Yap Pew-Thian, Dong Bin, Zhang Yong, Shen Dinggang
Department of Radiology and BRIC, University of North Carolina, Chapel Hill, U.S.A.
Beijing International Center for Mathematical Research, Peking University, Beijing, China.
Med Image Comput Comput Assist Interv. 2016 Oct;9902:561-569. doi: 10.1007/978-3-319-46726-9_65. Epub 2016 Oct 2.
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 -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 -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 , in two specific applications: denoising and super-resolution in -space using ℓ 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.
在扩散磁共振成像中，通过考虑在空间中用相邻波矢扫描的扩散加权图像的相关性，估计问题的结果通常可以得到改善。为此，我们在本文中提出采用在非平坦域上构造的紧小波框架，用于扩散信号的多尺度稀疏表示。这种表示非常适合于在空间中规则或不规则采样的信号，例如在网格或多个球壳上采样的信号。利用谱图理论，基于定义在图上的拟仿射系统（即有限小波函数集合的广义伸缩和平移）构造框架，图可视为流形的离散表示。相关的小波分析和合成变换可以高效、准确地计算，而无需对图拉普拉斯算子进行显式特征分解，从而能够扩展到非常大的问题。我们在两个具体应用中展示了使用我们所谓的生成的这种表示的有效性：在空间中使用ℓ正则化进行去噪和超分辨率。相关的优化问题仅涉及阈值处理和以迭代方式求解一个简单的逆问题。通过使用具有非中心卡方噪声的合成数据和具有重复扫描的真实数据进行评估，证实了图小波框架的有效性。
