1. Ultrasound Computed Tomography 何祚明 陳彥甫 2002/06/12

2. Introduction  Conventional X-ray image is the superposition of all the planes normal to the direction of propagation.  The tomography image is effectively an image of a slice taken through a 3-D volume.  An estimated 10-15% of breast cancers evade detection by mammography.  Poor differentiation of malignant tumors from highly common cysts (while ultrasound can do so with accuracies of 90-100%).  UCT can provide not only structural/density information, but also tissue compressibility and speed of sound maps

3. Tomography  Time of flight or intensity attenuation  Array transducer  D data (only t f )  Rotation : 2-D data (t f on range and angle  )  Scan : 3-D data (y position, t f and angle  ) A B

4. Tomography R  A reconstruct

5. Reconstruction Method  Attenuation method  Iterative method –Algebraic Reconstruction Technique (ART)  Direct reconstruction –Fourier transform  Alternative direct reconstruction –Back projection –Filtered back projection

6. Central Section Theorem f (x,y)g 90 (y’) x y y’

7. Ambiguity Angle  f(x,y) x y   R xcos  + ysin  = R 

8. Equations 1D FT of projection function 2D FT

9. Cont’d  2D Fourier transform  (u,v) in polar coordinates is (  co s ,  sin  )  2 D inverse Fourier Transform

10. f(x,y) x y  R u v   g  (R) G()G() Figures

11. Algorithm solve coordinate problems (polar to rectangular coordinates) 1.1D FT each of the projections g  (R)  G  (  ) 2.Integrate G  (  ) in  and  -- avoid coordinate transformation problems g  (R) G  (  ) =F(u,v) f (x, y)  1D FT

12. Back Projection Reconstruction  Back projection (at one angle)  Integration of all ambiguity angle 

13. Cont’d  Drawback of back projection method  So, we integrate this function after added 1/ 

14. Disadvantage  FT{1/    1/r : blurring function  f (x,y) convolves 1/r  Additional filter is needed  x yyy xx 1/r objectBackprojected object

15. Filter Design  Design filter at spatial frequency domain  Cone filter  Spatial filter u v 1/r   

16. Signal Processing Flow  Backprojection all angles, g  (R)  F b (  )  Forward 2D FT to get:  Apply filter on to get F (u,v)  Inverse 2D FT to f (x,y) Fb()Fb() F(u,v)f (x, y)  2D FT2D IFT g  (R)  Filtering

17. Algebraic Reconstruction Technique (ART)  The iterative process starts with all values set to a constant, such as the mean or 0.  At each iteration, the difference between the measured projection data for a given projection and the sum of all reconstructed elements along the line defining the projection is computed.  This difference is then evenly divided among the N reconstruction elements.

18. Algebraic Reconstruction Technique (ART) 1.Projection Step 2.Comparison Step 3.Backprojection Step 4.Update Step Additive ART :Multiplicative ART :

19. Algebraic Reconstruction Technique (ART) N elements per line f ij (calculated element) cross section g j (measured projection)

20. Simulation Results I. Using ART to reconstruct : Simulated DataReconstruction Result 102030405060708090100 10 20 30 40 50 60 70 80 90 100 102030405060708090100 10 20 30 40 50 60 70 80 90 100

21. Simulation Results I. 102030405060708090100 10 20 30 40 50 60 70 80 90 100 Simulated Data 1020304050607080 10 20 30 40 50 60 70 80 Reconstruction Result Using direct Fourier transform to reconstruct :

22. Simulation Results II. Using ART to reconstruct : Reconstruction ResultSimulated Data 102030405060708090100 10 20 30 40 50 60 70 80 90 100 102030405060708090100 10 20 30 40 50 60 70 80 90 100

23. Simulation Results II. Reconstruction ResultSimulated Data 102030405060708090100 10 20 30 40 50 60 70 80 90 100 5101520 2 4 6 8 10 12 14 16 18 20 Using direct Fourier transform to reconstruct :

24. Experiment Architecture Transducer Phantom Transducer y x Water Tank Transmitter Receiver 2 mm/step 2 degrees/rotate

25. Experiment Architecture

26. Breast Phantom

27. Eraser Phantom

28. Rubber Ball

29. Time-of-flight t1t1 t2t2 t1t1 t2t2 t3t3 t3t3

30. Reconstructed Image Breast phantom 1020304050607080 10 20 30 40 50 60 70 80 ARTDirect Fourier Transform 1020304050607080 10 20 30 40 50 60 70 80

31. Reconstructed Image Breast phantom, using ART 1020304050607080 10 20 30 40 50 60 70 80

32. Reconstructed Image Rubber ball 2468101214161820 2 4 6 8 10 12 14 16 18 20 5101520 2 4 6 8 10 12 14 16 18 20 ARTDirect Fourier Transform

33. Reconstructed Image Eraser 510152025303540 5 10 15 20 25 30 35 40 510152025303540 5 10 15 20 25 30 35 40 Direct Fourier TransformART

34. Conclusions  ART reconstruction is much faster than other reconstruction method but must notice the velocity profile normalization.  Direct fourier reconstruction is effective but less efficiency.  Backprojection reconstruction needed additional filter to cancel integration errors.

35. Future Works  Correct velocity profile estimation  Discuss backprojection method errors and additional filter design method  Discuss diffraction errors in ultrasound

36. References  [1] Albert Macovski, “Medical Imaging Systems”  [2] Matthew O’Donnell, “X-Ray Computed Tomography (CT)”, University of Michigan  [3] Douglas C. Noll, “Computed Tomography Notes”, University of Michigan, 2001  [4] Brian Borchers, “Tomography Lecture Notes”, New Mexico Tech., March 2000  [5] James F. Greenleaf et al., “Clinical Imaging with Transmissive Ultrasonic Computerized Tomography”, IEEE trans.on Biomedical Engineering vol. 28 no. 2 1981