MB-JASS D Reconstruction. Hannes Hofmann March 2006

Transcription

1 MB-JASS D Reconstruction Hannes Hofmann March 2006

2 Outline Projections Radon Transform Fourier-Slice-Theorem Filtered Backprojection Ramp Filter Mar 2006 Hannes Hofmann 2

3 Parallel Projections X-Rays are attenuated as they propagate through the human body Mar 2006 Hannes Hofmann 3

4 Radon Transform The signal p t is the Radon transform of the object p t = f x, y xcos ysin t dx dy Set of line integrals along the direction at the distance t from the origin Mar 2006 Hannes Hofmann 4

5 Radon Transform (2) A simple image (left) and the sinogram (right) produced by applying the Radon transform (180 projections with an angle of 1 degree) Mar 2006 Hannes Hofmann 5

6 Unfiltered BP Ok, that's bad. But we used only 18 projections Mar 2006 Hannes Hofmann 6

7 Unfiltered BP (2) Now we used all 180 projections. Still poor image quality Mar 2006 Hannes Hofmann 7

8 Fourier Slice Theorem t p t f x, y F u,v P w Projection under angle Slice under in freq. domain Mar 2006 Hannes Hofmann 8

9 Fourier Slice Theorem (2) Projection of f x, y in y direction is p =0 x = f x, y dy p =0 x

10 Fourier Slice Theorem (2) Projection of f x, y : p =0 x = The Fourier transform of f x, y is F u, v = f x, y e 2 i xu yv dx dy f x, y dy

11 Fourier Slice Theorem (2) Projection of f x, y : The Fourier transform of f x, y is F u, v = The slice s u is then s u =F u,0 = [ = = p =0 x = f x, y e 2 i xu yv dx dy which is just the Fourier transform of Formal derivation see e.g. Kak & Slaney f x, y e i2 xu dx dy f x, y dy]e i2 xu dx p =0 t e i2 tu dt f x, y dy p =0 x

12 Fourier Slice Theorem (3) t p t f x, y F u,v P w Projection under many angles Slices in frequency domain Mar 2006 Hannes Hofmann 12

13 FST - Regridding Projection data lies on circles (dots) and has to be interpolated to a Cartesian coordinate system Interpolation gets worse with increasing u & v v Errors in high-frequency regions, i.e. regions with high detail u Frequency domain Mar 2006 Hannes Hofmann 13

14 Unfiltered Backprojection Mar 2006 Hannes Hofmann 14

15 Solution: Filtering Ideal situation Fourier Slice Theorem Mar 2006 Hannes Hofmann 15

16 Solution: Filtering Ideal situation Fourier Slice Theorem The ideal is approximated by a weighting (highpass)

17 Filtered Backprojection Switch to polar coordinates 2 f x, y = 0 0 F w, e i2 xcos ysin w dw d = i2 xcos 0 0 F w, e ysin w dw d i2 0 0 F w, e xcos ysin w dw d

18 Filtered Backprojection Switch to polar coordinates 2 f x, y = 0 0 F w, e i2 xcos ysin w dw d = i2 xcos 0 0 F w, e ysin w dw d i2 0 0 F w, e xcos ysin w dw d 180 symmetry F u, =F u, f x, y = 0 [ F w, e i2 wt w dw]d t=xcos ysin Mar 2006 Hannes Hofmann 18

19 Filtered Backprojection (2) Apply Fourier Slice Theorem f x, y = 0 [ P w e i2 wt w dw]d Which can be written as f x, y = 0 Q t d Q t = P w e i2 wt w dw Q is the filtered projection with filter function w (in frequency domain) Mar 2006 Hannes Hofmann 19

20 Ramp Filter Convolution in frequency domain High-pass, emphasizes noise Mar 2006 Hannes Hofmann 20

21 Filter Results Attenuation profile of a cylinder p t Filtered attenuation profile p t K t convolution Mar 2006 Hannes Hofmann 21

22 Filtered BP: Result Reconstruction using ramp filtered backprojection (180 projections) Mar 2006 Hannes Hofmann 22

23 Sharp Kernel Bone image: use sharp kernel high resolution high noise Mar 2006 Hannes Hofmann 23

24 Soft Kernel Soft tissue: use soft kernel lower resolution less noise Mar 2006 Hannes Hofmann 24

25 Filtered Backprojection Mar 2006 Hannes Hofmann 25

26 Thanks Thank you for your attention. Do you have any questions? Mar 2006 Hannes Hofmann 26

27 References A. C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging. Society of Industrial and Applied Mathematics, 2001 S. Horbelt, M. Liebling, M. Unser: Filter design for filtered back-projection guided by the interpolation model, Proceedings of the SPIE International Symposium on Medical Imaging: Image Processing (MI'02), San Diego CA, USA, February 24-28, 2002, vol. 4684, Part II, pp J. Hornegger, D. Paulus: Slides to Medical Image Processing B. Heismann: Slides to Medical Imaging Slice_Reconstruction.html kprojection.htm Mar 2006 Hannes Hofmann 27