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Efficient estimation of three‐dimensional curves and their derivatives by free‐knot regression splines, applied to the analysis of inner carotid artery centrelines

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  • Laura M. Sangalli
  • Piercesare Secchi
  • Simone Vantini
  • Alessandro Veneziani

Abstract

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.

Suggested Citation

  • Laura M. Sangalli & Piercesare Secchi & Simone Vantini & Alessandro Veneziani, 2009. "Efficient estimation of three‐dimensional curves and their derivatives by free‐knot regression splines, applied to the analysis of inner carotid artery centrelines," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 58(3), pages 285-306, July.
    • Handle: RePEc:bla:jorssc:v:58:y:2009:i:3:p:285-306
    DOI: 10.1111/j.1467-9876.2008.00653.x
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    References listed on IDEAS

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    1. Gluhovsky, Ilya & Gluhovsky, Alexander, 2007. "Smooth Location-Dependent Bandwidth Selection for Local Polynomial Regression," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 718-725, June.
    2. Wenxin Mao & Linda H. Zhao, 2003. "Free‐knot polynomial splines with confidence intervals," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(4), pages 901-919, November.
    3. Zhang, Chunming, 2003. "Calibrating the Degrees of Freedom for Automatic Data Smoothing and Effective Curve Checking," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 609-628, January.
    4. Daniel Gervini, 2006. "Free‐knot spline smoothing for functional data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(4), pages 671-687, September.
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    Cited by:

    1. B. Ettinger & S. Perotto & L. M. Sangalli, 2016. "Spatial regression models over two-dimensional manifolds," Biometrika, Biometrika Trust, vol. 103(1), pages 71-88.
    2. Sangalli, Laura M. & Secchi, Piercesare & Vantini, Simone & Vitelli, Valeria, 2010. "k-mean alignment for curve clustering," Computational Statistics & Data Analysis, Elsevier, vol. 54(5), pages 1219-1233, May.
    3. Pigoli, Davide & Sangalli, Laura M., 2012. "Wavelets in functional data analysis: Estimation of multidimensional curves and their derivatives," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1482-1498.
    4. Aletti, Giacomo & May, Caterina & Tommasi, Chiara, 2016. "Best estimation of functional linear models," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 54-68.

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