Medical Imaging
Mohammad Dawood
Department of Computer Science
University of MünsterGermany
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Medical Imaging, SS-2010
Mohammad Dawood
Image Reconstruction
3
Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Law of Attenuation
4
Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Parallel projections of a plane
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Medical Imaging, SS-2010
Mohammad Dawood
x
y
r
s
θ
n
Reconstruction
Radon Transformation f
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Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Radon Transformation (Line Integrals at different angles)
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Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Radon Transformation
Original Sinogram (Radon Transform)
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Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Inverse Radon Transformation
H: Hilbert transform
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Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Filtered Back Projection
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Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Filtered Back Projection
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Medical Imaging, SS-2010
Mohammad Dawood
Projections
Backproject
Filter 1D
Filter 2D
Backproject
Image
Reconstruction
Filtered Back Projection
2D/3D filtering is costly
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Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Fourier Slice Theorem
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Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Fourier slice theorem
Take a two-dimensional function f(r), project it onto a line, and do a Fourier transform of that projection
Take that same function, but do a two-dimensional Fourier transform first, and then slice it through its origin parallel to the projection line
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Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Fourier Slice Theorem
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Medical Imaging, SS-2010
Mohammad Dawood
16
Medical Imaging, SS-2010
Mohammad Dawood
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Medical Imaging, SS-2010
Mohammad Dawood
1=Ram-Lak (ramp), 2=Shepp-Logan, 3=Cosine, and 4=Hamming
Reconstruction
FBP: Commonly used filters
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Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Iterative Reconstruction
b: measured valuesx: unknown attenuation coefficientsaij: weights
f1 f2 … fn
LOR1
LOR2
…
LORn
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Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Iterative Reconstruction
Kaczmarz Method (=ART: Algebraic Reconstruction Technique)
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Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Iterative Reconstruction
Kaczmarz Method (=ART: Algebraic Reconstruction Technique)
1. Start by setting x(0) = 0
2. Compute the forward projection from the n-th estimate, i.e. b(n) = A x(n)
3. Choose i and correct the current estimate x(n)
4. Iterate steps 2,3 until the difference between new forward projection b(n), computed in 2, and the old one is below tolerance
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Medical Imaging, SS-2010
Mohammad Dawood
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Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Iterative Reconstruction
EM (Expectation Maximization)
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Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Iterative Reconstruction
OSEM (Ordered Subset Expectation Maximization)
