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Tomography Reconstruction : Introduction and new results on Region of Interest reconstruction

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

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Tomography Reconstruction : Introduction and new results on Region of Interest reconstruction

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

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

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

  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. 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. x N0 f N 1. Introduction • X-ray and matter interaction: • Photoelectric absorption • Compton scattering • Rayleigh scattering Microscopic scale X-ray attenuation Beer-Lambert law: Macroscopic 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. L xin xout 1. Introduction X-ray and matter interaction : Patient X-ray sensor X-ray source

  8. 1. Introduction  ? s

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

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

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

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

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

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

  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. 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. 3. Reconstruction of a ROI from truncated data : new results Is it possible to reconstruct exactly a part of the object from an incomplete set of data? • 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…)

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

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

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

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

  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. 3. Reconstruction of a ROI from truncated data : new results The DBP algorithm : Differentiated backprojection xs Remind x1

  25. 3. Reconstruction of a ROI from truncated data : new results x2 The DBP algorithm +L fx1(x2) -L fx1(x2) 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)

  26. Merci de votre attention…Any questions?

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