Precision Learning: Reconstruction Filter Kernel ...€¦ · Results: Quantitative Evaluation...
Transcript of Precision Learning: Reconstruction Filter Kernel ...€¦ · Results: Quantitative Evaluation...
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Precision Learning: Reconstruction Filter KernelDiscretization
Christopher SybenPattern Recognition Lab, Friedrich-Alexander University of Erlangen-NürnbergSeptember 28, 2018
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Reconstruction Pipeline as a Neural Network
Already proposed• Filtered back-projection (FBP) algorithm as Neural Network1
• Compensation weights to reduce limited angle artifacts2
Benefits• data-driven knowledge-enhancing abilities3
• allows to exchange heuristically method
Question
→ Can we learn the reconstruction filter ?
1Tobias W\"urfl, Florin Cristian Ghesu, Vincent Christlein, and Andreas Maier, "Deep Learning Computed Tomography",in MICCAI 2016: 19th International Conference, Proceedings, Part III, 2016, vol. 3, pp. 432-440..
2Kerstin Hammernik, Tobias W\"urfl, Thomas Pock, and Andreas Maier,"A deep learning architecture for limited-angle computed tomography reconstruction", in BVM 2017 Heidelberg, 2017, pp. 92-97,Springer Berlin Heidelberg.
3Ge Wang, "A perspective on deep imaging", IEEE Access, vol.4, pp. 8914-8924, 2016.
Christopher Syben | Pattern Recognition Lab | PL: Ramp Filter Discretization September 28, 2018 1
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Recap: CT Reconstruction
Sinogram Filtered sinogram Reconstruction
p(s; θ) q(s; θ)h(s) ∗ p(s; θ) f (x; y) =
Rq(s; θ)dθ
where h(s) is the Ramp-Filter
Christopher Syben | Pattern Recognition Lab | PL: Ramp Filter Discretization September 28, 2018 2
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Cupping Artifacts
Line profile through phantom. Line profile through Ramp-Reco
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Deriving the Network Topology
Discrete reconstruction problem:
Ax = p
x = A>︸︷︷︸Back-projection
(AA>)−1︸ ︷︷ ︸Filter
p
substituting the inverse:x = A>FHKFp
where
A is the system matrix
x is the object
p is the sinogram
F,FH is the Fourier and inverse Fourier-transform
K is the filter in Fourier domain
Christopher Syben | Pattern Recognition Lab | PL: Ramp Filter Discretization September 28, 2018 4
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Deriving the Network Topology
Objective function:
f(K) =1
2‖A>FHKFp−x‖2
2
Derivative:∂ f(K)
∂ K= FA(ATFHKFp−x)(Fp)>
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Deriving the Network Topology
Objective function:
f(K) =1
2‖A>FHKFp−x‖2
2
Derivative:∂ f(K)
∂ K= FA(ATFHKFp−x)(Fp)>
Christopher Syben | Pattern Recognition Lab | PL: Ramp Filter Discretization September 28, 2018 5
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Deriving the Network Topology
Objective function:
f(K) =1
2‖A>FHKFp−x‖2
2
Derivative:∂ f(K)
∂ K= FA(ATFHKFp−x)︸ ︷︷ ︸
Error︸ ︷︷ ︸Back-propagation
( Fp︸︷︷︸l−1
)>
Reconstruction
x̂F K FH A>
FC layer weighting layer FC layer FC layer
p
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Experimental Setup
• K is initialized with the Ramp• For training 10 numerical disc phantoms (increasing radii)• Evaluation on real CT-dataset
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Results: Phantoms
Line profile through GT. Line profile throughRam-Lak-reco.
Line profile throughLearned-reco.
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Results: CT data
Reconstruction with Ram-Lakfilter.
Reconstruction with learnedfilter.
Christopher Syben | Pattern Recognition Lab | PL: Ramp Filter Discretization September 28, 2018 9
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Results: Quantitative Evaluation
Phantom data (absolute difference):
mean std. dev. min maxRamp-reco 0.235 0.07 0.001 0.596Ram-Lak-reco 0.01 0.031 0 0.41Learned-reco 0.023 0.03 6.76E-09 0.409
CT data:
mean std. dev. min maxRam-Lak-reco 66.99 61.401 6.10E-5 1634.82Learned-reco 83.53 68.06 8.39E-5 1685.70
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Conclusion
Outlook:• Apply noise models to the training data• Setup a complete CT Reconstruction pipeline
Take home message:• Derive the network topology from the continuous analytical problem
description• Neural network intrinsically compensate for discretization errors• Interesting link between neural network techniques and signal processing
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Thanks for listening.Any questions?