Tubular curvature filter: pointwise curvature calculation for tubular objects in images

Purpose: Accurate estimation of blood vessel tortuosity from medical images is an extremely important and challenging task. It is particularly relevant in the context of retinopathy of prematurity (ROP), where the staging of disease severity and consequent therapeutic approaches are heavily informed by the presence and prominence of vessel tortuosity. Existing methods based on centerline or skeleton curvature fail to capture curvature gradients across a rotating tubular structure, thereby limiting their effectiveness in the case of ROP. Design: A methodological study with validation on artificial and real images. Methods: This paper defines local tubular curvature and presents the Tubular Curvature Filter (TCF) method, which locally calculates the acceleration of curve bundles traversing a tubular object parallel to its centerline. These curves are parallel to the centerline in the sense that they are integral curves following the direction of the ridge. This is achieved by examining the directional rate of change in the eigenvectors of the Hessian matrix of a tubular intensity function in space. TCF implicitly calculates the local tubular curvature without the need to explicitly segment or extracting the centerline of the tubular object. Results: Experimental results demonstrate that TCF provides accurate estimates of local curvature at any point inside tubular structures. Results on two- and three-dimensional images show that TCF discerns curvature differences between the inner and outer sides of curved tubular objects, while centerline-based approaches cannot. Conclusion: Our findings highlight that TCF’s ability to discern between the inner and outer sides of curved tubular objects is particularly useful in medical fields that require vasculature curvature analysis from images, especially where vascular structures often have non-uniform diameters, such as in ROP.