1. 3
Bayes' theorem 2 … 1 1 2 3
Partial discreteness: a new type of prior knowledge for MRI reconstruction
Gabriel Ramos-Llordén1, Hilde Segers1, Willem J. Palenstijn1,
Arnold J. den Dekker1,2 and Jan Sijbers1
1iMinds-Vision Lab, University of Antwerp, Antwerp, Belgium.
2Delft Center for Systems and Control, Delft University of Technology, Delft, The Netherlands.

2. Introduction Some regions are approximately constant in intensity
Breast implant
Dental MRI
FLAIR sequences
Angiography
Some regions are approximately constant in intensity
Partial discrete images: piece-wise constant part + texture part
Partial discreteness as a prior for ill-posed reconstruction problems 1 2 3 4
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3. Partial discreteness model
test iy
𝒙 𝑑𝑖𝑠 = 𝑒 𝑖𝚽 ∘ 𝑘∈{1,2} 𝟏 𝒜 𝑘 𝜌 𝑘 + 𝒙 𝒜
𝒜= 𝑘∈{1,2} 𝒜 𝑘
𝜌 1
𝑐𝑎𝑟𝑑 𝒜 𝑘 ≫0
𝟏 𝒜 1 + Σ +
𝜌 2 +
𝟏 𝒜 2
𝑒 𝑖𝚽
Phase 5
𝒙 𝒜
Variant intensity class
Here another image
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4. Partial discreteness model
test iy
𝒙 𝑑𝑖𝑠 = 𝑒 𝑖𝚽 ∘ 𝑘∈{1,2} 𝟏 𝒜 𝑘 𝜌 𝑘 + 𝒙 𝒜
𝒜= 𝑘∈{1,2} 𝒜 𝑘
𝜌 1
𝑐𝑎𝑟𝑑 𝒜 𝑘 ≫0
𝟏 𝒜 1 + Σ +
𝜌 2 +
𝟏 𝒜 2
𝑒 𝑖𝚽
Phase 6
𝒙 𝒜
Variant intensity class
Here another image
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5. Penalized iterative reconstruction
image
Regularization parameter
k-space data
𝒙 (𝑡+1) =arg min 𝒙 𝒚−𝑨𝒙 2 +𝜆 𝑾 𝑡 𝒙− 𝒗 𝑡
Fourier matrix
Discreteness error
𝒗 𝑡 : iteratively estimated partial discrete image
𝑾 𝑡 : spatially-variant weight diagonal matrix
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6. Bayesian segmentation operator
Classification
𝒙 𝑑𝑖𝑠 (𝑡)
𝒎 𝑘 = 𝟏 𝒜 𝑘 ∘𝒙 (𝑡)
𝟏 𝒜 𝑘
𝑘=1,…,𝐾
Otsu’s method
𝒙 𝑝𝑟𝑜𝑏
Bayesian segmentation operator
test iy
Past characterization
|𝒙 (𝑡) | 2
Bayes' theorem … 3 1 1 3 2
K-Gaussian mixture
model fitting
[Caballero J., MICCAI 2014]
A posteriori probability maps
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7. Bayesian segmentation operator
Classification
𝒙 𝑑𝑖𝑠 (𝑡)
𝒎 𝑘 = 𝟏 𝒜 𝑘 ∘𝒙 (𝑡)
𝟏 𝒜 𝑘
𝑘=1,…,𝐾
Otsu’s method
𝒙 𝑝𝑟𝑜𝑏
Bayesian segmentation operator
test iy
Past characterization
|𝒙 (𝑡) | 2
Temporal regularization
𝜌 2 … 3 1 1 3 2
𝜌 1
𝑝 1 (𝑡)
𝑝 3 (𝑡)
𝑝 2 (𝑡)
𝒙 𝑝𝑟𝑜𝑏 = 𝜌 1 𝑝 1 (𝑡) + 𝜌 2 𝑝 2 (𝑡) + 𝑝 3 (𝑡) ∘| 𝒙 𝑡 |
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8. Bayesian segmentation operator
test iy
Estimated partially discrete image
Otsu thresholding
𝒜 2 𝒜
𝒙 𝑝𝑟𝑜𝑏
𝒙 𝒜 = 𝟏 𝒜 ∘𝒙 𝑝𝑟𝑜𝑏
𝒜 1
𝒗 (𝑡) = 𝑒 𝑖𝚽 ∘ 𝑘∈{1,2} 𝟏 𝒜 𝑘 𝜌 𝑘 + 𝒙 𝒜
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9. Bayesian segmentation operator
test iy
Discreteness error:
𝑾 𝑡 𝒙− 𝒗 𝑡
with 𝑾 𝑡 =diag( 𝒘 𝑡 )
Weights 𝒘 𝑡 determine where the discreteness error is considered
𝑝 1 (𝑡)
𝑝 2 (𝑡)
𝑝 3 (𝑡)
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10. Experiments Simulations with breast implant and angiography data
Single coil radial k-space sampling with varying
number of spokes,
Smoothly varying phase added
Comparison against Conjugate Gradient (CG) with smoothness prior and Total Variation (TV)
𝑘 𝑦
𝑘 𝑥
𝑁 𝑠𝑝𝑜𝑘𝑒𝑠
[Gai.J. et al. (Impatient Toolbox), ISMRM 2012]
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11. Results Breast implant experiment SNR=100 𝑵 𝒔𝒑𝒐𝒌𝒆𝒔 =𝟑𝟓
Recovered images and implant contour detection
SNR=100, 𝑵 𝒔𝒑𝒐𝒌𝒆𝒔 =𝟑𝟓
(a) CG + smoothness (b) CG + TV (c) Proposed
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12. Results
Breast implant experiment: segmentation metrics
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13. Results Angiography experiment
SNR=100, 𝑁 𝑠𝑝𝑜𝑘𝑒𝑠 =55
(a)Original (b)CG + smoothness (c)CG+TV (d)Proposed 
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14. Thanks for your attention!
Conclusions
Partial discreteness prior
More detailed reconstructed images
Segmentation benefits from partial discreteness
Thanks for your attention!
Contact:
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15. Image references Radiopedia.org http://www.drbicuspid.com/ 5.
options-and-upgrades/clinical-applications/advanced-angio
6. M Maijers, PhD Thesis