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Utilization of compressed sensingin perfusion MRI
Marie Dankova
4th SPLab Workshop
20th November 2014
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Perfusion imaging
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Lognormal model
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Sparse representation of signals
• model of signal y:y = Ax
• infinitely many solutions of this equations
y
=
A x
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Sparse representation of signals
• We want the sparsest possible, i.e.
minx‖x‖0 subject to Ax = y (P0)
y
=
A x
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`1-relaxation
`0:
1
Ax = y`1:
1
Ax = y`2:
1
Ax = y
• relaxed problem
minx‖x‖1 subject to Ax = y. (P1)
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Compressed sensing
• acceleration of the measurement
• longer computation time
minx
f(x) subject to Ax = y
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Compressed sensing• CS problem:
x := arg min ‖x‖0 subject to y =
A︷ ︸︸ ︷RΦ︸︷︷︸
P
Ψx︸︷︷︸z
(P1U)
• Ψ is orthonormal basis for signal z, coordinates x are sparse• Φ matrix N × N (in MRI Fourier transform)• R is random matrix performing selection of m rows (in MRI
trajectory in k-space)
y
=
R Φ Ψ x
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Reshaping video data
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L+S model
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Simulation
perfusion phantom – video
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Measurement matrices
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Random mask
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Radial halflines
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17
27
37
47
57
67
77
87
0 10 20 30 40 50 60 70 80 90 100
SNR
[d
B]
Measured samples [%]
random nonunif
random unif
radial lines
spiral
radial halflines
random nonunif
random
radial lines
spiral
radial halflines
10
0 ×
10
0 p
x 1
50
× 1
50
px
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THANK YOU FOR YOURATTENTION
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References
• R. Otazo, E. J. Candes and D. Sodickson. Low-rank andsparse matrix decomposition for accelerated dynamic MRIwith separation of background and dynamic components.Submitted to Magnetic Resonance in Medicine, Sept. 2013.
• V. Harabis, R. Kolar, M. Mezl, R. Jirik. Comparison andEvaluation of Indicator Dilution Models for Bolus ofUltrasound Contrast Agents. Physiol Meas. 2013, 34(2)
• M. Mezl, R. Jirık, V. Harabis. Acquisition and data processingin the ultrasound perfusion analysis (in Czech) In New trendsin biomedical engineering. Brno university of technology 2013,ISBN 978-80-214-4814-8.
• P. L. Combettes, J.-C. Pesquet. Proximal Splitting Methodsin Signal Processing. Fixed-Point Algorithms for InverseProblems in Science and Engineering. Springer, 2011.
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References
• Lustig, Donoho, Santos, Pauly: Compressed Sensing MRI,IEEE Signal Processing Magazine, 2008
• Bruckstein, Donoho, Elad: From Sparse Solutions of Systemsof Equations to Sparse Modeling of Signals and Images, SIAMReview, 2009
• Candes, Wakin: An Introduction to Compressive Sampling,IEEE, 2008
• http://fmri.mchmi.com
• http://spinwarp.ucsd.edu/neuroweb/Text/br-710epi.htm