ct reconstruction -outline
TRANSCRIPT
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Center for Fast Ultrasound ImagingDepartment of Health Technology
Radon Transform andFiltered Backprojection
Jørgen Arendt Jensen
October 25, 2021
Center for Fast Ultrasound Imaging, Build 349Department of Health TechnologyTechnical University of Denmark
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
CT reconstruction - outline
•CT scanners•Projection and Radon transform•Projection demo
•Fourier slice theorem•Inverse Radon transform – filtered backprojection
•Selection of filters•Filtered backprojection algorithm
•Reading material: Prince & Links chapter 6
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Modern CT system generations
From
: W. A
. Kal
ende
r; C
ompu
ted
Tom
ogra
phy,
Pub
licis
, 200
5
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
What do we measure?
• Intensity measured by detector:
• Conversion to attenuation:
• Attenuation values µ are scaled relative to water:
0
ln1II
x-=µ
1000´-
=water
watertissueHUµ
µµ
)exp( xII o ×-= µ
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Hounsfield units
From
: W. A
. Kal
ende
r; C
ompu
ted
Tom
ogra
phy,
Pub
licis
, 200
5
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Measurement of attenuation
From
: W. A
. Kal
ende
r; C
ompu
ter T
omog
raph
y, P
ublic
is, 2
005
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Parallel beam projection geometry
x’
y’
x
y
Patient coordinate system
CT coordinate system
Point
fy
q
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r
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Sinogram for point
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Shepp-Logan phantom
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Demo in: for_13/matlab_demo/proj_demo
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Fourier slice theorem
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Demo in: for_13/matlab_demo/ct_demo
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Filtered backprojection
• Perform for all projections:1. Make Fourier transform of
projected data2. Apply filter in Fourier domain
3. Make invers Fourier transform
4. Backproject and sum with previous image
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Influence from number of projections
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Transfer function of filters - Ideal
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Hanning weighted filter
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Shepp-Logan filter
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Filter transfer functions and impulse responses
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Comparison between filters
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Filtered backprojection
• Perform for all projection:1. Make Fourier transform of
projected data2. Apply filter in Fourier domain
3. Make invers transform
4. Backproject and sum with previous image
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Summary
• Parallel beam projection and Radon transform
• Fourier slice theorem• Filtered backprojection
reconstruction and choices
• P & L: Chapter 6
• Questions for CT assignments
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