Transcript Folie 1 - uni

Medical Imaging
Mohammad Dawood
Department of Computer Science
University of Münster
Germany
Medical Imaging, SS-2010
Image Reconstruction
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Medical Imaging, SS-2010
Reconstruction
Law of Attenuation
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Medical Imaging, SS-2010
Reconstruction
Parallel projections of a plane
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Medical Imaging, SS-2010
Reconstruction
y
s
Radon Transformation
f
n
r
θ
x
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Medical Imaging, SS-2010
Reconstruction
Radon Transformation (Line Integrals at different angles)
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Medical Imaging, SS-2010
Reconstruction
Radon Transformation
Original
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Sinogram (Radon Transform)
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Medical Imaging, SS-2010
Reconstruction
Inverse Radon Transformation
H: Hilbert transform
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Medical Imaging, SS-2010
Reconstruction
Filtered Back Projection
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Medical Imaging, SS-2010
Reconstruction
Filtered Back Projection
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Medical Imaging, SS-2010
Reconstruction
Filtered Back Projection
2D/3D filtering is costly
Backproject
Filter 2D
Projections
Image
Filter 1D
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Backproject
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Medical Imaging, SS-2010
Reconstruction
Fourier Slice Theorem
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Medical Imaging, SS-2010
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
Reconstruction
Fourier Slice Theorem
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Medical Imaging, SS-2010
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Medical Imaging, SS-2010
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Medical Imaging, SS-2010
Reconstruction
FBP: Commonly used filters
1=Ram-Lak (ramp), 2=Shepp-Logan,
3=Cosine, and 4=Hamming
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Medical Imaging, SS-2010
Reconstruction
Iterative Reconstruction
f1
b: measured values
x: unknown attenuation coefficients
aij: weights
f2
…
fn
LOR1
LOR2
…
LORn
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Medical Imaging, SS-2010
Reconstruction
Iterative Reconstruction
Kaczmarz Method (=ART: Algebraic Reconstruction Technique)
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Medical Imaging, SS-2010
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
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Medical Imaging, SS-2010
Reconstruction
Iterative Reconstruction
EM (Expectation Maximization)
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Medical Imaging, SS-2010
Reconstruction
Iterative Reconstruction
OSEM (Ordered Subset Expectation Maximization)
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