Blood vessel orientation as visualized in 3D medical images is an important descriptor of its geometry that can be used for centerline extraction and subsequent segmentation and visualization. Arteries appear at many scales and levels of tortuosity, and determining their exact orientation is challenging. Recent works have used 3D convolutional neural networks (CNNs) for this purpose, but CNNs are sensitive to varying vessel sizes and orientations. We present SIRE: a scale-invariant, rotation-equivariant estimator for local vessel orientation. SIRE is modular and can generalise due to symmetry preservation. SIRE consists of a gauge equivariant mesh CNN (GEM-CNN) operating on multiple nested spherical meshes with different sizes in parallel. The features on each mesh are a projection of image intensities within the corresponding sphere. These features are intrinsic to the sphere and, in combination with the GEM-CNN, lead to SO(3)-equivariance. Approximate scale invariance is achieved by weight sharing and use of a symmetric maximum function to combine multi-scale predictions. Hence, SIRE can be trained with arbitrarily oriented vessels with varying radii to generalise to vessels with a wide range of calibres and tortuosity. We demonstrate the efficacy of SIRE using three datasets containing vessels of varying scales: the vascular model repository (VMR), the ASOCA coronary artery set, and a set of abdominal aortic aneurysms (AAAs). We embed SIRE in a centerline tracker which accurately tracks AAAs, regardless of the data SIRE is trained with. Moreover, SIRE can be used to track coronary arteries, even when trained only with AAAs. In conclusion, by incorporating SO(3) and scale symmetries, SIRE can determine the orientations of vessels outside of the training domain, forming a robust and data-efficient solution to geometric analysis of blood vessels in 3D medical images.

翻译：摘要：三维医学图像中显示的血管方向是描述其几何形态的重要特征，可用于中心线提取及后续分割与可视化。动脉在不同曲度层级上呈现多尺度特性，确定其精确方向具有挑战性。已有研究采用三维卷积神经网络（CNN）实现该目标，但CNN对血管尺寸和方向变化敏感。我们提出SIRE：一种尺度不变、旋转等变的局部血管方向估计器。SIRE通过对称性保持实现模块化与泛化能力。SIRE包含一个规范等变网格CNN（GEM-CNN），并行处理多个不同尺寸的嵌套球面网格。每个网格上的特征是对应球体内图像强度的投影，这些特征内蕴于球面，并与GEM-CNN结合实现SO(3)等变性。通过权重共享和对称最大值函数整合多尺度预测，实现了近似尺度不变性。因此，SIRE可训练于任意方向且半径可变的血管，从而泛化至宽泛管径和曲度的血管。我们使用三个包含不同尺度血管的数据集验证SIRE的有效性：血管模型库（VMR）、ASOCA冠状动脉数据集和一组腹主动脉瘤（AAA）数据。将SIRE嵌入中心线追踪器后，无论训练数据如何，均可精确追踪AAA。此外，SIRE即使仅经AAA训练，也可用于追踪冠状动脉。总之，通过整合SO(3)与尺度对称性，SIRE能确定训练域外血管的方向，为三维医学图像中血管几何分析提供鲁棒且数据高效的解决方案。