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

2. Summary of Last Lecture X-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. Example = xray attenuation of 0 = xray attenuation of 2.5

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

13. This is called a sinogram

14. This is called a sinogram

15. This is called a sinogram

16. This is called a sinogram

17. This is called a sinogram

18. This is called a sinogram

19. This is called a sinogram

20. This is called a sinogram

21. This is called a sinogram

22. This is called a sinogram

23. This is called a sinogram

24. This is called a sinogram

25. This is called a sinogram

26. This is called a sinogram

27. This is called a 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
Given
and
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 l p
g(q,l) x y ??
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 as a 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 as a 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 as a slice of the 2D FT of the original object f(x,y)
f(x,y)
2D FT qo
F(u,v) qo

45. The Fourier Reconstruction Method
f(x,y)
F(u,v)
2D IFT qo q
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)

47. Image Reconstruction Using Filtered 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

52. Filtered Back Projection verses Back Projection
A. Back Projection
B. Filtered Back Projection

53. Filtered Back Projection Method
This always works!
Object
Digital Filter
take 1D FFT of projection
multiply by ramp filter
take 1D inverse FFT
make a back projection

54. Filtered Back Projection Method
Always works!
Object
Digital Filter
take 1D FFT of projection
multiply by ramp filter
take 1D inverse FFT
make a back projection

55. Filtered Back Projection Method
Always works!
Object
Digital Filter
take 1D FFT of projection
multiply by ramp filter
take 1D inverse FFT
make a back projection
We do this for every angle and then add together all the filtered back projected images

56. Filtered Back Projection verses Back Projection
A. Back Projection
Matlab Demo
Your Assignment
(b) Write a matlab function that reconstructs an image using the filtered back projection method
B. Filtered Back Projection

57. Convolution Back Projection
From the filtered back projection algorithm we get
It may be easier computationally to compute the inner 1D IFT using a convolution
recall

58. Convolution Back Projection
Let

59. Convolution Back Projection
The problem is
does not exist

60. Convolution Back Projection
The problem is
does not exist
The solution
where
is called a weighting function

61. Convolution Back Projection
Common window functions
Hamming window
Lanczos window (sinc function)
Simple rectangular window
Ram-Lak window
Kaiser window
Shepp-Logan window

62. Incorporated linear array of 30 detectors
More data acquired to improve image quality (600 rays x 540 views)
Shortest scan time was 18 seconds/slice
Narrow fan beam allows more scattered radiation to be detected

65. Number of detectors increased substantially (to more than 800 detectors)
Angle of fan beam increased to cover entire patient
Eliminated need for translational motion
Mechanically joined x-ray tube and detector array rotate together
Newer systems have scan times of ½ second

67. 3G 2G

68. Ring artifacts
The rotate/rotate geometry of 3rd generation scanners leads to a situation in which each detector is responsible for the data corresponding to a ring in the image
Drift in the signal levels of the detectors over time affects the t values that are backprojected to produce the CT image, causing ring artifacts

69. Ring artifacts

70. Designed to overcome the problem of ring artifacts
Stationary ring of about 4,800 detectors

71. Designed to overcome the problem of ring artifacts
Stationary ring of about 4,800 detectors

72. Developed specifically for cardiac tomographic imaging
No conventional x-ray tube; large arc of tungsten encircles patient and lies directly opposite to the detector ring
Electron beam steered around the patient to strike the annular tungsten target
Capable of 50-msec scan times; can produce fast-frame-rate CT movies of the beating heart

73. Helical CT scanners acquire data while the table is moving
By avoiding the time required to translate the patient table, the total scan time required to image the patient can be much shorter
Allows the use of less contrast agent and increases patient throughput
In some instances the entire scan be done within a single breath-hold of the patient

75. Computer Assignment
Write a MATLAB program that reconstructs an image from its projections using the back projection method. Your program should allow the user to input a phantom object and a set (e.g. vector) of projection angle. Your program should then: (a) compute the sinogram of the object (you can use Matlab’s radon.m command to do this), (b) compute the reconstructed image from the sinogram and vector of projection angles, (c) try your program out for several different objects and several different ranges of projection angles
Do the same as #1 using the filter back projection method.
(grad students only) Do the same with the convolution back projection method