Utilization of compressed sensingin perfusion MRI

Marie Dankova

4th SPLab Workshop

20th November 2014

Perfusion imaging

Lognormal model

Sparse representation of signals

• model of signal y:y = Ax

• infinitely many solutions of this equations

y

=

A x

Sparse representation of signals

• We want the sparsest possible, i.e.

minx‖x‖0 subject to Ax = y (P0)

y

=

A x

`1-relaxation

`0:

1

Ax = y`1:

1

Ax = y`2:

1

Ax = y

• relaxed problem

minx‖x‖1 subject to Ax = y. (P1)

Compressed sensing

• acceleration of the measurement

• longer computation time

minx

f(x) subject to Ax = y

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

Reshaping video data

L+S model

Simulation

perfusion phantom – video

Measurement matrices

Random mask

Radial halflines

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

THANK YOU FOR YOURATTENTION

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.

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