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

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Presentation on theme: "ELEG 479 Lecture #8 Mark Mirotznik, Ph.D. Associate Professor The University of Delaware."— Presentation transcript:

1 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

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4 Hounsfield’s Experimental CT

5 Lets look at how CT works!

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

7 Our First Projection

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9 Rotate and Take Another Projection

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11 This is called a sinogram

12 Sinogram This is called a sinogram

13 Sinogram This is called a sinogram

14 Sinogram This is called a sinogram

15 Sinogram This is called a sinogram

16 Sinogram This is called a sinogram

17 Sinogram This is called a sinogram

18 Sinogram This is called a sinogram

19 Sinogram This is called a sinogram

20 Sinogram This is called a sinogram

21 Sinogram This is called a sinogram

22 Sinogram This is called a sinogram

23 Sinogram This is called a sinogram

24 Sinogram This is called a sinogram

25 Sinogram This is called a sinogram

26 Sinogram This is called a sinogram

27 Sinogram This is called a sinogram

28 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? This is called a sinogram

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,  ) and need to determine f(x,y)  l 0  g  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  ) Step #2: Add all the back projected images together These are called back projected images

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

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

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

38 Projection-Slice Theorem This is a very important theorem in CT imaging First take the 1D Fourier transform a projection g(l,  Next we substitute the Radon transform for g(l,  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  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  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  the result is the same as  a slice of the 2D FT of the original object f(x,y) f(x,y) F(u,v) 2D FT  

45 The Fourier Reconstruction Method Take projections at all angles . 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) F(u,v) 2D IFT  

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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

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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 1)take 1D FFT of projection 2)multiply by ramp filter 3)take 1D inverse FFT 4)make a back projection

54 Filtered Back Projection Method Always works! Object Digital Filter 1)take 1D FFT of projection 2)multiply by ramp filter 3)take 1D inverse FFT 4)make a back projection

55 Filtered Back Projection Method Always works! Object Digital Filter 1)take 1D FFT of projection 2)multiply by ramp filter 3)take 1D inverse FFT 4)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 B. Filtered Back Projection Matlab Demo Your Assignment (b)Write a matlab function that reconstructs an image using the filtered back projection method Your Assignment (b)Write a matlab function that reconstructs an image using the filtered back projection method

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

58 Convolution Back Projection Let

59 Convolution Back Projection The problem isdoes not exist

60 Convolution Back Projection The problem isdoes 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

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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

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67 2G 3G

68 Ring artifacts The rotate/rotate geometry of 3 rd 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

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75 Computer Assignment 1.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 2.Do the same as #1 using the filter back projection method. 3.(grad students only) Do the same with the convolution back projection method


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