Papadopoulo Théo, Ghosh Aurobrata, Deriche Rachid
Med Image Comput Comput Assist Interv. 2014;17(Pt 3):233-40. doi: 10.1007/978-3-319-10443-0_30.
Invariants play a crucial role in Diffusion MRI. In DTI (2nd order tensors), invariant scalars (FA, MD) have been successfully used in clinical applications. But DTI has limitations and HARDI models (e.g. 4th order tensors) have been proposed instead. These, however, lack invariant features and computing them systematically is challenging. We present a simple and systematic method to compute a functionally complete set of invariants of a non-negative 3D 4th order tensor with respect to SO3. Intuitively, this transforms the tensor's non-unique ternary quartic (TQ) decomposition (from Hilbert's theorem) to a unique canonical representation independent of orientation - the invariants. The method consists of two steps. In the first, we reduce the 18 degrees-of-freedom (DOF) of a TQ representation by 3-DOFs via an orthogonal transformation. This transformation is designed to enhance a rotation-invariant property of choice of the 3D 4th order tensor. In the second, we further reduce 3-DOFs via a 3D rotation transformation of coordinates to arrive at a canonical set of invariants to SO3 of the tensor. The resulting invariants are, by construction, (i) functionally complete, (ii) functionally irreducible (if desired), (iii) computationally efficient and (iv) reversible (mappable to the TQ coefficients or shape); which is the novelty of our contribution in comparison to prior work. Results from synthetic and real data experiments validate the method and indicate its importance.
不变量在扩散磁共振成像中起着至关重要的作用。在扩散张量成像（DTI，二阶张量）中，不变标量（如分数各向异性（FA）、平均扩散率（MD））已成功应用于临床。但DTI存在局限性，因此有人提出了高角分辨率扩散成像（HARDI）模型（如四阶张量）。然而，这些模型缺乏不变特征，系统地计算它们具有挑战性。我们提出了一种简单且系统的方法，用于计算相对于特殊正交群SO3的非负三维四阶张量的一组功能完备的不变量。直观地说，这将张量的非唯一三元四次（TQ）分解（源自希尔伯特定理）转换为与方向无关的唯一规范表示——不变量。该方法包括两个步骤。第一步，我们通过正交变换将TQ表示的18个自由度（DOF）减少3个自由度。此变换旨在增强三维四阶张量选择的旋转不变性。第二步，我们通过坐标的三维旋转变换进一步减少3个自由度，以得到张量相对于SO3的一组规范不变量。通过构造，所得不变量具有以下特点：（i）功能完备；（ii）功能不可约（如果需要）；（iii）计算效率高；（iv）可逆（可映射到TQ系数或形状）；与先前工作相比，这是我们贡献的新颖之处。合成数据和真实数据实验的结果验证了该方法并表明了其重要性。
