COMPRESSIVE SENSING IN DT-MRICOMPRESSIVE SENSING IN DT-MRIDaniele Perrone1, Jan Aelterman1, Aleksandra Pižurica1,

Alexander Leemans2 and Wilfried Philips1

1 Ghent University - TELIN - IPI – IBBT - St.-Pietersnieuwstraat 41, B-9000 Gent, Belgium

daniele.perrone@telin.ugent.be

Diffusion Tensor Magnetic Resonance Imaging (DT- MRI) can infer the orientation of fibre bundles within brain white matter tissue using so-called Fiber Tractography (FT) algorithms. The data capture step force to choose a trade off between data quality and acquisition time.

What is Diffusion Tensor MRI?

MRI scanner

The color image

It is not possible to acquire fewer data without losing information and eventually generating corruption in images...

…is it possible to discard only “superfluous” information?

Focusing on the problem

No artifacts

Shearlets can represent natural image optimallywith only a few significant (non-zero) coefficients, while noise and corruptions would need more coefficients.

We can force a noise and artifact free reconstruction.

Why shearlet transform?

Results for DT MRI

Split Bregman technique

Considering just the Fourier samples that are not attenuated:

• the number of samples is less than the number of image pixels an infinite number of possible image reconstructions;

• naively filling in the missing data in Fourier space leads to artifacts in MRI and therefore in FT reconstruction;

• proposed: Compressive Sensing (CS) reconstruction we have an infinite number of possible image reconstructions we do want to maintain fidelity to the acquired data

we can choose the image that is representable with the LOWEST number of coefficients possible in the shearlet domain ( that corresponds to the minimum number of image structures )

Speeding up with Compressive Sensing

2 Image Sciences Institute - University Medical Center - Heidelberglaan 100, 3584 Utrecht, the Netherlands.

Full brain acquisition( axial view )

Formalization of the problem :

* N ( N different directions )

Gibbs ringing artifact in IMAGE DOMAIN

Wrong - even negative ! - eigenvalues inDIFFUSION TENSOR DOMAIN

FiberTract.

IFFT

IFFT

IFFT

FiberTract.

FiberTract.

FiberTract.

- number of fiber tracts- mean length of the fiber tracts- fractional anisotropy- apparent diffusion coefficient- mean value and variance of the three eigenvalues

Compr. SensingK-space

reconstruction

Promising quantitative improvements considering :

FFT

50%FFT

Fiber Tractography SEED(corpus callosum - sagittal view)

Brain tracts( coronal view )

Brain tractography( coronal view )

Split Bregman algorithm :

Image domain: experiment50% of the coefficients usedPSNR of the reconst. image: 40 dBGibbs artifact : ABSENTTheoretical acquisition speedup: 2

f is C² (edges:piecewise C²)

Approximation error:

Fourier basis : ε M ≤ C * M -1/2

Wavelet basis : ε M ≤ C * M -1

Shearlet tight frame : ε M ≤ C * (log(M))3 * M-2

x* = argmin |Sx| so that ||y – Fx||2

x

xi+1 = argmin λ/2||Fx - yi||2 + μ/2||di - Sx - μi||2

di+1 = argmin |d| + μ/2||d - Sxi+1 - μi||2

μ = μi + Sxi+1 - di+1

goto 1 until convergence of μi+1

yi+1 = yi + y - Fxi+1

goto 1 until convergence of yi+1

x

d

1

2

3

4

5

6

CS