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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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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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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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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

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

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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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