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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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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 • Reconstruction in 2D tomography : standard algorithms • Reconstruction of a Region Of Interest from truncated data : new results.
1. Introduction Computer Tomography : a non-destructive imaging technique for interior inspection. Waste inspection CT scanner Some applications…
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) • …
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.
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) …
L xin xout 1. Introduction X-ray and matter interaction : Patient X-ray sensor X-ray source
1. Introduction ? s
p(,s) s t f(x) 2. Reconstruction in 2D tomography : standard algorithms Notations
2. Reconstruction in 2D tomography : standard algorithms The Radon transform : p(,s) We note : s t f(x)
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)
2. Reconstruction in 2D tomography : the BackProjection p(2,s) p(1,s) p(3,s) x x We note :
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
2. Reconstruction in 2D tomography : the FBP algorithm 1. Projection filtering For k=1:N pf(,s)=(pr ) (,s) where R()=| | End 2. Backprojection f=R* pf Ramp filter
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).
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?
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…)
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
3. Reconstruction of a ROI from truncated data : virtual fan-beam Inverse Radon transform and the Hilbert transform : the filtering step Remind : Then
a s a 3. Reconstruction of a ROI from truncated data : virtual fan-beam Rebinning formula: Let us introduce :
a s a 3. Reconstruction of a ROI from truncated data : virtual fan-beam Rebinning formula: Let us define : Hilbert rebinning formula :
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
3. Reconstruction of a ROI from truncated data : new results The ROI that can be exactly reconstructed using the virtual fan-beam algorithm
3. Reconstruction of a ROI from truncated data : new results The DBP algorithm : Differentiated backprojection xs Remind x1
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)
