Medical Imaging

Mohammad Dawood

Department of Computer Science

University of MünsterGermany

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Medical Imaging, SS-2010

Mohammad Dawood

Image Reconstruction

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Medical Imaging, SS-2010

Mohammad Dawood

Reconstruction

Law of Attenuation

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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

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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)