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CT Imaging: Reconstruction and Filtered Backprojection

This lecture provides an overview of the principles and methods used in CT imaging, including the reconstruction of an unknown image from sinogram data using the Radon transform and the filtered backprojection technique. The lecture also discusses the limitations and challenges associated with the Fourier reconstruction method.

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CT Imaging: Reconstruction and Filtered Backprojection

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  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

  3. Hounsfield’s Experimental CT

  4. Lets look at how CT works!

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

  6. Our First Projection

  7. Our First Projection

  8. Rotate and Take Another Projection

  9. Rotate and Take Another Projection

  10. This is called a sinogram

  11. This is called a sinogram 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 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?

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

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

  30. 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)

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

  32. 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

  33. 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

  34. 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.

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

  36. 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)

  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) Next we do a little rearranging

  38. 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?

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

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

  41. 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)

  42. 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)

  43. 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)

  44. 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)

  45. Image Reconstruction Using Filtered Backprojection Filter Backprojection

  46. 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

  47. 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

  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. We can write this as Since you can show which can be rewritten as

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