SIRE: scale-invariant, rotation-equivariant estimation of artery
  orientations using graph neural networks
Dieuwertje Alblas Julian Suk Christoph Brune Kak Khee Yeung Jelmer M. Wolterink
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.