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Center for Fast Ultrasound ImagingDepartment of Electrical Engineering

Radon Transform andFiltered Backprojection

Jørgen Arendt Jensen

October 23, 2019

Center for Fast Ultrasound Imaging, Build 349Department of Health TechnologyTechnical University of Denmark

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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

•Reading material: Prince & Links chapter 6

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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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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

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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

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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

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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

fy

q

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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

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 Electrical Engineering Technical 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 Electrical Engineering Technical 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 Electrical Engineering Technical University of Denmark

Influence from number of projections

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Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark

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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Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark

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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Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark

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 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 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

• Questions for assignments

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