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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

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. Copyright (c) 2009 Royal Statistical Society.

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.
  • Handle: RePEc:bla:jorssc:v:58:y:2009:i:3:p:285-306
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    File URL: http://www.blackwell-synergy.com/doi/abs/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. 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.
    3. 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.
    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.
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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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