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.

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 1/12

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 2/12

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 3/12

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 4/12

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 5/12

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 | 5/12

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 6/12

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) 7/12

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] 8/12

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   9/12

Results Breast implant experiment: segmentation metrics 10/12

Results Angiography experiment SNR=100, 𝑁 𝑠𝑝𝑜𝑘𝑒𝑠 =55   (a)Original (b)CG + smoothness (c)CG+TV (d)Proposed    11/12

Thanks for your attention! Conclusions Partial discreteness prior More detailed reconstructed images Segmentation benefits from partial discreteness Thanks for your attention! Contact: http://visielab.uantwerpen.be/people/gabriel-ramos-llorden 12/12

Image references Radiopedia.org http://www.drbicuspid.com/ www.reviewofoptometry.com https://www.healthcare.siemens.com/ 5. https://www.healthcare.siemens.com/magnetic-resonance-imaging/ options-and-upgrades/clinical-applications/advanced-angio 6. M Maijers, PhD Thesis