Multilevel Optimization for Multi-Modal X-ray Data Analysis

Transcription

1 Multilevel Optimization for Multi-Modal X-ray Data Analysis Zichao (Wendy) Di Mathematics & Computer Science Division Argonne National Laboratory May 25, 2016

2 2 / 35 Outline Multi-Modality Imaging Example: X-ray Tomography Forward Model Optimization Algorithms Numerical Results Multilevel-Based Acceleration Introduction of MG/OPT Application of Multilevel in Tomographic Reconstruction Numerical Results Summary and Outlook

3 3 / 35 Outline Multi-Modality Imaging Example: X-ray Tomography Forward Model Optimization Algorithms Numerical Results Multilevel-Based Acceleration Introduction of MG/OPT Application of Multilevel in Tomographic Reconstruction Numerical Results Summary and Outlook

4 More is Better 1 CT:hard-tissue; T2(MRI):soft tissue; PET: functional characteristics. 1 Boss, A. and et al. (2010). Hybrid PET/MRI of Intracranial Masses: Initial Experiences and Comparison to PET/CT, JNM 4 / 35

5 5 / 35 Outline Multi-Modality Imaging Example: X-ray Tomography Forward Model Optimization Algorithms Numerical Results Multilevel-Based Acceleration Summary and Outlook

6 6 / 35 Tomographic Image Reconstruction Non-invasive imaging technique to visualize internal structures of object. Tomography applications: physics, chemistry, astronomy, geophysics, medicine, etc. Tomographic imaging modalities: X-ray transmission, ultrasound, magnetic resonance, X-ray fluorescence, etc. Task: estimate distribution of physical quantities in sample from measurements. Limited angle tomography reconstruction is naturally ill-conditioned.

7 Schematic Experimental Setup with Two Modalities 7 / 35

8 8 / 35 Reconstruction Approaches Traditional: filtered backprojection Restricted to simple tomographic model Requires a large number of projections Alternatively, iterative reconstruction from single data modality Requires much less data acquisition, results in higher accuracy Our Goal: Formulate a joint inversion integrating XRF and XRT data to improve the reconstruciton quality of elemental map.

9 9 / 35 Outline Multi-Modality Imaging Example: X-ray Tomography Forward Model Optimization Algorithms Numerical Results Multilevel-Based Acceleration Summary and Outlook

10 10 / 35 X-ray Transmission (XRT) Traditionally, the XRT projection of the object from beam line (θ, τ) is modeled as { F T θ,τ ( µ E ) = I 0 exp } L v µ E v. v to directly solve the linear attenuation coefficient µ E v for each voxel v, In our approach,notice µ E v = e W v,e µ E e, { F T θ,τ (W) = I 0 exp } L v µ E e W v,e v,e I 0 : incident photon flux µ E e : mass attenuation coefficient of element e E at beam incident energy E W = W v,e: tensor denoting how much of element e is in voxel v L = [L v ]: tensor of intersection length of beam line (θ, τ) with the voxel v

11 X-ray Fluorescence (XRF) 11 / 35

12 12 / 35 Mathematical Model of XRF First, we obtain the unit fluorescence spectrum ( { })) 1 x M e = F (F(I 1 2 e ) F σ 2π exp 2σ 2 Then, the XRF spectrum, F R θ,τ = L v W v,e M e exp n d ( W v,e µ E e L v I v U v + µee v,e,d v,e e P v,v,d ) P = [P v,v,d ]: tensor of intersection length of fluorescence detectorlet path d with the voxel v

13 13 / 35 Outline Multi-Modality Imaging Example: X-ray Tomography Forward Model Optimization Algorithms Numerical Results Multilevel-Based Acceleration Summary and Outlook

14 14 / 35 Resulting Optimization: Joint Reconstruction (JRT) Goal: Find W so that F R θ,τ (W) = DR θ,τ and FT θ,τ (W) = DT θ,τ min φ(w) = ( 1 F R θ,τ (W) D R θ,τ 2 + β F T θ,τ (W) D T θ,τ 2) W θ,τ where D R θ,τ Rn E : measurement data of XRF signal detected at angle θ from light beam τ R: the measurement data of XRT signal detected at angle θ from light beam τ D T θ,τ β > 0: a scaling parameter to balance the two modalities Note: Optimization differs on how F R θ,τ and FT θ,τ are combined

15 How Multimodality Can Help Consider an overdetermined (more equations than the number of unknowns) but rank deficient system which has an infinitude of solutions: 1 2 [ ] 2 4 x1 = x Another such rank-deficient (not full-rank) system: [ ] [ ] [ ] 2 3 x1 1.5 = However, combine these two can form a full rank and consistent system with a unique solution x [ ] x1 2 = 2 3 x x 2 15 / 35

16 16 / 35 Optimization Solver In experiments, we use truncated-newton (TN) method with preconditioned conjugate gradient (PCG) to provide a search direction: To satisfy the bound constraints, the projected PCG is applied to the reduced Newton system One TN iteration typically requires κ(o(10)) PCG iterations Main expense of each outer iteration is κ + 2 function-gradient evaluations.

17 17 / 35 Outline Multi-Modality Imaging Example: X-ray Tomography Forward Model Optimization Algorithms Numerical Results Multilevel-Based Acceleration Summary and Outlook

18 JRT versus Single XRF Reconstruction (Synthetic) 1 Element Unit: g/µm 2 1 Di, Leyffer, and Wild (2016). An Optimization-Based Approach for Tomographic Inversion from Multiple Data Modalities, SIAMIS 18 / 35

19 Convergence 19 / 35

20 20 / 35 Addressing Self-Absorption Effect Sample: Solid glassrod (Silicon) with 2 wires (Tungsten, Gold).

21 21 / 35 Outline Multi-Modality Imaging Example: X-ray Tomography Forward Model Optimization Algorithms Numerical Results Multilevel-Based Acceleration Introduction of MG/OPT Application of Multilevel in Tomographic Reconstruction Numerical Results Summary and Outlook

22 22 / 35 Motivation N = 9 2, flops = N = 5 2, flops = N = 3 2, flops = N = 2 2, flops =

23 23 / 35 Outline Multi-Modality Imaging Multilevel-Based Acceleration Introduction of MG/OPT Application of Multilevel in Tomographic Reconstruction Numerical Results Summary and Outlook

24 Introduction of MG/OPT Multigrid optimization algorithm (MG/OPT) is a general framework to accelerate a traditional optimization algorithm 1. Recursively use coarse problems to generate search directions for fine problems. MG/OPT can deal with more general problems in an optimization perspective, in particular, it is able to handle inequality constraints in a natural way. Multiple options to design the hierarchy of the problem: through image space. through data space. 1 Nash, S. G. (2000). A Multigrid Approach to Discretized Optimization Problems, OMS 24 / 35

25 MG/OPT Given: High-resolution model f h (z h ), easier-to-solve low-resolution model f H (z H ) z + OPT(f (z), z, k) A restriction operator Ih H and an interpolation operator IH h An initial estimate z 0 h of the solution z h on the fine level Integers k 1 and k 2 satisfying k 1 + k 2 > 0 25 / 35

26 26 / 35 Outline Multi-Modality Imaging Multilevel-Based Acceleration Introduction of MG/OPT Application of Multilevel in Tomographic Reconstruction Numerical Results Summary and Outlook

27 27 / 35 Interpolation/Restriction Operators Restriction on 2D parameter W h R 2 to produce W H using full weighted matrix: Restriction on gradient ÎH h = C IH h where C balances the order difference between φ h ( W h ) and φ H (Ih H W h ) Interpolation operator: I h H = 4(IH h )T.

28 28 / 35 Coarse Grid Surrogate Model Shift experimental data for coarse level: D D D θ,τ = D θ,τ (F,h θ,τ ( W h ) F,H θ,τ (IH h W h )) The surrogate model: φ H (W H ) = θ,τ ( 1 2 F,H θ,τ (W H ) D D D θ,τ 2 ) ( φ H (Ih H W h ) ÎH h φ h( W T h )) WH

29 29 / 35 Outline Multi-Modality Imaging Multilevel-Based Acceleration Introduction of MG/OPT Application of Multilevel in Tomographic Reconstruction Numerical Results Summary and Outlook

30 30 / 35 MG/OPT Search Directions versus Error Problem Size: [33 17] Fine Level Error Interpolated Coarse Level Error

31 31 / 35 2-level MG/OPT Reconstruction 10 2 Single Level Multigrid Reconstruction Error Reduction Equivalent number of f g evaluations

32 32 / 35 Outline Multi-Modality Imaging Example: X-ray Tomography Forward Model Optimization Algorithms Numerical Results Multilevel-Based Acceleration Introduction of MG/OPT Application of Multilevel in Tomographic Reconstruction Numerical Results Summary and Outlook

33 33 / 35 Summary Established a link between X-ray transmission and X-ray fluorescence datasets by reformulating their corresponding physical models. Developed a simultaneous optimization approach for the joint inversion, and achieved a dramatic improvement of reconstruction quality with no extra computational cost. Proposed a multigrid-based optimization framework to further reduce the computational cost of the reconstruction problem. Preliminary results show that coarsening in voxel space improves accuracy/speed.

34 34 / 35 Making More of More Extend our multimodal analysis tool to data from different instruments, involving varying spatial resolution and contrast mechanisms. Guided by the hierarchical nature of our multilevel algorithm, we will investigate new data acquisition strategies and allow for flexible and adaptive sampling approaches. Enable a true real-time feedback.

35 35 / 35 Acknowledgement This work was funded under the project The Tao of Fusion: Pathways for Big-data Analysis of Energy Materials at Work and Base optimization (DOE-ASCR DE-AC02-06CH11357).