Polynomial smoothing and curvature estimation of noisy biological surfaces with application to aortic morphology

We present a polynomial-based smoothing and curvature estimation pipeline for extracting geometric descriptors from noisy manifolds produced by segmentation of biological surfaces, with application to thoracic aortic geometry from computed tomography angiography (CTA). The approach is a mesh-adapted variant of local polynomial regression: we fit low-order polynomials on Monge patches to smooth segmentation noise while preserving volume, then estimate curvatures from the same locally defined polynomials. Conceptually, our smoothing step generalizes classic Savitzky-Golay least-squares fitting to irregularly sampled surface neighborhoods in the style of Locally Estimated Scatterplot Smoothing, while the curvature step follows standard differential geometry on Monge patches. Per-facet curvatures are aggregated into scale-invariant integrated quantities including the total squared and integrated Gaussian curvatures, which serve as global shape descriptors. The result is a transparent, reproducible workflow that preserves interpretable surface features and yields physically meaningful descriptors suitable for downstream modeling. We validate on a cohort of 267 CTA scans from normal and dissected aortas. As an illustrative downstream task, we show that integrated curvature descriptors derived from our pipeline recover outcome-group separability comparable to prior analyses, while providing a richer per-facet curvature field for future modeling.