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Dictionary- & Model-based methods (in quantitative MRI reconstruction)
Mariya DonevaPhilips Research, Hamburg
14.10.2019
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Outline
• Brief introduction to MRI
• Image reconstruction as an inverse problem
• Model-based reconstruction for MR parameter mapping
• Dictionary-based methods
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A very brief introduction to MRI
PolarizationPut a person in a strong magnetic field B0
ExcitationApply an RF pulse B1 to flip the magnetization in the transverse plane
Spatial encodingApply a magnetic field gradients
Acquisition„Listen“ to the signal using RF coils
ReconstructionDecode the measured signal to obtain an image
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MRI data acquisition
Pulse sequence k-space
Spin echo
RF
Greadout
Gphase
Scan Time = TR × (No. phase encodes) ×(No. averages)
Data in k-space are acquired sequentially
TR
image space
Data acquisition is just a small portion of the MR scan
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• Assumption: The measured signal is a Fourier transform of the image
Standard MRI reconstruction
• In many cases this model is too simplified
k-space image space
iFFT
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Which factors influence the MR images?
T1 T2 PD
B0B1+/-
Diffusion
Perfusion
Chemical shift(Tissue composition)
Physiological motion
MRI signal strength is a function of the tissue parameters, system parameters, and sequence parameters
Magnetic susceptibility
TR TETI
…
Magnetization transfer
radial spiral
randomrosette lissajous
propeller
Many ways to sample the k-spacePrecise knowledge of gradient waveforms becomes important
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Model-based Reconstruction
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Forward model:
• 𝑦 measured data
• 𝐹 forward transform (linear or non-linear)
• 𝑥 image (T1W, T2W, parameter map,…)
• 𝑛 noise
Image reconstruction as an inverse problem
𝒚 = 𝑭𝒙 + 𝒏
Reconstruction problem: Recover 𝑥 based on the measurements 𝑦
Cause(parameter, unknown)
Effect(observation,
data)
inverse problem
forward problem
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Well-posed/ill-posed inverse problems
Well-posed problem
• A solution exists
• The solution is unique
• The inverse mapping 𝒚 → 𝒙 is well conditioned
Image reconstruction in MRI is always an ill-posed inverse problem
• No exact solution (data inconsistencies, noise)
• Solution is not unique (incomplete data)
• Sometimes the problem is ill-conditioned (noise amplification)
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• 𝑭𝒙 = 𝒚 has no exact solution
Least squares solution
• Solution is not unique, many choices of 𝒙 lead to the same 𝒚
Minimum norm solution
The Moore-Penrose pseudoinverse gives the minimum L2 norm solution
Solving ill-posed inverse problems
𝒙 = argmin 𝑭𝒙 − 𝒚 𝟐𝟐
minimize 𝒙subject to 𝑭𝒙 = 𝒚
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• Ill-conditioned problem: add regularization
Regularized inverse problem
• Regularization adds prior knowledge to stabilize the solution
Regularization
𝒙 = argmin𝒙
𝑭𝒙 − 𝒚 𝟐𝟐 + 𝜆𝑅(𝒙)
data consistency regularization
Examples:
• L2 norm 𝑅 𝒙 = 𝒙 𝟐𝟐
• Total Variation 𝑅 𝒙 = Δ𝒙
• L1 norm 𝑅 𝒙 = 𝒙Propagated data error
Approximation error
𝝀
Total error
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Modelling signal and system properties
• Ignoring many things, the measured signal in MRI is:
𝑦(𝑡) = 𝜌(𝑟)𝑒−2𝜋𝑖𝑘 𝑡 ∙𝑟 ⅆ𝑟
𝑦(𝑡) = 𝑒−𝑅2 𝑟 𝑡𝜌(𝑟) 𝑒−2𝜋𝑖𝑘 𝑡 ∙𝑟 ⅆ𝑟
𝑦(𝑡) = 𝑒𝑖Δ𝐵0 𝑟 𝑡𝜌(𝑟)𝑒−2𝜋𝑖𝑘 𝑡 ∙𝑟 ⅆ𝑟
• Include T2 relaxation
• Include off-resonance
• Include CSM 𝑦𝑗(𝑡) = 𝑐𝑗 𝑟 𝜌(𝑟)𝑒−2𝜋𝑖𝑘 𝑡 ∙𝑟 ⅆ𝑟
• Other effects: chemical shift, motion, flow, diffusion, …
• More accurate, more complete description of the measurements
• Allows estimating additional physical properties from the data
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Model-based reconstruction for quantitative MR parameter mapping
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Quantitative MRI• Improved tissue contrast
• Robust tissue segmentation
• Detection of diffuse disease
• Improved data consistency and comparability
• Synthetic MRI/ Single protocol exam
T2W FLAIR T1W
T1 map T2 map PD map
WM GM CSF
segmentation synthetic MRI
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Generalized View on Quantitative MRI
• Collect multiple images for different acquisition parameters, each of which is “weighted” by a specific tissue property
• Use a model to extract the value of the tissue property from the weighted images
𝑠
𝑝
Parameter Map
Challenge: we need to acquire multiple images
Acquisition parameter 𝑝 𝑠 = 𝑓(𝑝, Θ) Θ
Solution: undersample the data (and use model-based reconstruction)
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Model-based reconstruction for MR parameter mapping
Generalized signal model
𝑥 = 𝑓(𝑝, Θ)
k-p measurements
Acqusition parameters
Sampling operator
Θ = argmin1
2
𝑝
Φ𝑝𝑥𝑝(Θ) − 𝑦𝑝 2
2
Tissue parameters
Model-based reconstruction problem
Obtain the parameter maps directly from the undersampled data, solving a non-linear (and possibly non-convex) problem
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Model-based reconstruction for MR parameter mapping
Θ = argmin1
2
𝑝
Φ𝑝𝑥𝑝 Θ − 𝑦𝑝 2
2+
𝑖
𝜆𝑖𝑅𝑖 𝑥𝑝, Θ
Add regularization
𝜌 𝑇2
= argmin1
2 𝑐 𝑇𝐸𝑗
𝑀𝑇𝐸𝑗ℱ𝐶𝑐𝜌 𝑟 𝑒−
𝑇𝐸𝑗
𝑇2 𝑟 − 𝑦𝑇𝐸𝑗,𝑐 2
2
+ 𝑖 𝜆𝑖𝑅𝑖 𝑥𝑝, Θ
Example: T2 mapping
Block KT et al. IEEE Trans Med Imaging 28 (2009): 1759-1769Sumpf TJ et al. JMRI 2011
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Proton density R2 relaxivity
Radial 2D FSE sequence
32 x 16 echoes, ∆TE = 10 ms, TR = 7000 ms, 224 x 224 pixels (acceleration R=11)
Model based T2 mapping
Block KT et al. IEEE Trans Med Imaging 28 (2009): 1759-1769
Optimization performed with non-linear CG algorithm (CG-DESCENT)
Slide Courtesy of Dr. T Block
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Proton density R2 relaxivity
Radial 2D FSE sequence
32 x 16 echoes, ∆TE = 10 ms, TR = 7000 ms, 224 x 224 pixels (acceleration R=11)
Model based T2 mapping
Block KT et al. IEEE Trans Med Imaging 28 (2009): 1759-1769
Optimization performed with non-linear CG algorithm (CG-DESCENT)
Slide Courtesy of Dr. T Block
Provides quantitative T2 & PD map from single radial FSE dataset
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Radial Projections Consistent Model ResultModelMagnetization Preparation
&Look-Locker Acquisition
Termination
Criterion
n Single Projections
n Complete k-Spaces
Pixel-wise
Model Fit
Reinsert
Projections
Repeat Iteratively
Reconstruction Scheme[2]
[1] Tran-Gia, MRM 2013. [2]According to Doneva, MRM 2010. [3] Tran-Gia, PLOS ONE 2015.
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↔ T1 map[3]
Slide Courtesy of Dr. Johannes Tran-Gia
MAP (Model-based Acceleration of Parameter mapping)
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Model-based Regularization and Dictionaries
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Model-based sparsity constraint in a compressed sensing reconstruction
1. Create a training data set using the model2. Learn a sparsifying transform from the training data 3. Use the constraint in the reconstruction
f(p;θ1)
p
f(p;θ2)
p
f(p;θ3)
p
f(p;θn)
p
s1 s2 s3 sn
Training dataset S = [s1, s2, s3,... sn]
Discrete vector of sampling locations p
Set of parameter values {θ1 ,...,θn}
Data modelf(p;θ1)
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Data adapted sparse representations: Dictionaries
Dictionary Dx
s
= .
Each atom is a basic unit that is used to compose larger units atoms
0.6
0.4
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Model-based sparsity constraint: Orthogonal transform
𝑥 = argmin1
2
𝑐
𝑝
𝑀𝑝ℱ𝐶𝑐𝑥𝑝 − 𝑦𝑝,𝑐 2
2+ 𝜆1 𝑈𝐻𝑥 1
𝑅 = 𝑆𝑆𝐻 = 𝑈Σ𝑈𝐻
𝑧 = 𝑈𝐻𝑥
• 𝑈𝐻 is a linear sparsifying transform for the measurements 𝑥
• Include the data model in the regularization term
• PCA-based constraint
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Model-based subspace projection (REPCOM, T2 shuffling)
𝑥 = argmin1
2
𝑐
𝑝
𝑀𝑝ℱ𝐶𝑐𝑈𝑅𝑧𝑝 − 𝑦𝑇𝐸𝑗,𝑐 2
2
𝑅 = 𝑆𝑆𝐻 = 𝑈Σ𝑈𝐻
𝑧 = 𝑈𝑅𝐻𝑥
• 𝑈𝑅𝐻 projection to the subspace of the first R principal components
• Reconstruct the compressed image series
• PCA-based constraint
1) Huang et al MRM 2012 2) Tamir et al MRM 2017
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Model-based sparsity constraint: Over-complete dictionary
• Improve the sparsity: use over-complete dictionary
Obtain overcomplete dictionary by training (K-SVD2)
basis Dx
=
s
.
Dictionary Dx
s
= .
1) Doneva et al MRM 2010 2) Aharon M et. al, IEEE Trans Signal Process 2006
Orthogonal transform(e.g. PCA)
Over-complete dictionary
minimize 𝑥 − 𝐷𝑠 2, s. t. 𝑠 0 ≤ 𝐾
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Model-based sparsity constraint
Generate signal prototypes from
model
Dictionary training (K-SVD)
Apply dictionary in CS reconstruction
Original 2 Atoms 10 Atoms
Doneva et al MRM 2010
𝒙 = argmin𝒙
𝑭𝒙 − 𝒚 𝟐𝟐 + 𝝀 𝒙 − 𝑫𝒔 𝟐
𝟐,
s. t. 𝒔 0 ≤ 𝐾
Reconstruction
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Undersampled T1 mapping (CS)
multi-echo IR brain data TE = 1.9ms, TR = 3.8ms, α = 10°, FOV = 250 mm, 224×224 matrix, 40 images
0.055 0.086 0.106 0.131
0.051 0.062 0.083 0.113
NRMSE
NRMSE
1x 2x 4x 6x 8x
Doneva et al MRM 2010
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Dictionaries for higher dimensional signals
• Dictionaries for 2D image representation
– 2D image patches
Ravishankar S et al IEEE Trans Med Imag 2011; 30(5): 1028-41Caballero J et al. ISMRM 2014 #1560Katscher U et al ISMRM2017 #3641
minimize 𝑅𝑖𝑗𝑥 − 𝐷𝑠𝑖𝑗 2, s. t. 𝑠𝑖𝑗 0
≤ 𝐾
• Reconstruction
𝑅𝑖𝑗 - operator that extracts a patch centered at position i,j
Each atom in the dictionary is a 2D patch Image patch based dictionary
• Spatiotemporal dictionaries
– Spatio-temporal blocks (3D, 4D)
𝒙 = argmin𝒙
𝑭𝒙 − 𝒚 𝟐𝟐 + 𝝀
𝒊𝒋
𝑅𝑖𝑗𝑥 − 𝐷𝑠𝑖𝑗 𝟐
𝟐, s. t. 𝑠𝑖𝑗 0
≤ 𝐾
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More Dictionary-based techniques(extremely sparse representation)
“In a dictionary with infinitely many atoms, the signal can be ultimately represented by a single atom. This is equivalent to fitting the signal to the model”
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T2 mapping: model-based reconstruction with EMC
0
50
100
150
[ms]
22 min
Cartesian
Multi Spin-Echo Acquisitions
a e g
Single Spin-Echo
Cartesian
b f hT 2
Re
laxa
tio
n
Map
s
c
d 3:10 min
[Exponential fit] [EMC fit][Exponential fit]
Ben-Eliezer etl al, Magn Reson Med (2015) 73(2): 809-17.
Note: Parameter maps are discretized
0 2 4 6 8 10 12 140
5
10
15
20
25
30Echo train modulation
T2 = 30:2:55; 73:2:103
Echo train length = 13
29 simulations• The Echo Modulation Curve (EMC) Algorithm
1
• Compute signal evolution using Bloch simulations
Echo-modulation curves database EMC (T2, B1, …)
𝑦(𝑡) = 𝐸𝑀𝐶(𝐵1, 𝑇2)𝜌(𝑟) 𝑒−2𝜋𝑖𝑘 𝑡 ∙𝑟 ⅆ𝑟
• Fit experimental decay curve to simulated EMC database
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Magnetic Resonance Fingerprinting
time (ms)
sign
al in
ten
sity
B0 map
M0 map
T2 map
T1 map
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Magnetic Resonance Fingerprinting
full sampling undersamplingspiral read-out
Is the dictionary matching in MRF the same as model-based reconstruction for MR parameter mapping?
• Direct matching in MRF seems to work quite well, but it is only the first iteration of an iterative model-based reconstruction
• For long sequences and VD spiral sampling 1 iteration might be enough
• In the general case, iterative reconstruction is needed
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Model-based reconstruction for MRF
• Davies M et al. A Compressed Sensing Framework for Magnetic Resonance Fingerprinting SIAM J. Imaging Sci., 2014, 7(4), 2623–2656.
• Pierre E et al. Multiscale Reconstruction for MR Fingerprinting MagnReson Med. 2016 Jun;75(6):2481-92
• Zhao B, et al. Maximum Likelihood Reconstruction for Magnetic resonance Fingerprinting IEEE Trans Med Imaging. 2016 Aug;35(8):1812-23. doi:10.1109/TMI.2016.2531640
• Doneva M et al. Matrix Completion-based reconstruction for undersampled magnetic resonance fingerprinting data Magn ResonImaging. 2017 Mar 3 doi:10.1016/j.mri.2017.02.007
• Assländer J et al. Low rank alternating direction method of multipliers reconstruction for MR fingerprinting Magn Reson Med. 2017 Mar 5. doi: 10.1002/mrm.26639.
direct matching iterative recon
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MR-STATmulti-parametric model-based reconstruction (non-linear inversion)
Time-data Signal model
Slide Courtesy of Dr. Alessandro. Sbrizzi
T1
T2
True Recon
|B1+|
ΔB0
P.D.
Tx/Rxphase
True Recon
• Large scale non-linear inverse problem for all relevant tissue and system parameters
– Computationally very intensive
– Careful initialization is required
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Summary
• Model-based reconstruction
– Generalized framework for MR reconstruction
– Modelling of physical properties, reduced artifacts, QMRI
– Insert prior knowledge as regularization
• Dictionaries
– Provide sparse(r) signal representation
– Convert a non-linear inverse problem to a linear search
– Can be learned from training data
• Examples in MR Parameter mapping
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Acknowledgements
• Tobias Block
• Johannes Tran-Gia
• Noam Ben-Eliezer
• Alessandro Sbrizzi