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Singular value decomposition analysis of back projection operator of maximum likelihood expectation maximization PET image reconstruction

Vencel Somai

Vencel Somai

Institute of Nuclear Techniques, Budapest University of Technology and EconomicsBudapest, Hungary
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Somai, Vencel
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David Legrady

David Legrady

Institute of Nuclear Techniques, Budapest University of Technology and EconomicsBudapest, Hungary
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Legrady, David
 and 
Gabor Tolnai

Gabor Tolnai

Institute of Nuclear Techniques, Budapest University of Technology and EconomicsBudapest, Hungary
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Tolnai, Gabor
  
Mar 24, 2018
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Article Category: Research Article
Published Online: Mar 24, 2018
Page range: 337 - 345
Received: Aug 28, 2017
Accepted: Feb 22, 2018
DOI: https://doi.org/10.2478/raon-2018-0013
Keywords
© 2018 Vencel Somai, David Legrady, Gabor Tolnai, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Figure 1

Top view of the reconstruction of the cylinder-ring mathematical phantom of the full system with faithful modelling in the back projection. Light grey area represents water, dark grey area represents bone material. Underestimated activity and increased full width at half maximum (FWHM) / full width at tenth maximum (FWTM) can be seen for voxels located in water. FWHM and FWTM are calculated along the ring. Red line indicates the phantom ideal FWHM.
Top view of the reconstruction of the cylinder-ring mathematical phantom of the full system with faithful modelling in the back projection. Light grey area represents water, dark grey area represents bone material. Underestimated activity and increased full width at half maximum (FWHM) / full width at tenth maximum (FWTM) can be seen for voxels located in water. FWHM and FWTM are calculated along the ring. Red line indicates the phantom ideal FWHM.

Figure 2

Top view of the reconstruction of the cylinder-ring mathematical phantom of the full system with simplified modelling in the back projection. Homogeneous activity estimate and full width at half maximum (FWHM) can be seen along the ring, neglect of positron range in theback projection abolished the artefact of Figure 1 and phantom ideal FWHM is reached.
Top view of the reconstruction of the cylinder-ring mathematical phantom of the full system with simplified modelling in the back projection. Homogeneous activity estimate and full width at half maximum (FWHM) can be seen along the ring, neglect of positron range in theback projection abolished the artefact of Figure 1 and phantom ideal FWHM is reached.

Figure 3

Mathematical phantom and system geometry for 1D model. 1-128 voxels are located in bone material, 129–256 voxels are located in water.
Mathematical phantom and system geometry for 1D model. 1-128 voxels are located in bone material, 129–256 voxels are located in water.

Figure 4

Singular values of the system matrix for positron range neglecting and modelling case. Increased values for the former imply the faster convergence of the corresponding (first 133) basis components of the activity estimate.
Singular values of the system matrix for positron range neglecting and modelling case. Increased values for the former imply the faster convergence of the corresponding (first 133) basis components of the activity estimate.

Figure 5

One of the voxel space singular vectors of the system matrices corresponding to positron range neglect (left – back projection posrange OFF) and positron range modelling (right – back projection posrange ON). Back projection posrange OFF reflects only the symmetries of the geometry. Back projection posrange ON accounts for the material map as well, increased position uncertainty due to longer average positron free path implies smaller space-frequency in water area.
One of the voxel space singular vectors of the system matrices corresponding to positron range neglect (left – back projection posrange OFF) and positron range modelling (right – back projection posrange ON). Back projection posrange OFF reflects only the symmetries of the geometry. Back projection posrange ON accounts for the material map as well, increased position uncertainty due to longer average positron free path implies smaller space-frequency in water area.

Figure 6

Absolute value of the spectral coefficients of the measurement-forward projection Hadamard ratio in the sinogram basis corresponding to positron range neglect (left – back projection posrange OFF) and positron range modelling (right – back projection posrange ON). Faster decay means less information gathered as the coefficients of the horizontal plateau are corrupted by noise thus it represents an error level estimate. Due to one to one correspondence property of SVD between sinogram and voxel space singular (basis) vectors these basis coefficients of the activity are not hoped to be correctly estimated
Absolute value of the spectral coefficients of the measurement-forward projection Hadamard ratio in the sinogram basis corresponding to positron range neglect (left – back projection posrange OFF) and positron range modelling (right – back projection posrange ON). Faster decay means less information gathered as the coefficients of the horizontal plateau are corrupted by noise thus it represents an error level estimate. Due to one to one correspondence property of SVD between sinogram and voxel space singular (basis) vectors these basis coefficients of the activity are not hoped to be correctly estimated

Figure 7

The L2-norm of the difference between the activity distribution and the current estimate after a given number of iterations. Smaller value means better agreement. Subfigure on the left shows the result of the noiseless test case where the convergence of back projection posrange ON setting to the exact solution and the convergence of back projection posrange OFF setting to an other fix point is presented. Subfigure on the right shows the result of a simulated reconstruction with 106 positron used for measurement generation and in both forward and back projection Monte Carlo simulations. After slower initial convergence back projection posrange ON reaches much better activity estimate. Back projection posrange OFF converges to a different fix point similarly to noiseless case.
The L2-norm of the difference between the activity distribution and the current estimate after a given number of iterations. Smaller value means better agreement. Subfigure on the left shows the result of the noiseless test case where the convergence of back projection posrange ON setting to the exact solution and the convergence of back projection posrange OFF setting to an other fix point is presented. Subfigure on the right shows the result of a simulated reconstruction with 106 positron used for measurement generation and in both forward and back projection Monte Carlo simulations. After slower initial convergence back projection posrange ON reaches much better activity estimate. Back projection posrange OFF converges to a different fix point similarly to noiseless case.

Figure 8

The L2-norm of the difference between the activity distribution and the current estimate after a given number of iterations. Smaller value means better agreement. Reconstruction with SVD filter outperforms the best setting so far in terms of faster initial convergence and the farther starting point of increasing discrepancy due to semi-convergence. Also the faster initial convergence of positron range neglecting back projection can be seen compared to positron range modelling back projection.
The L2-norm of the difference between the activity distribution and the current estimate after a given number of iterations. Smaller value means better agreement. Reconstruction with SVD filter outperforms the best setting so far in terms of faster initial convergence and the farther starting point of increasing discrepancy due to semi-convergence. Also the faster initial convergence of positron range neglecting back projection can be seen compared to positron range modelling back projection.
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