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6
Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Radon Transformation (Line Integrals at different angles)
7
Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Radon Transformation
Original Sinogram (Radon Transform)
8
Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Inverse Radon Transformation
H: Hilbert transform
11
Medical Imaging, SS-2010
Mohammad Dawood
Projections
Backproject
Filter 1D
Filter 2D
Backproject
Image
Reconstruction
Filtered Back Projection
2D/3D filtering is costly
13
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
17
Medical Imaging, SS-2010
Mohammad Dawood
1=Ram-Lak (ramp), 2=Shepp-Logan, 3=Cosine, and 4=Hamming
Reconstruction
FBP: Commonly used filters
18
Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Iterative Reconstruction
b: measured valuesx: unknown attenuation coefficientsaij: weights
f1 f2 … fn
LOR1
LOR2
…
LORn
19
Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Iterative Reconstruction
Kaczmarz Method (=ART: Algebraic Reconstruction Technique)
20
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
22
Medical Imaging, SS-2010
Mohammad Dawood
Reconstruction
Iterative Reconstruction
EM (Expectation Maximization)