Efficient Estimation of Three-Dimensional Curves and their Derivatives by Free-Knot Regression Splines, Applied to the Analysis of Inner Carotid Artery Centrelines

Summary We deal with the problem of efficiently estimating a three-dimensional 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 three-dimensional images and measures, such as three-dimensional 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 three-dimensional angiographies. This need has emerged within the AneuRisk project, a scientific endeavour which aims to investigate 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 three-dimensional curves and their derivatives. We thoroughly compare this technique with a classical regression method, local polynomial smoothing, showing that three-dimensional free-knot regression splines yield more accurate and efficient estimates.