COMPRESSIVE SENSING IN DT-MRI
Daniele
Jan
Aleksandra
2
1
Alexander Leemans and Wilfried Philips
1
Perrone ,
1 Ghent
2 Image
1
Aelterman ,
1
Pižurica ,
University - TELIN - IPI – IBBT - St.-Pietersnieuwstraat 41,
B-9000 Gent, Belgium
Sciences Institute - University Medical Center - Heidelberglaan 100,
3584 Utrecht, the Netherlands.
daniele.perrone@telin.ugent.be
What is Diffusion Tensor MRI?
Why shearlet transform?
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.
MRI scanner
Full brain acquisition
( axial view )
Brain tractography
( coronal view )
Fiber
Tract.
f is C² (edges:piecewise C²)
Shearlets can represent
natural image optimally
with only a few significant
(non-zero) coefficients, while
noise and corruptions would
need more coefficients.
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
We can force a noise and
artifact free reconstruction.
Split Bregman technique
|Sx| so that ||y – Fx||2
Formalization of the problem : x* = argmin
x
IFFT
*N
( N different directions )
Split Bregman
algorithm :
Focusing on the problem
1
xi+1 = argmin λ/2||Fx - yi||2 + μ/2||di - Sx - μi||2
2
di+1 = argmin |d| + μ/2||d - Sxi+1 - μi||2
3
μ = μi + Sxi+1 - di+1
4
goto 1 until convergence of μi+1
x
d
5
yi+1 = yi + y - Fxi+1
6
goto 1 until convergence of yi+1
CS
No artifacts
The color image
IFFT
Gibbs ringing artifact in
IMAGE DOMAIN
IFFT
Image domain: experiment
50% of the coefficients used
PSNR of the reconst. image: 40 dB
Gibbs artifact : ABSENT
Theoretical acquisition speedup: 2
Results for DT MRI
Fiber Tractography SEED
(corpus callosum - sagittal view)
Wrong - even negative ! - eigenvalues in
DIFFUSION TENSOR DOMAIN
It is not possible to acquire fewer data without losing
information and eventually generating corruption in images...
Brain tracts
( coronal view )
…is it possible to discard only
“superfluous” information?
Fiber
Tract.
FFT
Speeding up with Compressive Sensing
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 )
50%
FFT
Fiber
Tract.
Compr. Sensing
K-space
reconstruction
Fiber
Tract.
Promising quantitative improvements considering :
- number of fiber tracts
- mean length of the fiber tracts
- fractional anisotropy
- apparent diffusion coefficient
- mean value and variance of the three eigenvalues