Rekonstruktionsmetoder inom PET

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Rekonstruktionsmetoder inom PET

• Anna Olsson
• Sjukhusfysiker
• Nuklearmedicin, Linköping
• anna.olsson_at_lio.se

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Tomografiska Rekonstruktionsmetoder

• Analytiska (t.ex. FBP)
• Iterativa
• Algebraiska
• Statistiska
• Poisson (t.ex.. ML-EM)
• Least squares

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Insamlingsdata 2D, 3D

• 3D ?2D (rebinnig)
• Fully 3D
• List mode

• Projektioner
• List mode

2D
3D
Timothy Turkington, Ph.D. Department of
Radiology, Duke University Medical Center,
Durham, North Carolina, USA
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Insamling och Rekonstruktion
Data Acquisition
Reconstructed Image
Sinogram (raw data)
Timothy Turkington, Ph.D. Department of
Radiology, Duke University Medical Center,
Durham, North Carolina, USA
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Analytiska metoder
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2D FBP
Zeng GL, Comput. Med. Imag. Graph. 2597-103,
2001.
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Convolution kernel
Kak AC, Slaney M, Principles of Computerized
Tomographic Imaging, 1988.
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The Ramp filter
Kak AC, Slaney M, Principles of Computerized
Tomographic Imaging, 1988.
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The Fourier slice theorem
Kak AC, Slaney M, Principles of Computerized
Tomographic Imaging, 1988.
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Frequency domain mapping
Kak AC, Slaney M, Principles of Computerized
Tomographic Imaging, 1988.
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The Ramp filter
Kak AC, Slaney M, Principles of Computerized
Tomographic Imaging, 1988.
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Reconstruction filter
Bendriem B, Townsend DW, The theory and practice
of 3D PET, 1998.
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2D filtered backprojection (2D-FBP)

• where p(?) projection data (2D), h1(?)
  kernel (1D), f(?) image (2D), ?
  projection angle, e?? unit vector
  perpendicular to direction of projection,
  ? convolution operator.

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Reconstruction demo
Zeng GL, Comput. Med. Imag. Graph. 2597-103,
2001.
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3D filtered backprojection (3D-FBP)

• where p(?) projection data (4D), h2(?)
  kernel (2D), f(?) image (3D), e?,? unit
  vector in direction of projection, ?
  convolution operator.

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Missing data
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3D FBP - The reprojectionalgorithm3DRP
Bendriem B, Townsend DW, The theory and
practice of 3D PET, 1998.
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Rebinning 3D?2D projection data
z
z
3D Recon
2D Recon
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Rebinning algorithms
Single slice rebinning
Fourier rebinning
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Fourier rebinning
Defrise M et al., IEEE TMI 16145-158, 1997.
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Iterativa metoder
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Byggstenar

• Data model
• Image model
• Cost function
• Iterative algorithm

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Byggstenar

• Data model
• Image model
• Cost function
• Iterative algorithm

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Byggstenar

• Data model
• Image model
• Cost function
• Iterative algorithm

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ML-EM Maximum likelihood expectation
maximization(Shepp Vardi 1982, Lange Carson
1984)
Uppmätta projektioner
BP av ( )
Normerad
Bild(k1)
Bild(k)


Projektioner av bilden(k)
Cost function poisson likelihood
Bra beskriven av Bruyant, JNM 2002
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OS-EM Ordered Subset Expectation Maximization
(Hudson Larkin, IEEE TMI 1994)
ML-EM
OS-EM
Sn data subset
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OSEM problem

• Subset imbalance
• Convergence problem

xn
xk
arg max f1(x)
x
arg max f2(x)
Fessler, IEEE MIC 2001
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Rescaled block iterative EMML (RBI - EMML)
(Byrne, IEEE TIP 1996)

• Removes subset imbalance.

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Row action maximum likelihood algorithm (RAMLA)
(Browne De Pierro, IEEE TMI 1996)

• where ?k is a relaxation parameter 0 lt ?k ?
  1 k ? ? ? ?k ? 0 ? ?k ?

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EM reconstruction
1 2
5
comparison with projections
10 15
20
50 100
150
comparison with actual object
Courtesy Brian Hutton, UCL London
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Regularisation

• Regularisation (noise reduction)
• stop early
• stopping criteria based on statistical hypothesis
  testing
• filtering
• post-reconstruction or between-iterations
• penalised cost function
• Bayesian methods
• Bayesian methods
• maximum a posteriori (MAP) or penalised methods
• Penalty function
• quadratic, Huber, median
• Structural information
• avoid smoothing across anatomical boundaries

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MAP
penalty function (quadratic)

• regularisation parameter (hyperparameter)
• Nj neighbourhood of voxel j
• ?l weight factors

Bra beskriven av Bruyant, JNM 2002
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Penalty functions
Fessler, IEEE MIC 2001
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Maximum a-posteriori (MAP)
Quadratic Huber
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Iterative algorithms

• Advantages
• physical model of measurement
• Normalization (detector efficiency)
• Randoms (measured)
• Scatter (model)
• Attenuation (transmission scan)
• statistical model of measurement (Poisson)
• a priori information (non-negativity)
• structural information (from CT/MRI)
• incomplete data set (limited angle, truncation)
• Disadvantages
• complexity (modelling)
• computation time
• non-linearity f(ab) ? f(a) f(b)
• Result depends on statistics and activity
  distribution!

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Vad har vi lärt oss idag?

• 2D FBP
• 3D FBP
• MLEM
• OSEM
• RBI EMML
• RAMLA
• MAP

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SLUT