Barmpoutis Angelos, Jian Bing, Vemuri Baba C, Shepherd Timothy M
Computer and Information Science and Engineering, University of Florida, Gainesville, FL 32611, USA.
Inf Process Med Imaging. 2007;20:308-19. doi: 10.1007/978-3-540-73273-0_26.
In Diffusion Weighted Magnetic Resonance Image (DW-MRI) processing a 2nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the DW-MRI data. It is now well known that this 2nd-order approximation fails to approximate complex local tissue structures, such as fibers crossings. In this paper we employ a 4th order symmetric positive semi-definite (PSD) tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the DW-MRI data guaranteeing the PSD property. There have been several published articles in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positive semi-definite constraint, which is a fundamental constraint since negative values of the diffusivity coefficients are not meaningful. In our methods, we parameterize the 4th order tensors as a sum of squares of quadratic forms by using the so called Gram matrix method from linear algebra and its relation to the Hilbert's theorem on ternary quartics. This parametric representation is then used in a nonlinear-least squares formulation to estimate the PSD tensors of order 4 from the data. We define a metric for the higher-order tensors and employ it for regularization across the lattice. Finally, performance of this model is depicted on synthetic data as well as real DW-MRI from an isolated rat hippocampus.
在扩散加权磁共振成像（DW-MRI）处理中，二阶张量通常用于近似DW-MRI数据每个格点处的扩散率函数。现在众所周知，这种二阶近似无法近似复杂的局部组织结构，如纤维交叉。在本文中，我们采用四阶对称半正定（PSD）张量近似来表示扩散率函数，并提出一种从DW-MRI数据估计这些张量的新技术，以保证PSD特性。文献中已有几篇关于扩散率函数高阶张量近似的文章，但它们都没有保证半正定约束，而这是一个基本约束，因为扩散系数的负值没有意义。在我们的方法中，我们通过使用线性代数中所谓的Gram矩阵方法及其与希尔伯特三元四次方程定理的关系，将四阶张量参数化为二次型的平方和。然后，这种参数表示用于非线性最小二乘公式中，以从数据估计四阶PSD张量。我们为高阶张量定义了一个度量，并将其用于格点上的正则化。最后，在合成数据以及来自孤立大鼠海马体的真实DW-MRI上展示了该模型的性能。
