Tensor fields are useful for modeling the structure of biological tissues. The challenge to measure tensor fields involves acquiring sufficient data of scalar measurements that are physically achievable and reconstructing tensors from as few projections as possible for efficient applications in medical imaging. In this paper, we present a filtered back-projection algorithm for the reconstruction of a symmetric second-rank tensor field from directional X-ray projections about three axes. The tensor field is decomposed into a solenoidal and irrotational component, each of three unknowns. Using the Fourier projection theorem, a filtered back-projection algorithm is derived to reconstruct the solenoidal and irrotational components from projections acquired around three axes. A simple illustrative phantom consisting of two spherical shells and a 3D digital cardiac diffusion image obtained from diffusion tensor MRI of an excised human heart are used to simulate directional X-ray projections. The simulations validate the mathematical derivations and demonstrate reasonable noise properties of the algorithm. The decomposition of the tensor field into solenoidal and irrotational components provides insight into the development of algorithms for reconstructing tensor fields with sufficient samples in terms of the type of directional projections and the necessary orbits for the acquisition of the projections of the tensor field.

中文翻译：

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从获得的关于三个轴的投影的张量层析成像的分析算法

张量场可用于模拟生物组织的结构。测量张量场的挑战涉及获取足够的物理上可实现的标量测量数据，并从尽可能少的投影中重建张量，以便在医学成像中有效应用。在本文中，我们提出了一种滤波反投影算法，用于从三个轴的定向 X 射线投影重建对称二阶张量场。张量场被分解为一个螺线管和无旋分量，每个分量都是三个未知数。使用傅立叶投影定理，推导了一种过滤反投影算法，以根据围绕三个轴获取的投影重建螺线管和无旋分量。由两个球壳组成的简单说明性模型和从离体心脏的扩散张量 MRI 获得的 3D 数字心脏扩散图像用于模拟定向 X 射线投影。仿真验证了数学推导并证明了算法的合理噪声特性。将张量场分解为螺线管和无旋分量，可以深入了解用于根据方向投影的类型和获取张量场投影的必要轨道，用足够的样本重建张量场的算法的发展。仿真验证了数学推导并证明了算法的合理噪声特性。将张量场分解为螺线管和无旋分量，可以深入了解用于根据方向投影的类型和获取张量场投影的必要轨道，用足够的样本重建张量场的算法的发展。仿真验证了数学推导并证明了算法的合理噪声特性。将张量场分解为螺线管和无旋分量，可以深入了解用于根据方向投影的类型和获取张量场投影的必要轨道，用足够的样本重建张量场的算法的发展。