Tight graph framelets for sparse diffusion MRI q-space representation

In diffusion MRI,the outcome of estimation problems can often be improved by taking into account the correlation of diffusionweighted 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 quasiaffine 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.