Combined Image Reconstruction for Combined PET-MR Imaging Matthias J. Ehrhardt University of Cambridge, UK with: Arridge, Atkinson, Barnes, Duncan, Hutton, Liljeroth, Markiewicz Ourselin, Pizarro, Thielemans (UCL, London) Kolehmainen (Kuopio, Finland) m.j.ehrhardt@damtp.cam.ac.uk
Positron Emission Tomography and Magnetic Resonance Imaging m.j.ehrhardt@damtp.cam.ac.uk
Positron Emission Tomography (PET) m.j.ehrhardt@damtp.cam.ac.uk
Positron Emission Tomography (PET) m.j.ehrhardt@damtp.cam.ac.uk
Positron Emission Tomography (PET) m.j.ehrhardt@damtp.cam.ac.uk
Positron Emission Tomography (PET) m.j.ehrhardt@damtp.cam.ac.uk
Magnetic Resonance Imaging (MRI) m.j.ehrhardt@damtp.cam.ac.uk
Magnetic Resonance Imaging (MRI) m.j.ehrhardt@damtp.cam.ac.uk
Magnetic Resonance Imaging (MRI) m.j.ehrhardt@damtp.cam.ac.uk
Magnetic Resonance Imaging (MRI) m.j.ehrhardt@damtp.cam.ac.uk
Magnetic Resonance Imaging (MRI) m.j.ehrhardt@damtp.cam.ac.uk
Combined PET-MR Imaging m.j.ehrhardt@damtp.cam.ac.uk
Combined PET-MR Imaging m.j.ehrhardt@damtp.cam.ac.uk
Combined PET-MR Imaging m.j.ehrhardt@damtp.cam.ac.uk
Part I: Utilizing Resolution of MRI Part II: Joint PET-MRI Reconstruction m.j.ehrhardt@damtp.cam.ac.uk
Part I: Utilizing Resolution of MRI Part II: Joint PET-MRI Reconstruction m.j.ehrhardt@damtp.cam.ac.uk
PET Reconstruction ? PET data MLEM TV MRI m.j.ehrhardt@damtp.cam.ac.uk
MAP reconstruction and Total Variation MAP reconstruction u ∗ ∈ argmin � � L ( Au + r , b ) + α R ( u ) u ◮ total variation Rudin, Osher, Fatemi 1992 � R ( u ) = TV( u ) = |∇ u | Ω � β 2 + |∇ u | 2 � 1 / 2 � R ( u ) = TV β ( u ) = Ω edge-preserved reconstruction m.j.ehrhardt@damtp.cam.ac.uk
MAP reconstruction and Anatomical Information MAP reconstruction with Anatomical Information u ∗ ∈ argmin � � L ( Pu + r , b ) + α R ( u | v ) u We want 1) Convexity: R ( u | v ) should be convex in u 2) No Segmentation: should not need a segmentation of v 3 ∗ ) Total Variation: R ( u | v = const) = TV( u ) PET with MRI/CT: Leahy and Yan 1991 , Baete et al 2004 , Pedemonte et al 2011 , Bowsher et al 2004 , Kazantsev et al 2014 , Nuyts 2007 , Somayayula et al 2005 2011 , Tang and Rahmim 2009 2015 (Mutual information/ Entropy), Jiao et al 2015 EIT with CT: Kaipio et al 1999 Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Parallel Level Set Prior Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Parallel Level Set Prior Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Parallel Level Set Prior Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Parallel Level Set Prior Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Parallel Level Set Prior Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Parallel Level Set Prior �∇ u , ∇ v � = cos( θ ) |∇ u ||∇ v | Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Parallel Level Set Prior �∇ u , ∇ v � = cos( θ ) |∇ u ||∇ v | Measure Similar Structures |∇ u | 2 − �∇ u , ∇ v / |∇ v |� 2 � 1 / 2 � S ( u ) := Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Parallel Level Set Prior �∇ u , ∇ v � = cos( θ ) |∇ u ||∇ v | Measure Similar Structures |∇ u | 2 − �∇ u , ξ � 2 � 1 / 2 � S ( u ) := |∇ v | 2 + η 2 , ◮ ξ := ∇ v / |∇ v | η , � |∇ v | η := η > 0 Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Parallel Level Set Prior �∇ u , ∇ v � = cos( θ ) |∇ u ||∇ v | Measure Similar Structures |∇ u | 2 − �∇ u , ξ � 2 � 1 / 2 � S ( u ) := |∇ v | 2 + η 2 , ◮ ξ := ∇ v / |∇ v | η , � |∇ v | η := η > 0 ◮ 0 ≤ S ( u ) ≤ |∇ u | ◮ S ( u ) = 0 ⇔ u ∼ v ( ∇ u � ∇ v ) Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Asymmetric Parallel Level Sets |∇ u | 2 − �∇ u , ξ � 2 � 1 / 2 � S ( u ) := Asymmetric Parallel Level Sets Prior � β 2 + |∇ u | 2 − �∇ u , ξ � 2 � 1 / 2 � P ( u | v ) := , β > 0 Ω This is convex, does not need a segmentation and reduces to total variation. Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Numerical Results m.j.ehrhardt@damtp.cam.ac.uk
Other Methods for Anatomical Information � β 2 + |∇ u | 2 + γ |∇ v | 2 � 1 / 2 � TV J ( u | v ) := , γ > 0 Ω Sapiro and Ringach IEEE TIP 1996 ; Haber and Holtzman-Gazit Surveys in Geophysics 2013 ; Ehrhardt et al Inv Probl 2015, Lu et al Phys Med Bio 2015 B ( u | v ) := 1 � � ω i , j ( v )( u i − u j ) 2 , k ∈ N 2 i j ∈ N ( i ) Bowsher et al IEEE NSS-MIC 2004 � β 2 + |∇ u | 2 � 1 / 2 � D ( u | v ) := − �∇ u , ξ � Ω Kazantsev et al Sensing and Imaging 2014 K ( u | v ) := 1 � |∇ u | 2 − �∇ u , ξ � 2 2 Ω Kaipio et al Inv Prob 1999 m.j.ehrhardt@damtp.cam.ac.uk
Summary of Methods TV J B D K P reduces to total variation ✓ ✗ ✓ ✗ ✓ edge location dependent ✓ ✓ ✓ ✓ ✓ edge orientation dependent ✗ ✗ ✓ ✓ ✓ allows negative edge correlation - - ✗ ✓ ✓ Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Data a) b) c) m.j.ehrhardt@damtp.cam.ac.uk
Software Phantom: Quantitative Results whole phantom whole phantom 50 relative ℓ 2 -error [%] SSIM [%] 80 40 30 70 grey matter right hot lesion relative ℓ 2 -error [%] relative ℓ 2 -error [%] 40 40 30 30 20 20 low high low high regularization P † TV TV B D K MLEM J m.j.ehrhardt@damtp.cam.ac.uk
Software Phantom: Normal Recon v Anatomical Prior PET gr. truth MRI MLEM TV P Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Software Phantom: Compare Anatomical Priors PET gr. truth MRI TV J B D K P Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Software Phantom: Close-Ups TV B D K P † MRI side info PET ground truth J Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Software Phantom: Bias vs SD whole phantom grey matter reg → 0 0.03 0.09 reg → ∞ 0.02 0.06 0.01 0.03 mean standard deviation 0 0 0 0.01 0.02 0.03 0 0.03 0.06 0.09 white matter lesions 0.09 0.39 right hot 0.06 0.26 cold left hot 0.03 0.13 0 0 0 0.03 0.06 0.09 0 0.13 0.26 0.39 mean absolute bias TV TV B D K P † MLEM J Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Software Phantom: Bias vs SD MLEM TV TV B D K P † J ≥ 0.5 bias ≤ -0.5 standard deviation ≥ 0.25 0 Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Software Phantom: Bias vs SD P † MLEM TV K ≥ 0.5 bias ≤ -0.5 standard deviation ≥ 0.25 0 Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Hardware Phantom: Compare Anatomical Priors MRI TV J B D K P Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Hardware Phantom: Close-Ups P † MLEM TV TV B D K MRI side info J Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Patient Data: Normal Recon v Anatomical Prior MRI MLEM TV P Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Patient Data: Compare Anatomical Priors MRI TV J B D K P Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Patient Data: Close-Ups P † MLEM TV TV B D K MRI side info J Ehrhardt et al 2016 (under review) m.j.ehrhardt@damtp.cam.ac.uk
Conclusions of Part I ◮ new prior that can incorporate anatomical structure ◮ convex, no segmentation and reduces to total variation ◮ based on directions, not only magnitude ◮ handles arbitrary intensities, no need for positive correlation ◮ better in quality measures ( ℓ 2 -error, SSIM, bias-vs-SD trade-off) ◮ reduces bias of total variation (similar to Bregman iterations) MLEM P m.j.ehrhardt@damtp.cam.ac.uk
Part I: Utilizing Resolution of MRI Part II: Joint PET-MRI Reconstruction m.j.ehrhardt@damtp.cam.ac.uk
Data Acquisition in MRI ◮ sequential sampling of Fourier coefficients ◮ less data ⇒ shorter acquisition time ⇒ motion, patient comfort, money ◮ higher spatial resolution m.j.ehrhardt@damtp.cam.ac.uk
Joint Reconstruction m.j.ehrhardt@damtp.cam.ac.uk
Joint Reconstruction ? Ehrhardt et al Inverse Problems 2015 m.j.ehrhardt@damtp.cam.ac.uk
Joint Reconstruction ? Ehrhardt et al Inverse Problems 2015 m.j.ehrhardt@damtp.cam.ac.uk
Joint Reconstruction ? ? Ehrhardt et al Inverse Problems 2015 m.j.ehrhardt@damtp.cam.ac.uk
Joint Reconstruction ? ? Ehrhardt et al Inverse Problems 2015 m.j.ehrhardt@damtp.cam.ac.uk
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