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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. [email protected] 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