Reconstruction in 2D tomography : the FBP algorithm Comments : To compute the single value f(x) at x, all the projections are needed as the filtering step is not local  if one data is missing, all the reconstruction (for all x) is affected by the FBP algorithm. FBP is very efficient (standard from 30 years).

16. 16 3. Reconstruction of a ROI from truncated data : new results Truncated data : only the lines that intersect the circle are measured Not measured measured Is it possible to reconstruct exactly a part of the object from the incomplete set of data?

17. 17 3. Reconstruction of a ROI from truncated data : new results Solution : the answer is no if FBP is used yes for some ROI using - virtual fan-beam algorithm (2004) -Differentiated Backprojection with truncated Hilbert Inverse (2004) (two-step, DBP, chord…) Is it possible to reconstruct exactly a part of the object from an incomplete set of data?

18. 18 3. Reconstruction of a ROI from truncated data : new results 1.Virtual fan-beam 1.The ramp filter and the Hilbert transform 2.Fan-beam projection 3.Rebining (the Hilbert transform) 2.DBP 1.Differentiated Backprojection 2.Truncated Hilbert Inverse

19. 19 3. Reconstruction of a ROI from truncated data : virtual fan-beam Inverse Radon transform and the Hilbert transform : the filtering step Remind : Then

20. 20 3. Reconstruction of a ROI from truncated data : virtual fan-beam Rebinning formula: Let us introduce : a  a  s

21. 21 3. Reconstruction of a ROI from truncated data : virtual fan-beam Rebinning formula: Let us define : a  Hilbert rebinning formula : a  s

22. 22 3. Reconstruction of a ROI from truncated data : new results Not measured measured Is it possible to reconstruct exactly a part of the object from the incomplete set of data? Yes, by selecting a switable virtual fan-beam projection s a

23. 23 3. Reconstruction of a ROI from truncated data : new results The ROI that can be exactly reconstructed using the virtual fan-beam algorithm

24. 24 3. Reconstruction of a ROI from truncated data : new results The DBP algorithm : Differentiated backprojection Remind x1x1 xsxs

25. 25 3. Reconstruction of a ROI from truncated data : new results The DBP algorithm f x1 (x 2 ) can be reconstructed where a vertical line, crossing the support of f, can be found, assuming backprojection of the line points is possible. NB: Generalization for all the direction (not only the vertical line) -L +L f x1 (x 2 ) x2x2

26. 26 Merci de votre attention… Any questions?