Direct Fourier Reconstruction
-
Upload
sebastian-camacho -
Category
Documents
-
view
25 -
download
2
description
Transcript of Direct Fourier Reconstruction
![Page 1: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/1.jpg)
1
Direct Fourier Reconstruction
Medical imagingGroup 1
Members: Chan Chi Shing Antony
Chang Yiu Chuen, Lewis Cheung Wai Tak Steven
Celine Duong Chan Samson
![Page 2: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/2.jpg)
2
Abstract
Shepp-Logan Head Phantom
ModelRadon Transform
1D Fourier transformed
projection slices of different angles
Convert from polar to Cartesian coordinate
Inverse 2D Fourier
transform.
Reconstructed image
![Page 3: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/3.jpg)
3
Not that simple!!!Problem 1: Continuous Fourier Transform is impracticalSolution: Discrete Fourier Transform
Problem 2: DFT is slowSolution: Fast Fourier Transform
Problem 3: FFT runs faster when number of samples is a power of twoSolution: Zeropad
Problem 4: F1D Radon Function (polar) Cartesian coordinate
but the data now does not have equal spacing, which needs for IF2D
Solution: Interpolation
![Page 4: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/4.jpg)
4
Agenda1.Theory 1.1. Central Slice Theorem (CST) 1.1.1 Continuous Time Fourier Transform (CTFT) - > Discrete Time Fourier Transform (DTFT) -> Discrete Fourier Transform (DFT) -> Fast Fourier Transform (FFT) 1.2. Interpolation
2. Experiments 2.1. Basic 2.1.1. Number of sensors 2.1.2. Number of projection slices 2.1.3. Scan angle (<180, >180) 2.2. Advanced 2.2.1. Noise 2.2.2. Sensor Damage
3. Conclusion
4. References
![Page 5: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/5.jpg)
5
1. Theory – 1.1. Central Slice Theorem (CST)
Name of reconstruction method:
Direct Fourier Reconstruction The Fourier Transform of a projection at an angle q is a line in the Fourier transform of the image at the same angle. If (s, q) are sampled sufficiently dense, then from g (s, q) we essentially know F(u,v) (on the polar coordinate), and by inverse transform can obtain f(x,y)[1].
![Page 6: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/6.jpg)
6
1. Theory – 1.1. Central Slice Theorem (CST) – 1.1.1 Continuous Time Fourier Transform (CTFT) - > Discrete Time Fourier Transform (DTFT) -> Discrete Fourier Transform (DFT) -> Fast Fourier Transform (FFT)
• CTFT -> DTFTDescription: DTFT is a discrete time sampling version of CTFT Reasons: fast and save memory space
• DTFT -> DFTDescription: DFT is a discrete frequency sampling version of DTFTReasons: fast and save memory space sampling all frequencies are not possible
• DFT -> FFTDescription: Faster version of FFTReasons: even faster
![Page 7: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/7.jpg)
7
Con’t
• DFT -> FFTSpecial requirement : Number of samples should be a power of twoSolution: Zeropad
How to make zeropad?In the sinogram, add black lines evenly on top and bottom
Physical meaning?Scan the sample in a bigger space!
1. Theory – 1.1. Central Slice Theorem (CST) – 1.1.1 CTFT - > DTFT -> DFT -> FFT
![Page 8: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/8.jpg)
8
1. Theory – 1.2. Interpolation
Why we need interpolation?Reasons : Equal spacing for x and y coordinates are required for IF2D
Reasons?• 1D Fourier Transform of Radon function is in polar coordinate• Convert to 2D Cartesian coordinate system, x = rcos q and y = rsin q
Solution: Interpolation
![Page 9: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/9.jpg)
9
1. Theory – 1.2. Interpolation (con’t)
![Page 10: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/10.jpg)
10
1. Theory – 1.2. Interpolation (con’t)
![Page 11: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/11.jpg)
11
2. Experiment – 2.1. Basic – 2.1.1. Number of sensors
The number of sensors decreases
The resolution of the reconstructed images
decreases and with low contrast
![Page 12: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/12.jpg)
12
2. Experiment – 2.1. Basic – 2.1.1. Number of sensors (con’t)
![Page 13: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/13.jpg)
13
2. Experiment – 2.1. Basic – 2.1.1. Number of sensors (con’t)
![Page 14: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/14.jpg)
14
2. Experiment – 2.1. Basic – 2.1.1. Number of sensors (con’t)
![Page 15: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/15.jpg)
15
2. Experiment – 2.1. Basic – 2.1.2. Number of projection slices• As the number of projection slices decreases, the reconstructed
images become blurry and have many artifacts• The resolution can be better by using more slices
![Page 16: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/16.jpg)
16
2. Experiment – 2.1. Basic – 2.1.2. Number of projection slices (con’t)
![Page 17: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/17.jpg)
17
2. Experiment – 2.1. Basic – 2.1.2. Number of projection slices (con’t)
![Page 18: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/18.jpg)
18
2. Experiment – 2.1. Basic – 2.1.3. Scan angle (<180, >180)• The image resolution increases as the scanning angle
increases • Meanwhile artifacts reduced
![Page 19: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/19.jpg)
19
2. Experiment – 2.2. Advanced – 2.2.1. Noise• The noise is added on the sinogram• The more the noise, the more the data being distorted
![Page 20: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/20.jpg)
20
2. Experiment – 2.2. Advanced – 2.2.2. Sensor Damage• From the sinogram, each s value in the vertical axis corresponds to a
sensor• If there is a sensor damaged, then it will appear as a semi-circle artifact
![Page 21: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/21.jpg)
21
2. Experiment – 2.2. Advanced – 2.2.2. Sensor Damage (con’t)
• The more the damage sensors, the lower the quality of the reconstructed images
How to deal with damaged sensors?1. Replace those sensors2. Scan the object by 360o instead of 180o, so, if previously, a positive s value
sensor is damaged, after scanning 180o, it can now be covered by the corresponding negative s value sensor (same s value but opposite sign), except for the case that the same s value sensors with opposite sign are also damaged
3. Scan the object once, covering 0o to 179o. Then rotates the object by 180o and starts another scanning
![Page 22: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/22.jpg)
22
3. Conclusion
• Direct Fourier Reconstruction is able to use short computation time to give a good quality image, with all details in the Phantom can be conserved
• The resolution is high and even there is little artifact, it is still acceptable. • To make the reconstructed images better, we can
1) use more sensors2) use more projection slices3) scan the Phantom more than 180o
4) avoid noise appearing5) prevent using damaged sensors.
Shepp-Logan Head Phantom
Model
Radon Transform
1D Fourier transformed
projection slices of different angles
Convert from polar to Cartesian coordinate
Inverse 2D Fourier
transform.
Reconstructed image
![Page 23: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/23.jpg)
23
4. Reference1. Yao Wang, 2007, Computed Tomography, Polytechnic University2. Forrest Sheng Bao, 2008, FT, STFT, DTFT, DFT and FFT, revisited, Forrest Sheng Bao, http://narnia.cs.ttu.edu/drupal/node/46
![Page 24: Direct Fourier Reconstruction](https://reader036.fdocuments.net/reader036/viewer/2022062517/568132c4550346895d9988ee/html5/thumbnails/24.jpg)
24
Thank you