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Center for Fast Ultrasound Imaging Department of Electrical Engineering

Filtered backprojection and Radon transforms Jørgen Arendt Jensen October 20, 2014 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

• Advise 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

Modern CT system generations

From

: W. A

. Kal

ende

r; C

ompu

ted

Tom

ogra

phy,

Pub

licis

, 200

5

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

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

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x

y

x’

y’

Patient coordinate system

CT coordinate system Point

φ ψ

Φ

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

Fourier slice theorem

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

Filtered backprojection •  Perform for all projection:

•  Make Fourier transform of projected data

•  Apply filter in Fourier domain •  Make invers transform

•  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:

•  Make Fourier transform of projected data

•  Apply filter in Fourier domain •  Make invers transform

•  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

•  Next lecture: PhD, MD Thomas Kristensen, Department of Radiology, Rigshospitalet on usage of the latest CT technology

•  P & L: chapter 6

•  Now: Exercise 5 questions Questions for assignments

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

Exercise 5: Image processing

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

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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