Optimal selection of diffusion-weighting gradient waveforms using compressed sensing and dictionary learning

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Optimal selection of diffusion-weighting gradient

waveforms using compressed sensing and dictionary

learning

Raphaël Truffet, Emmanuel Caruyer

To cite this version:

Raphaël Truffet, Emmanuel Caruyer. Optimal selection of diffusion-weighting gradient waveforms

using compressed sensing and dictionary learning. AI days 2018, Apr 2018, Rennes, France. pp.1.

�inserm-01939066v2�

Optimal selection of diffusion-weighting

gradient waveforms using compressed

sensing and dictionary learning

Raphaël Truffet

supervised by

Emmanuel Caruyer

I

Introduction

Data generation

II

Signal undersampling

Results

IV

Concluding remarks

and further work

VI

Dictionary learning

V

Oscillating gradient Generalized waveform

• Diffusion MRI measures the movement of water molecules and gives information about white matter microstructure.

• The acquisition sequences rely on magnetic field gradients. • While pulsed gradient waveforms are the most used because of their simplicity, it has been shown that oscillating arbitrary waveforms provide better estimation of microstructure parameters(1)_.

• Since every function of the time that respect a few constraints provide a possible gradient waveform, the sampling remains largely unexploited.

• Here are several possible gradient waveforms:

• Magnetic Resonance Imaging (MRI) is a non-invasive technique for the observation of the tissue in vivo.

• Many signals are simulated for several gradient waveforms and several microstructure parameters using CAMINO(2)_.

• 180 different microstructure are generated with parallel fibers and different densities, radii distributions. These microstructure are rotated to

represent several orientations. • 2600 gradients are used in the

simulations. Their direction is constant and they are piecewise constant with 4 steps of time.

• The gradients are spread over 40 directions that cover the unit sphere.

III

• We want to find a sparse representation for the signal.

0 10 20 30 40 50 60 70 80 40 30 20 10 0 10 20 30 40 10 10 10 10 10 10 10 10 10 magnetic fi leld gradien t (mT/ m) time (ms) time (ms) time (ms) time (ms)

.

signal dictionary _{representation} sparse

• Dictionary learning is made over the families of signals previously generated.

• The learning is made by updating alternatively the dictionary and the sparse signals to minimize:

{

{

fidelity sparsity

~

~

• We select some lines of dictionary (which correspond to gradients) in several ways:

• We can restrict the dictionary to the selected gradients.

• We can construct a sparse signal using only a few measurements (with compressed sensing techniques, in particular, -minimization)

.

=

using only a few measurements • We use only the measures associated to the selected gradients.

3.77

atom 50

+ 1.64 + 0.51 ₊

...=

atom 80 atom 183 reconstructed signal

0 500 1000 1500 2000 2500 0.0 0.1 0.2 0.3 0.4 0.5 0.6 sampled signal 0 25 50 75 100125 150175200 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 sparse representation 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 10 15 20 25 30 Fitting error (root mean squares)

(sparsity parameter) used for reconstruction

number of samples used

overfitting sparse

over-Pulsed gradient

• Encouraging results that show an efficient reconstruction • Our gradient selection heuristic performs better than randomness (often used in compressed sensing) • We can still improve :

- The gradients given in input

- The learning and reconstruction parameters

- The criterion to optimize for the subsampling (for example, the incoherence of the columns

magnetic fi leld gradien t (mT/ m) - randomly

- uncorrelated lines (minimizing the norm of the restricted correlation matrix)

-minimization

(2) P. A. Cook, Y. Bai, S. Nedjati-Gilani, K. K. Seunarine, M. G. Hall, G. J. Parker, D. C. Alexander. "Camino: Open-Source Diffusion-MRI Reconstruction and Processing", 14th Scientific Meeting of the International Society for Magnetic Resonance in Medicine, Seattle, WA, USA, (2006) p. 2759.

(1) Ivana Drobnjak, Bernard Siow, and Daniel C. Alexander. "Optimizing gradient waveforms for microstructure sensitivity in diffusion-weighted MR". In: Journal of Magnetic Resonance 206.1 (2010), pp. 41-51.

minimizing the lines correlation minimizing the columns correlation random selection 35 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.000 0.002 0.004 0.006 0.008 0.010

minimizing the lines correlation minimizing the columns correlation random selection

- minimizining the coherence

• The dictionary learning is performed by the tool SPAMS in python.