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Center for Fast Ultrasound Imaging Department of Electrical Engineering
Radon Transform and Filtered Backprojection Jørgen Arendt Jensen October 13, 2016 Center for Fast Ultrasound Imaging, Build 349 Department of Electrical Engineering Technical University of Denmark
Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical 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
• Exercise 5 • Questions for the assignments
• Reading material: Prince & Links chapter 6
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Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark
Question for 2D signal processing: Is phase or amplitude most important?
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Demo in: for_13/matlab_demo/phase_demo Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical 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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2
Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical 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
5/x Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark
Hounsfield units
From
: W. A
. Kal
ende
r; C
ompu
ted
Tom
ogra
phy,
Pub
licis
, 200
5
6/x
Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark
Measurement of attenuation
From
: W. A
. Kal
ende
r; C
ompu
ter T
omog
raph
y, P
ublic
is, 2
005
7/x Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark
Parallel beam projection geometry
x’
y’
x
y
Patient coordinate system
CT coordinate system
Point
φ
ψ
θ
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Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark
Sinogram for point
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Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark 12/x
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Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark 13/x
Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark 14/x
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Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark 16/x
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Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark
Shepp-Logan phantom
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Demo in: for_13/matlab_demo/proj_demo
Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark
Fourier slice theorem
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Demo in: for_13/matlab_demo/cd_demo Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark
Filtered backprojection • Perform for all projection:
1. Make Fourier transform of projected data
2. 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 Electrical Engineering Technical University of Denmark
Influence from number of projections
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Transfer function of filters - Ideal
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Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark
Hanning weighted filter
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Shepp-Logan filter
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Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark
Filter transfer functions and impulse responses
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Comparison between filters
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Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark
Filtered backprojection • Perform for all projection:
1. Make Fourier transform of projected data
2. Apply filter in Fourier domain 3. Make invers transform
4. Backproject and sum with previous image
27/x Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark
Summary
• Parallel beam projection and Radon transform
• Fourier slice theorem • Filtered backprojection
reconstruction and choices
• P & L: chapter 6
• Now: Exercise 5 questions
• Questions for assignments
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Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark
Exercise 5: Image processing
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1. Show Shepp-Logan phantom images. 2. Shepp-Logan phantom gray scale mapping. 3. Clinical images of the brain and its gray scale mapping. 4. Make a two-dimensional Fourier transform of the
sh_black image, and make a mesh plot of the amplitude spectrum with the command mesh. Plot the spectrum with the correct spatial frequency axis. Study the symmetry relations for the Fourier transform.
5. Make a low-pass filter with a circularly symmetric transfer function that removes all frequencies above a value of 116 m-1.
6. Use an edge enhancement filter given as [-1 -1 -1; -1 9 -1; -1 -1 -1] to enhance the edges in the image sh_black.
7. Try the above mentioned image processing on the clinical images downloaded previously.
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