CT Imaging: Reconstruction and Filtered Backprojection

1. ELEG 479Lecture #8 Mark Mirotznik, Ph.D. Associate Professor The University of Delaware

2. Summary of Last LectureX-ray Radiography • Overview of different systems for projection radiography • Instrumentation • Overall system layout • X-ray sources • grids and filters • detectors • Imaging Equations • Basic equations • Geometrical distortions • More complicated imaging equations

4. Hounsfield’s Experimental CT

5. Lets look at how CT works!

6. = xray attenuation of 2.5 = xray attenuation of 0 Example = xray attenuation of 5

7. Our First Projection

8. Our First Projection

9. Rotate and Take Another Projection

10. Rotate and Take Another Projection

11. This is called a sinogram

12. This is called a sinogram Sinogram

13. This is called a sinogram Sinogram

14. This is called a sinogram Sinogram

15. This is called a sinogram Sinogram

16. This is called a sinogram Sinogram

17. This is called a sinogram Sinogram

18. This is called a sinogram Sinogram

19. This is called a sinogram Sinogram

20. This is called a sinogram Sinogram

21. This is called a sinogram Sinogram

22. This is called a sinogram Sinogram

23. This is called a sinogram Sinogram

24. This is called a sinogram Sinogram

25. This is called a sinogram Sinogram

26. This is called a sinogram Sinogram

27. This is called a sinogram Sinogram

28. This is called a sinogram The sinogram is what is measured by a CT machine. The real trick is how do we reconstruct the unknown image from the sinogram data?

29. Radon Transform and Given In CT we measure and need to find using

30. Radon Transform In CT we measure and need to find We use

31. Reconstruction The Problem In imaging we measure g(l,q) and need to determine f(x,y) q y p ?? g(q,l) x l 0 f(x,y)

32. Back Projection Method A little trick that almost works! Object

33. Back Projection Method A little trick that almost works! Object We do this for every angle and then add together all the back projected images

34. Back Projection Method Step #1: Generate a complete an image for each projection (e.g. for each angle q) These are called back projected images Step #2: Add all the back projected images together

35. Back Projection Method Original object Reconstructed object Kind of worked but we need to do better than this. Need to come up with a better reconstruction algorithm.

36. Projection-Slice Theorem This is a very important theorem in CT imaging First take the 1D Fourier transform a projection g(l,q)

37. Projection-Slice Theorem This is a very important theorem in CT imaging First take the 1D Fourier transform a projection g(l,q) Next we substitute the Radon transform for g(l,q)

38. Projection-Slice Theorem This is a very important theorem in CT imaging First take the 1D Fourier transform a projection g(l,q) Next we substitute the Radon transform for g(l,q) Next we do a little rearranging

39. Projection-Slice Theorem This is a very important theorem in CT imaging Next we do a little rearranging Applying the properties of the delta function What does this look like?

40. Projection-Slice Theorem This is a very important theorem in CT imaging What does this look like? This looks a lot like with

41. Projection-Slice Theorem This is a very important theorem in CT imaging So what does this mean?

42. Projection-Slice Theorem This is a very important theorem in CT imaging Question: So what does this mean? Answer: If I take the 1D FT of a projection at an angle q the result is the same asa slice of the 2D FT of the original object f(x,y)

43. Projection-Slice Theorem This is a very important theorem in CT imaging So what does this mean? If I take the 1D FT of a projection at an angle q the result is the same asa slice of the 2D FT of the original object f(x,y)

44. Projection-Slice Theorem If I take the 1D FT of a projection at an angle q the result is the same asa slice of the 2D FT of the original object f(x,y) 2D FT qo F(u,v) qo f(x,y)

45. The Fourier Reconstruction Method 2D IFT q F(u,v) qo Take projections at all angles q. Take 1D FT of each projection to build F(u,v) one slice at a time. Take the 2D inverse FT to reconstruct the original object based on F(u,v) f(x,y)

47. Image Reconstruction Using Filtered Backprojection Filter Backprojection

48. Filtered Back Projection The Fourier method is not widely used in CT because of the computational issues with creating the 2D FT from projections. However, the method does lead to a popular technique called filtered back projection. In polar coordinates the inverse Fourier transform can be written as with

49. Filtered Back Projection The Fourier method is not widely used in CT because of the computational issues with creating the 2D FT from projections. However, the method does lead to a popular technique called filtered back projection. In polar coordinates the inverse Fourier transform can be written as with From the projection theorem We can write this as

50. Filtered Back Projection The Fourier method is not widely used in CT because of the computational issues with creating the 2D FT from projections. However, the method does lead to a popular technique called filtered back projection. We can write this as Since you can show which can be rewritten as