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Multilevel Optimization for Multi-Modal X-rayData Analysis
Zichao (Wendy) Di
Mathematics & Computer Science DivisionArgonne National Laboratory
August 8, 2017
Outline
Multi-Modality ImagingExample: X-ray TomographyForward ModelOptimization AlgorithmsNumerical Results
Multilevel-Based AccelerationOther Applications of Multilevel MethodsIntroduction of MG/OPTApplication of Multilevel in Tomographic ReconstructionNumerical Results
Summary and Outlook
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Outline
Multi-Modality ImagingExample: X-ray TomographyForward ModelOptimization AlgorithmsNumerical Results
Multilevel-Based AccelerationOther Applications of Multilevel MethodsIntroduction of MG/OPTApplication of Multilevel in Tomographic ReconstructionNumerical Results
Summary and Outlook
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More is Better
CT:hard-tissue; T2(MRI):soft tissue; PET: functional characteristics 1
1Boss, A. and et al. (2010). Hybrid PET/MRI of Intracranial Masses: InitialExperiences and Comparison to PET/CT, JNM
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Outline
Multi-Modality ImagingExample: X-ray TomographyForward ModelOptimization AlgorithmsNumerical Results
Multilevel-Based Acceleration
Summary and Outlook
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Tomographic Image Reconstruction
I Non-invasive imaging technique to visualize internalstructures of object.
I Tomography applications: physics, chemistry, astronomy,geophysics, medicine, etc.
I Tomographic imaging modalities: X-ray transmission,ultrasound, magnetic resonance, X-ray fluorescence, etc.
I Task: estimate distribution of physical quantities in samplefrom measurements.
I Limited angle tomography reconstruction is naturallyill-conditioned.
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Schematic Experimental Setup with Two Modalities
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Reconstruction Approaches
I Traditional: filtered backprojectionI Restricted to simple tomographic modelI Requires a large number of projections
I Alternatively, iterative reconstruction from single datamodality
I Requires much less data acquisition, results in higheraccuracy
Our Goal:Formulate a joint inversion integrating XRF and XRT data toimprove the reconstruciton quality of elemental map.
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Outline
Multi-Modality ImagingExample: X-ray TomographyForward ModelOptimization AlgorithmsNumerical Results
Multilevel-Based Acceleration
Summary and Outlook
9 / 37
X-ray Transmission (XRT)
I Traditionally, the XRT projection of theobject from beam line (θ, τ) is modeled as
FTθ,τ (µE ) = I0 exp
{−∑
v
Lv µEv
}.
to directly solve the linear attenuationcoefficient µE
v for each voxel v ,
I In our approach,notice µEv =
∑e
Wv ,eµEe ,
FTθ,τ (WWW) = I0 exp
{−∑v ,e
LvµEeWv ,e
} I0: incident photon flux
µEe : mass attenuation coefficient of element
e ∈ E at beam incident energy E
WWW =Wv,e : tensor denoting how muchof element e is in voxel v
L = [Lv ]: tensor of intersection lengthof beam line (θ, τ) with the voxel v
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X-ray Fluorescence (XRF)
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Mathematical Model of XRF
I First, we obtain the unit fluorescencespectrum
Me = F−1(
F (Ie) ∗ F(
1σ√
2πexp
{−x2
2σ2
}))
I Then, the XRF spectrum, FRθ,τ
=∑v ,e,d
LvWv ,eMe
ndexp
−∑v ′,e′Wv ′,e′
(µE
e′Lv ′Iv ′∈Uv + µEee′ Pv ,v ′,d
)P = [Pv,v′,d ]: tensor of intersection length of fluorescence detectorlet path d with the voxel v′
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Outline
Multi-Modality ImagingExample: X-ray TomographyForward ModelOptimization AlgorithmsNumerical Results
Multilevel-Based Acceleration
Summary and Outlook
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Resulting Optimization: Joint Reconstruction (JRT)Goal:FindWWW so that FR
θ,τ (WWW) = DRθ,τ and FT
θ,τ (WWW) = DTθ,τ
minWWW≥0
φ(WWW) =∑θ,τ
(12
∥∥∥FRθ,τ (WWW)− DR
θ,τ
∥∥∥2+β1
2
∥∥∥FTθ,τ (WWW)− β2DT
θ,τ
∥∥∥2)
where
I DRθ,τ ∈ RnE : measurement data of XRF signal detected at angle
θ from light beam τ
I DTθ,τ ∈ R: the measurement data of XRT signal detected at angle
θ from light beam τ
I β1, β2 > 0 are scaling parameters to balance the ability of eachmodality to fit the data, and detect the relative variability betweenthe data sources.
Note: Optimization differs on how FRθ,τ and FT
θ,τ are combined14 / 37
How Multimodality Can HelpConsider an overdetermined (more equations than thenumber of unknowns) but rank deficient system which has aninfinitude of solutions: 1 2
2 40.5 1
[x1x2
]=
12
0.5
Another such rank-deficient (not full-rank) system:[
2 34 6
] [x1x2
]=
[1.53
]However, combine these two can form a full rank andconsistent system with a unique solution x
1 22 4
0.5 12 34 6
[x1x2
]=
12
0.51.53
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Optimization Solver
In experiments, we use truncated-Newton (TN) method withpreconditioned conjugate gradient (PCG) to provide a searchdirection:
I To satisfy the bound constraints, the projected PCG isapplied to the reduced Newton system
I One TN iteration typically requires κ(O(10)) PCGiterations
Main expense of each outer iteration is κ+ 2 function-gradientevaluations.
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Outline
Multi-Modality ImagingExample: X-ray TomographyForward ModelOptimization AlgorithmsNumerical Results
Multilevel-Based Acceleration
Summary and Outlook
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JRT versus Single XRF Reconstruction (Synthetic)1
Element Unit: g/µm2
1Di, Leyffer, and Wild (2016). An Optimization-Based Approach forTomographic Inversion from Multiple Data Modalities, SIAMIS
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Convergence
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Addressing Self-Absorption Effect 1
Sample: Solidglassrod (Silicon)
with 2 wires(Tungsten, Gold).
1Di and et al. (2017). Joint reconstruction of x-ray fluorescence andtransmission tomography, Optics Express
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Outline
Multi-Modality ImagingExample: X-ray TomographyForward ModelOptimization AlgorithmsNumerical Results
Multilevel-Based AccelerationOther Applications of Multilevel MethodsIntroduction of MG/OPTApplication of Multilevel in Tomographic ReconstructionNumerical Results
Summary and Outlook
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Motivation
N = 92, flops = 2×107
N = 52, flops = 6×106
N = 32, flops = 3×106
N = 22, flops = 8×105
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Outline
Multi-Modality Imaging
Multilevel-Based AccelerationOther Applications of Multilevel MethodsIntroduction of MG/OPTApplication of Multilevel in Tomographic ReconstructionNumerical Results
Summary and Outlook
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Multilevel Acceleration of Image Registration
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Outline
Multi-Modality Imaging
Multilevel-Based AccelerationOther Applications of Multilevel MethodsIntroduction of MG/OPTApplication of Multilevel in Tomographic ReconstructionNumerical Results
Summary and Outlook
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Introduction of MG/OPT
I Multigrid optimization algorithm (MG/OPT) is a generalframework to accelerate a traditional optimizationalgorithm1.
I Recursively use coarse problems to generate searchdirections for fine problems.
I MG/OPT can deal with more general problems in anoptimization perspective, in particular, it is able to handleinequality constraints in a natural way.
I Multiple options to design the hierarchy of the problem:I through image space.I through data space.
1Nash, S. G. (2000). A Multigrid Approach to Discretized OptimizationProblems, OMS
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MG/OPTGiven:
I High-resolution model fh(zh), easier-to-solve low-resolution modelfH(zH)
I z+ ← OPT(f (z), z, k)
I A restriction operator IHh and an interpolation operator Ih
H
I An initial estimate z0h of the solution z∗h on the fine level
I Integers k1 and k2 satisfying k1 + k2 > 0
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Outline
Multi-Modality Imaging
Multilevel-Based AccelerationOther Applications of Multilevel MethodsIntroduction of MG/OPTApplication of Multilevel in Tomographic ReconstructionNumerical Results
Summary and Outlook
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Interpolation/Restriction Operators
I Restriction on 2D parameterWWWh ∈ R2 to produce WWWHusing full weighted matrix:
I Restriction on gradient IHh = C IH
h where C balances theorder difference between φh(WWWh) and φH(IH
h WWWh)
I Interpolation operator: IhH = 4(IH
h )T .
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Coarse Grid Surrogate Model
I Shift experimental data for coarse level:
DDDθ,τ = Dθ,τ − (F,hθ,τ (WWWh)− F,Hθ,τ (IHh WWWh))
The surrogate model:
φH(WWWH) =∑θ,τ
(12
∥∥∥F,Hθ,τ (WWWH)− DDDθ,τ
∥∥∥2)
−(∇φH(IH
h WWWh)− IHh ∇φh(WWWh)
)TWWWH
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Outline
Multi-Modality Imaging
Multilevel-Based AccelerationOther Applications of Multilevel MethodsIntroduction of MG/OPTApplication of Multilevel in Tomographic ReconstructionNumerical Results
Summary and Outlook
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MG/OPT Search Directions versus Error
Problem Size: [33 17]
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2-level MG/OPT Reconstruction
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Outline
Multi-Modality ImagingExample: X-ray TomographyForward ModelOptimization AlgorithmsNumerical Results
Multilevel-Based AccelerationOther Applications of Multilevel MethodsIntroduction of MG/OPTApplication of Multilevel in Tomographic ReconstructionNumerical Results
Summary and Outlook
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Summary
I Established a link between X-ray transmission and X-rayfluorescence datasets by reformulating their correspondingphysical models.
I Developed a simultaneous optimization approach for thejoint inversion, and achieved a dramatic improvement ofreconstruction quality with no extra computational cost.
I Proposed a multigrid-based optimization framework tofurther reduce the computational cost of the reconstructionproblem.
I Preliminary results show that coarsening in voxel spaceimproves accuracy/speed.
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Making More of More
I Extend our multimodal analysis tool to data from differentinstruments, involving varying spatial resolution andcontrast mechanisms.
I Guided by the hierarchical nature of our multilevelalgorithm, we will investigate new data acquisitionstrategies and allow for flexible and adaptive samplingapproaches.
I Enable true real-time feedback.
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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)”.
Thanks toSven Leyffer, Stefan Wild, andcollaborators at APS:Francesco De Carlo, Si Chen, Doga Gursoy, Young Pyo Hong,Chris Jacobsen, Stefan Vogt
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