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RADON TRANSFORM

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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 coordinates x, y • Input function f(x, y) • Output space coordinates a, s • Output function F(a, s)

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

  6. Full circle in RT

  7. Shifted full circle in RT

  8. Empty circle in RT

  9. Shifted empty circle in RT

  10. Thin stick in RT

  11. Shifted thin stick in RT

  12. Full triangle in RT

  13. Shifted full triangle in RT

  14. Full square in RT

  15. Shifted full square in RT

  16. Empty square in RT

  17. Shifted empty square in RT

  18. | x |2/3 + | y |2/3≤ 1 in RT

  19. | x | + | y | ≤ 1 in RT

  20. | x |3/2 + | y |3/2≤ 1 in RT

  21. | x |2 + | y |2≤ 1 in RT

  22. | x |6 + | y |6≤ 1 in RT

  23. | x |n + | y |n≤ 1forn  in RT

  24. 2D Gaussian in RT

  25. Shifted 2D Gaussian in RT

  26. Six 2D Gaussians in RT

  27. Smooth elliptic object in RT

  28. 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

  29. 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 • Image of n points is a set ofn • sine wave lines • Image of a line is a point (2D Dirac • function) • Image of polygon contour is a point set

  30. 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

  31. 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

  32. 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);

  33. Reconstruction from 32 angles

  34. Reconstruction from 64 angles

  35. Reconstruction from 96 angles

  36. Reconstruction from 128 angles

  37. Reconstruction from 180 angles

  38. Reconstruction from 256 angles

  39. Reconstruction from 360 angles

  40. Reconstruction from 512 angles