Bayesian Trajectory Optimization for Magnetic Resonance Imaging Sequences

Transcription

1 Bayesian Trajectory Optimization for Magnetic Resonance Imaging Sequences Matthias Seeger Saarland University and MPI for Informatics, Saarbrücken Joint work with Hannes Nickisch, Rolf Pohmann, Bernhard Schölkopf Max Planck Institute for Biological Cybernetics, Tübingen 26 February 2009 UNIVERSITÄT DES SAARLANDES Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

2 Sparse Linear Model y = X }{{} Design Signal {}}{ Noise {}}{ u + ε Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

3 Sparse Estimation Fixed X, y. û = argmax P(y u) u Sparsity Prior {}}{ P(u) Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

4 Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

5 Sparse Elimination Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

6 Sparse Elimination Improved Sensing Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

7 Improved Sensing Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

8 Improved Sensing Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

9 Improved Sensing Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

10 Improved Sensing Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

11 Improved Sensing Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

12 Design Optimization Needs Uncertainty Design Score How informative is X for reconstruction of u? Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

13 Design Optimization Needs Uncertainty Sparse Estimation Design Score How informative is X for reconstruction of u? Point estimate not enough Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

14 Design Optimization Needs Uncertainty Sparse Estimation Design Score How informative is X for reconstruction of u? Point estimate not enough Reconstruction uncertainty How good are you? How could you improve? Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

15 Design Optimization Needs Uncertainty Insufficient for Uncertainty Bayesian Posterior Uncertainty representation (from same input) P(u y ) P(y u)p(u) µ σ µ µ+σ Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

16 Sparse Estimation vs Sparse Inference Sparse Estimation Sparse Inference Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

17 Sparse Estimation vs Sparse Inference Sparse Estimation Sparse Inference Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

18 Sparse Estimation vs Sparse Inference Sparse Estimation Sparse Inference Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

19 Sparse Estimation vs Sparse Inference Sparse Estimation Sparse Inference Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

20 Sparse Estimation vs Sparse Inference Sparse Estimation Sparse Inference Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

21 Sparse Estimation vs Sparse Inference Sparse Estimation Sparse Inference Easily scaled up Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

22 Sparse Estimation vs Sparse Inference Sparse Estimation Sparse Inference Easily scaled up Uncertainty eliminated (no fix: it s just gone) Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

23 Sparse Estimation vs Sparse Inference Sparse Estimation Sparse Inference Easily scaled up Uncertainty eliminated (no fix: it s just gone) Design optimization, etc. Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

24 Sparse Estimation vs Sparse Inference Sparse Estimation Sparse Inference Easily scaled up Uncertainty eliminated (no fix: it s just gone) Design optimization, etc. No exact sparsity Harder to scale up Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

25 Sequential Optimization of MRI Trajectories Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

26 Sequential Optimization of MRI Trajectories Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

27 Sequential Optimization of MRI Trajectories Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

28 Sequential Optimization of MRI Trajectories Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

29 Sequential Optimization of MRI Trajectories Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

30 Large Scale Sparse Inference Variational relaxation 1 Bound potentials by Gaussians Convex criterion to minimize Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

31 Large Scale Sparse Inference Variational relaxation 1 Bound potentials by Gaussians Convex criterion to minimize 2 Decouple criterion (Fenchel duality) Scalable algorithm φ π Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

32 Large Scale Sparse Inference Variational relaxation 1 Bound potentials by Gaussians Convex criterion to minimize 2 Decouple criterion (Fenchel duality) Scalable algorithm Double loop (d.c.) inference algorithm Inner loops: Iteratively reweighted least squares ( smooth MAP ) φ π Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

33 Large Scale Sparse Inference Variational relaxation 1 Bound potentials by Gaussians Convex criterion to minimize 2 Decouple criterion (Fenchel duality) Scalable algorithm Double loop (d.c.) inference algorithm Inner loops: Iteratively reweighted least squares ( smooth MAP ) Once per outer loop (usually 2 6): Approximate covariance eigendecomposition (Lanczos) φ π Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

34 Large Scale Sparse Inference Variational relaxation 1 Bound potentials by Gaussians Convex criterion to minimize 2 Decouple criterion (Fenchel duality) Scalable algorithm Double loop (d.c.) inference algorithm Inner loops: Iteratively reweighted least squares ( smooth MAP ) Once per outer loop (usually 2 6): Approximate covariance eigendecomposition (Lanczos) Score computation needs Lanczos as well φ π Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

35 Large Scale Sparse Inference Variational relaxation 1 Bound potentials by Gaussians Convex criterion to minimize 2 Decouple criterion (Fenchel duality) Scalable algorithm Double loop (d.c.) inference algorithm Inner loops: Iteratively reweighted least squares ( smooth MAP ) Once per outer loop (usually 2 6): Approximate covariance eigendecomposition (Lanczos) Score computation needs Lanczos as well φ π Scales up without exact sparsity: Orders of magnitude faster than sparse inference previously. u R 65536, sparsity potentials [same model as Lustig et.al., MRM 07] Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

36 Spiral Trajectories Random: Err=31.90 Equi spaced: Err= Optimized: Err= k space k space k space log of L 2 Error 10 5 # arms Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

37 Spiral Trajectories Random: Err=24.76 Equi spaced: Err= Optimized: Err= k space k space k space log of L 2 Error 10 5 # arms Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

38 Spiral Trajectories Random: Err=23.21 Equi spaced: Err= Optimized: Err= k space k space k space log of L 2 Error 10 5 # arms Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

39 Spiral Trajectories Random: Err=17.59 Equi spaced: Err= Optimized: Err= k space k space k space log of L 2 Error 10 5 # arms Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

40 Spiral Trajectories Random: Err=13.38 Equi spaced: Err= Optimized: Err= k space k space k space log of L 2 Error 10 5 # arms Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

41 Spiral Trajectories Random: Err=13.39 Equi spaced: Err= Optimized: Err= k space k space k space log of L 2 Error 10 5 # arms Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

42 Does It Generalize? L 2 reconstruction error rd eq op Other slices (sagittal, TE 88ms) N shot, Number of spiral arms Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

43 Does It Generalize? L 2 reconstruction error rd 25 eq 20 op N shot, Number of spiral arms L 2 reconstruction error rd eq op Other subjects (sagittal, TE 92ms) N shot, Number of spiral arms Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

44 Does It Generalize? rd 30 rd 25 eq 25 eq 20 op 20 op L 2 reconstruction error N shot, Number of spiral arms L 2 reconstruction error N shot, Number of spiral arms L 2 reconstruction error rd eq op TE 11ms, sagittal (other subjects) N shot, Number of spiral arms Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

45 Does It Generalize? rd 30 rd 25 eq 25 eq 20 op 20 op rd eq op L 2 reconstruction error L 2 reconstruction error L 2 reconstruction error N shot, Number of spiral arms N shot, Number of spiral arms N shot, Number of spiral arms L 2 reconstruction error rd eq op Axial, other subjects (TE 92ms) N shot, Number of spiral arms Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

46 Does It Generalize? rd eq op rd eq op rd eq op rd eq op L 2 reconstruction error L 2 reconstruction error L 2 reconstruction error L 2 reconstruction error N shot, Number of spiral arms N shot, Number of spiral arms N shot, Number of spiral arms N shot, Number of spiral arms L 2 reconstruction error rd eq op TE 11ms, axial (other subjects) N shot, Number of spiral arms Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

47 What s Wrong with Random? Optimized on Tested on Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

48 Conclusions Real-world : Largest improvements through design optimization Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

49 Conclusions Real-world : Largest improvements through design optimization Design optimization needs uncertainty (not point) estimation: Sparse scalable variational inference Related to MAP (convex optimization, scalable algorithms) Different from MAP (uncertainty, not elimination) Little attention from theorists so far Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

50 Conclusions Real-world : Largest improvements through design optimization Design optimization needs uncertainty (not point) estimation: Sparse scalable variational inference Related to MAP (convex optimization, scalable algorithms) Different from MAP (uncertainty, not elimination) Little attention from theorists so far and CS Non-standard trajectories for rapid scanning Delicate application (fine details crucial) Sparse, but not too sparse Seeger (MMCI) Bayesian MRI Optimization 26 February / 10

51 Conclusions Real-world : Largest improvements through design optimization Design optimization needs uncertainty (not point) estimation: Sparse scalable variational inference Related to MAP (convex optimization, scalable algorithms) Different from MAP (uncertainty, not elimination) Little attention from theorists so far and CS Non-standard trajectories for rapid scanning Delicate application (fine details crucial) Sparse, but not too sparse Future work Design across multiple slices Parallel imaging (multiple sources) Dynamic MRI Seeger (MMCI) Bayesian MRI Optimization 26 February / 10