Efficient estimation of 3-dimensional curves and their derivatives by free knot regression splines, applied to the analysis of inner carotid artery centerlines

We deal with the problem of efficiently estimating a 3D curve and its derivatives, starting from a discrete and noisy observation of the curve. This problem is now arising in many applicative contexts, thanks to the advent of devices that provide 3D images and measures, such as 3D scanners in medical diagnostics. Our research, in particular, stems from the need for accurate estimation of the curvature of an artery, from image reconstructions of 3D angiographies. This need has emerged within AneuRisk Project, a scientific endeavor which aims at investigating the role of vessel morphology, blood fluid-dynamics, and biomechanical properties of the vascular wall, on the pathogenesis of cerebral aneurysms. We develop a regression technique that exploits free knot splines in a novel setting, to estimate 3D curves and their derivatives. We thoroughly compare this technique to a classical regression method, local polynomial smoothing, showing that 3D free knot regression splines yield more accurate and efficient estimates.