1. COMPRESSIVE SENSING IN DT-MRI Daniele Perrone 1, Jan Aelterman 1, Aleksandra Pižurica 1, Alexander Leemans 2 and Wilfried Philips 1 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 optimally with 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 in DIFFUSION TENSOR DOMAIN Fiber Tract. IFFT Fiber Tract. Fiber Tract. Fiber Tract. - number of fiber tracts - mean length of the fiber tracts - fractional anisotropy - apparent diffusion coefficient - mean value and variance of the three eigenvalues Compr. Sensing K-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: experiment 50% of the coefficients used PSNR of the reconst. image: 40 dB Gibbs artifact : ABSENT Theoretical 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 x i+1 = argmin λ/2||Fx - y i || 2 + μ/2||d i - Sx - μ i || 2 d i+1 = argmin |d| + μ/2||d - Sx i+1 - μ i || 2 μ = μ i + Sx i+1 - d i+1 goto 1 until convergence of μ i+1 y i+1 = y i + y - Fx i+1 goto 1 until convergence of y i+1 x d 1 2 3 4 5 6 CS