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1. RADON TRANSFORM A small introduction to RT, its inversion and applications Jaromír Brum Kukal, 2009

2. Johann Karl August Radon • Born in Děčín (Austrian monarchy, now North Bohemia, CZ) in 1887 • Austrian mathematician living in Vienna • Discover the transform and its inversion in 1917 as pure theoretical result • No practical applications during his life • Died in 1956 in Vienna

3. Actual applications of inverse Radon transform • CT – Computer Tomography • MRI – Magnetic Resonance Imaging • PET – Positron Emission Tomography • SPECT – Single Photon Emission Computer Tomography

4. Geometry of 2D Radon transform • Input space coordinatesx, y • Input function f(x, y) • Output space coordinatesa, s • Output function F(a, s)

5. Formulas of pure RT and IRT Radon transform Inverse Radon transform

7. Full circle in RT

8. Shifted full circle in RT

9. Empty circle in RT

10. Shifted empty circle in RT

11. Thin stick in RT

12. Shifted thin stick in RT

13. Full triangle in RT

14. Shifted full triangle in RT

15. Full square in RT

16. Shifted full square in RT

17. Empty square in RT

18. Shifted empty square in RT

19. | x |2/3 + | y |2/3≤ 1 in RT

20. | x | + | y | ≤ 1 in RT

21. | x |3/2 + | y |3/2≤ 1 in RT

22. | x |2 + | y |2≤ 1 in RT

23. | x |6 + | y |6≤ 1 in RT

24. | x |n + | y |n≤ 1forn  in RT

25. 2D Gaussian in RT

26. Shifted 2D Gaussian in RT

27. Six 2D Gaussians in RT

28. Smooth elliptic object in RT

30. Radon transform properties • Image of any f + g is F + G • Image of cf is cF for any real c • Rotation of f causes translation of F in α • Scaling of f in (x,y) causes scaling of F in s • Image of a point (2D Dirac function) is sine wave line

31. Radon transform properties 6. Image of n points is a set ofn sine wave lines 7. Image of a line is a point (2D Dirac function) 8. Image of polygon contour is a point set

32. Radon transform realization • Space domain: • Pixel splitting into four subpixels • 2D interpolation in space domain • 1D numeric integration along lines • Frequency domain: • 2D FFT of original • Resampling to polar coordinates • 2D interpolation in frequency domain • Inverse 2D FFT brings result

33. Inverse transform realization • Filtered back projection in space domain: • 1D HF filtering of 2D original along s • Additional 1D LF filtering along s • 2D interpolation in space domain • 1D integration along lines brings result • Frequency domain: • 2D FFT of original • Resampling to rectangular coordinates • 2D interpolation in frequency domain • 2D LF filtering in frequency domain • Inverse 2D FFT brings result

34. RT and IRT in Matlab • Original as a square matrix D(2n2n) • of nonnegative numbers • Vector of angles alpha • Basic rangealpha = 0:179 • Digital range is better • alpha = (0:2^N -1)*180/2^N • Extended rangealpha = 0:359 • Output matrix R of nonnegative numbers • Angles alpha generates columns of R • R = radon(D,alpha); • D = iradon(R,alpha); • D = iradon(R,alpha,metint,metfil);

36. Reconstruction from 32 angles

37. Reconstruction from 64 angles

38. Reconstruction from 96 angles

39. Reconstruction from 128 angles

40. Reconstruction from 180 angles

41. Reconstruction from 256 angles

42. Reconstruction from 360 angles

43. Reconstruction from 512 angles

44. Radon transform applications Natural transform as result of measurement: Gamma ray decay from local density map Extinction from local concentration map Total radioactivity from local concentration map Total echo from local nuclei concentration map 3D reality is investigated via 2D slices Artificial realization: Noise – RT – noise – IRT simulations Image decryption as a fun TSR invariant recognition of objects