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Med Image Comput Comput Assist Interv. 2016 Oct;9902:561-569. doi: 10.1007/978-3-319-46726-9_65. Epub 2016 Oct 2.

Tight Graph Framelets for Sparse Diffusion MRI q-Space Representation.

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Department of Radiology and BRIC, University of North Carolina, Chapel Hill, U.S.A.
Beijing International Center for Mathematical Research, Peking University, Beijing, China.
Department of Psychiatry & Behavioral Sciences, Stanford University, U.S.A.


In diffusion MRI, the outcome of estimation problems can often be improved by taking into account the correlation of diffusion-weighted images scanned with neighboring wavevectors in q-space. For this purpose, we propose in this paper to employ tight wavelet frames constructed on non-flat domains for multi-scale sparse representation of diffusion signals. This representation is well suited for signals sampled regularly or irregularly, such as on a grid or on multiple shells, in q-space. Using spectral graph theory, the frames are constructed based on quasi-affine systems (i.e., generalized dilations and shifts of a finite collection of wavelet functions) defined on graphs, which can be seen as a discrete representation of manifolds. The associated wavelet analysis and synthesis transforms can be computed efficiently and accurately without the need for explicit eigen-decomposition of the graph Laplacian, allowing scalability to very large problems. We demonstrate the effectiveness of this representation, generated using what we call tight graph framelets, in two specific applications: denoising and super-resolution in q-space using ℓ0 regularization. The associated optimization problem involves only thresholding and solving a trivial inverse problem in an iterative manner. The effectiveness of graph framelets is confirmed via evaluation using synthetic data with noncentral chi noise and real data with repeated scans.

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