Dynamic Imaging from Incomplete Data Applications in MRI and PET

Transcription

1 Dynamic Imaging from Incomplete Data Applications in MRI and PET Martin Benning, and many many others Department of Applied Mathematics & Theoretical Physics Challenges in Dynamic Imaging Isaac Newton Institute

2 Outline Applications in MRI and PET Spatial vs temporal resolution Variational regularisation methods Challenges in dynamic imaging Conclusions and outlook

3 Outline Applications in MRI and PET Spatial vs temporal resolution Variational regularisation methods Challenges in dynamic imaging Conclusions and outlook

4 Application: velocity-encoded magnetic resonance imaging (MRI) in chemical engineering Flowing water in tubes filled with glass beads (model application to understand fluid dynamics in reactors) z ~ 8 mm 40 mm Flowing water Steady flow (low Reynolds number) Velocity constant over time k y measured k-space (Fourier) data k x MB et al. "Phase reconstruction from velocity-encoded MRI measurements a survey of sparsitypromoting variational approaches." Journal of Magnetic Resonance 238 (2014):

5 Application: velocity-encoded magnetic resonance imaging (MRI) in chemical engineering Flowing water in tubes filled with glass beads (model application to understand fluid dynamics in reactors) z ~ 8 mm 40 mm Flowing water y z-velocity image Steady flow (low Reynolds number) Velocity constant over time x MB et al. "Phase reconstruction from velocity-encoded MRI measurements a survey of sparsitypromoting variational approaches." Journal of Magnetic Resonance 238 (2014):

6 Application: velocity-encoded magnetic resonance imaging (MRI) in chemical engineering y x z in-plane velocity: 14.4 cm s z-velocity (cm s -1 ) Goal: observe physics of flow phenomena z x Tayler et al. (2012) Phys. Rev. Lett. 108,

7 Rising bubbles z y x in-plane velocity: 26.7 cm s -1 Tayler et al. (2012) Phys. Rev. Lett. 108,

8 27 mm 17 mm Application: dynamic contrast enhanced (DCE) MRI Flow rate: 5 ml/min Flowing water Still water y 5 mm 27 mm x Resolution: 421 μm, 5 frames/min Dimension: 64 x 64 x 128 x 50 x y z # of frames 54 mm z Tracer FLASH Pulse-Sequence T E = 0.98 ms T R = 3.0 ms Data acquisition: D. Allen, J. Beech, V. Kersemans, S. Smart (Uni. Oxford), M. Mantle (Uni. Cambridge 5

9 Intensity Intensity Intensity 27 mm 27 mm DCE-MRI 54 mm 1 Flow Ground Truth Zero Filling Reconstruction 1 Steady Water Ground Truth Zero Filling Reconstruction 1 Ground Truth Zero Filling Reconstruction Transition Frames Frames Frames 6

10 DCE-MRI Real medical imaging data: Transverse view Coronal view Sagittal view Data acquisition: D. Allen, J. Beech, V. Kersemans, S. Smart (Uni. Oxford) 7

11 Application: Positron-Emission-Tomography (PET) Courtesy of wikimedia commons Courtesy of Frank Wübbeling, WWU Münster 8

12 Dynamic H 2 15 O PET Inputcurve h(t) th frame Time in seconds 26 frames ~ 6 minutes Coronal view Transverse view Sagittal view MB, Pia Heins and Martin Burger, "A solver for dynamic PET reconstructions based on forward-backward-splitting." Aip Conference Proceedings. Vol No

13 Outline Applications in MRI and PET Spatial vs temporal resolution Variational regularisation methods Challenges in dynamic imaging Conclusions and outlook

14 Spatial vs temporal resolution sampling = spatio/temporal function = spatial sampling function = temporal sampling function Example: PET 10

15 Spatial vs temporal resolution = spatio/temporal function = spatial sampling function = temporal sampling function 11

16 Spatial vs temporal resolution Sinogram Filtered Backprojection (FBP) Coarser temporal sampling usually leads to finer spatial resolution and better signalto-noise ratio (SNR) Not an option for dynamic imaging, as we lose important information about the dynamics of the system 11

17 Spatial vs temporal resolution Sinogram Filtered Backprojection (FBP) Coarser temporal sampling usually leads to finer spatial resolution and better signalto-noise ratio (SNR) Not an option for dynamic imaging, as we lose important information about the dynamics of the system but: incomplete spatial information 11

18 Outline Applications in MRI and PET Spatial vs temporal resolution Variational regularisation methods Challenges in dynamic imaging Conclusions and outlook

19 Variational regularisation methods Fidelity term Regularisation parameter Regularisation term Fidelity term examples Notation: 2-Norm data fidelity Kullback-Leibler data fidelity image sequence measurement data linear matrices Sub-sampled Fourier transform, Radon transform etc. 12

20 Variational regularisation methods Regularisation term examples Fidelity term Regularisation parameter Regularisation term Isotropic TV in space/anisotropic TV in time image sequence Joint sparsity Additive regularisation Infimal convolution regularisation Many many other possible regularisation energies! 13

21 Inexact alternating direction method of multipliers How to solve numerically? Many many options; inexact ADMM is one of them: Consider, for Inexact ADMM with Globally convergent Easy to implement Only point-wise operations and matrix multiplications Gabay 1983; Gabay, Mercier 1976; Glowinski, Le Tallec 1989; Arrow, Hurwicz, Uzawa 1958; Douglas, Rachford 1956; Zhang, Burger, Osher 2011; Esser, Zhang, Chan, 2010; Chambolle, Pock 2011; Goldstein, Osher 2009; Burger, Sawatzky, Steidl 2015; and many more 14

22 27 mm Example reconstruction DCE-MRI reconstruction, α = 0.7, 50 iterations 27 mm 54 mm 15

23 How to choose the regularisation parameter α? The regularisation parameter α controls the amount of regularity Many different approaches for choosing alpha: L-curve method, cross-validation, etc. Hansen, Per Christian. "Analysis of discrete ill-posed problems by means of the L-curve." SIAM review 34.4 (1992): Hansen, Per Christian, and Dianne Prost O'Leary. "The use of the L-curve in the regularization of discrete ill-posed problems." SIAM Journal on Scientific Computing 14.6 (1993):

24 Bregman iteration & discrepancy principle Alternative: Choose α large and solve for,, until the discrepancy principle is satisfied. Example: σ 2 variance of the noise S. Osher et al, SIAM Multiscale Model. Simul. 4 (2005), V. Morozov, Soviet Math. Dokldy 7 (1966),

25 Bregman iteration & discrepancy principle 18

26 Intensity Intensity Intensity 27 mm 27 mm Application to DCE-MRI 20 % Line-Subsampling 54 mm 1 Flow Ground Truth Zero Filling Reconstruction 1 Steady Water Ground Truth Zero Filling Reconstruction 1 Ground Truth Zero Filling Reconstruction Transition Frames Frames Frames 19

27 27 mm Application to DCE-MRI mm 54 mm 0 Ground Truth Zero Filling Bregman it. Reconstruction 20

28 Discrepancy principle quantitative comparison Different sampling schemes Quantitative Bregman iteration & discrepancy principle comparison for subsampled velocity-encoded MRI data Colour indicates the reconstruction quality in terms of a structure similarity index* Continuous white line shows violation of the discrepancy principle Dashed black line shows optimal SSIM value MB et al. "Phase reconstruction from velocity-encoded mri measurements a survey of sparsity-promoting variational approaches." JMR 238 (2014): * Z. Wang, et al., Image quality assessment: from error visibility to structural similarity, IEEE Trans. Image Processing 13 (2004) , 21

29 Outline Applications in MRI and PET Spatial vs temporal resolution Variational regularisation methods Challenges in dynamic imaging Conclusions and outlook

30 Challenges in dynamic imaging Big data: how to address very large datasets? Parallelisation (operations of each inexact ADMM iterate are highly parallelisable) Domain decomposition methods [Fornasier, Langer, Schönlieb 2010; Langer, Osher, Schönlieb, 2013] (Randomised) Coordinate descent methods [Nesterov 2012; Richtárik, Takáč, 2014; Fercoq, Richtárik, preprint; Qu, Richtárik, Zhang, preprint; ] Stochastic proximal methods and other Multi-modal imaging: how to combine dynamic MRI and PET (or other modalities)? Enhance dynamic reconstructions by combining data from different modalities Lots of research for the static case [Knoll, Kösters et al. 2014; Ehrhardt et al. 2015; ] 22

31 Challenges in dynamic imaging Big data: how to address very large datasets? Parallelisation (operations of each inexact ADMM iterate are highly parallelisable) Domain decomposition methods [Fornasier, Langer, Schönlieb 2010; Langer, Osher, Schönlieb, 2013] (Randomised) Coordinate descent methods [Nesterov 2012; Richtárik, Takáč, 2014; Fercoq, Richtárik, preprint; Qu, Richtárik, Zhang, preprint; ] Stochastic proximal methods and other Multi-modal imaging: how to combine dynamic MRI and PET (or other modalities)? Enhance dynamic reconstructions by combining data from different modalities Lots of research for the static case [Knoll, Kösters et al. 2014; Ehrhardt et al. 2015; ] 22

32 Challenges in dynamic imaging Non-linearity and non-convexity: how to extend variational regularisation to nonlinear operators or non-convex functions? Nonlinearities in MRI: velocity reconstruction, susceptibility artefact correction, integration of Bloch equations into reconstruction, etc. Nonlinearities in PET: nonlinear matrix factorisation, direct reconstruction of kinetic parameters, motion correction, etc. Plenty of current research: [Valkonen 2014; Zhao et al. 2012; Ochs et al. 2014; Bachmayr, Burger 2009; Möller, MB, Schönlieb, Cremers preprint;...] Learning: how to learn optimal parameters based on training data? Learn optimal regularisation parameters, transforms, inverse operators Static imaging: [Chung, Chung, O Leary 2011; De Los Reyes, Schönlieb 2013; Chen, Ranftl, Pock 2014; Chen, Yu, Pock 2015; ] 23

33 Challenges in dynamic imaging Non-linearity and non-convexity: how to extend variational regularisation to nonlinear operators or non-convex functions? Nonlinearities in MRI: velocity reconstruction, susceptibility artefact correction, integration of Bloch equations into reconstruction, etc. Nonlinearities in PET: nonlinear matrix factorisation, direct reconstruction of kinetic parameters, motion correction, etc. Plenty of current research: [Valkonen 2014; Zhao et al. 2012; Ochs et al. 2014; Bachmayr, Burger 2009; Möller, MB, Schönlieb, Cremers preprint;...] Learning: how to learn optimal parameters based on training data? Learn optimal regularisation parameters, transforms, inverse operators Static imaging: [Chung, Chung, O Leary 2011; De Los Reyes, Schönlieb 2013; Chen, Ranftl, Pock 2014; Chen, Yu, Pock 2015; ] 23

34 Outline Applications in MRI and PET Spatial vs temporal resolution Variational regularisation methods Challenges in dynamic imaging Conclusions and outlook

35 Conclusions & outlook Conclusions: we have presented numerous applications of dynamic imaging in MRI and PET discussed the issue of temporal vs spatial resolution considered variational regularisation to improve the temporal resolution while maintaining important spatial features discussed Bregman iteration for iterative contrast-enhancement Outlook: we should address the challenges mentioned on the previous slides (and other or newly emerging challenges) 24

36 Thank you for your attention Acknowledgement MRRC group, 2014 Cambridge Image Analysis group,