Methods for capturing images in three dimensions are now widely available, with stereo-photogrammetry and laser scanning being two common approaches. In anatomical studies, a number of landmarks are usually identified manually from each of these images and these form the basis of subsequent statistical analysis. However, landmarks express only a very small proportion of the information available from the images. Anatomically defined curves have the advantage of providing a much richer expression of shape. This is explored in the context of identifying the boundary of breasts from an image of the female torso and the boundary of the lips from a facial image. The curves of interest are characterised by ridges or valleys. Key issues in estimation are the ability to navigate across the anatomical surface in three-dimensions, the ability to recognise the relevant boundary and the need to assess the evidence for the presence of the surface feature of interest. The first issue is addressed by the use of principal curves, as an extension of principal components, the second by suitable assessment of curvature and the third by change-point detection. P-spline smoothing is used as an integral part of the methods but adaptations are made to the specific anatomical features of interest. After estimation of the boundary curves, the intermediate surfaces of the anatomical feature of interest can be characterised by surface interpolation. This allows shape variation to be explored using standard methods such as principal components. These tools are applied to a collection of images of women where one breast has been reconstructed after mastectomy and where interest lies in shape differences between the reconstructed and unreconstructed breasts. They are also applied to a collection of lip images where possible differences in shape between males and females are of interest.

Institute of Mathematics and Statistics Masaryk University Brno Czech Republic

MRC CSO Social and Public Health Sciences Unit The University of Glasgow UK

School of Mathematics and Statistics The University of Glasgow UK

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Andresen P., Bookstein F., Conradsen K., Ersboll B., Marsh J., Kreiborg S. Surface-bounded growth modeling applied to human mandibles. IEEE Trans. Med. Imaging. 2000;19:1053–1063. PubMed

Ayoub A., Bell A., Simmons D., Bowman A., Brown D., Lo T.W., Xiao Y. 3d assessment of lip scarring and residual dysmorphology following surgical repair of cleft lip and palate: a preliminary study. Cleft Palate Craniofac. J. 2011;48:379–387. PubMed

Banfield J., Raftery A. Ice-floe identification in satellite images using mathematical morphology and clustering about principal curves. J. Amer. Statist. Assoc. 1992;87:7–16.

Barber, C.B., Habel, K., Grasman, R., Gramacy, R.B., Stahel, A., Sterratt, D.C., 2012. geometry: Mesh generation and surface tesselation, CRAN.

Bollaerts K., Eilers P.H.C., van Mechelen I. Simple and multiple p-splines regression with shape constraints. Br. J. Math. Stat. Psychol. 2006;59:451–469. PubMed

Bookstein F. Principal warps: thin-plate splines and the decomposition of deformations. IEEE Trans. Pattern Anal. Mach. Intell. 1989;11:567–585.

Bowman A.W., Pope A., Ismail B. Detecting discontinuities in nonparametric regression curves and surfaces. Stat. Comput. 2006;16:377–390.

Caffo B.S., Crainiceanu C.M., Deng L., Hendrix C.W. A case study in pharmacologic colon imaging using principal curves in single-photon emission computed tomography. J. Amer. Statist. Assoc. 2008;103:1470–1480. PubMed PMC

Canny J. A computational approach to edge-detection. IEEE Trans. Pattern Anal. Mach. Intell. 1986;8:679–698. PubMed

Carlstein E., Müller H.G., Sigmund D. IMS; Hayward, CA: 1994. Change-point Problems.

Cleveland W. Robust locally weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc. 1979;74:829–836.

Cootes T.F., Hill A., Taylor C.J., Haslam J. Use of active shape models for locating structure in medical images. Image Vis. Comput. 1994;12:355–365.

Delmas, P., Coulon, P.Y., Fristot, V., 1999. Automatic snakes for robust lip boundaries extraction, in: ICASSP ’99: 1999, IEEE International Conference On Acoustics, Speech, and Signal Processing, Proceedings I-VI, pp. 3069–3072.

Dryden I., Mardia K. Wiley; New York: 1998. The Statistical Analysis of Shape.

Duffy S., Noar J., Evans R., Sanders F. Three-dimensional analysis of the child cleft face. Cleft Palate Craniofac. J. 2000;37:137–144. PubMed

Eilers P., Marx B. Flexible smoothing with b-splines and penalties. Statist. Sci. 1996;11:89–102.

Eveno, N., Caplier, A., Coulon, P., 2001. A new color transformation for lips segmentation, in: Dugelay, J., Rose, K. (Eds.), 2001 IEEE Fourth Workshop on Multimedia Signal Processing, pp. 3–8. Cannes, France.

Eveno, N., Caplier, A., Coulon, P., 2002. Key points based segmentation of lips, in: IEEE International Conference on Multimedia and Expo, Vol I and II, pp. A125–A128. Lausanne, Switzerland.

Fan J., Gijbels I. Chapman and Hall; London: 1996. Local Polynomial Modelling and its Applications.

Gasser T., Sroka L., Jennen-Steinmetz C. Residual variance and residual pattern in nonlinear-regression. Biometrika. 1986;73:625–633.

Green P., Silverman B. Chapman and Hall/CRC; London: 1994. Nonparametric Regression and Generalized Linear Models.

Hall P., Peng L., Rau C. Local likelihood tracking of fault lines and boundaries. J. R. Stat. Soc. Ser. B Stat. Methodol. 2001;63:569–582.

Hall P., Titterington DM, D. Edge-preserving and peak-preserving smoothing. Technometrics. 1992;34:429–440.

Hammond P., Hutton T., Allanson J., Buxton B., Campbell L., Smith J., Donnai D., Smith A., Metcalfe K., Murphy K., Patton M., Pober B., Prescott K., Scambler P., Shaw A., Smith A., Stevens A., Temple I., Hennekam R., Tassabehji M. Discriminating power of localized three-dimensional facial morphology. Am. J. Hum. Genet. 2005;77:999–1010. PubMed PMC

Hastie T., Stuetzle W. Principal curves. J. Amer. Statist. Assoc. 1989;84:502–516.

Hastie, T., Weingessel, A., 2011. princurve: fits a principal curve in arbitrary dimension (S original by Trevor Hastie and R port by Andreas Weingessel), CRAN.

Hennessy R.J., Baldwin P.A., Browne D.J., Kinsella A., Waddington J.L. Frontonasal dysmorphology in bipolar disorder by 3d laser surface imaging and geometric morphometrics: Comparisons with schizophrenia. Schizophr. Res. 2010;122:63–71. PubMed PMC

Henseler H., Khambay B.S., Bowman A., Smith J., Siebert J.P., Oehler S., Ju X., Ayoub A., Ray A.K. Investigation into accuracy and reproducibility of a 3d breast imaging system using multiple stereo cameras. J. Plastic Reconstr. Aesthetic Surg. 2011;64:577–582. PubMed

Henseler H., Smith J., Bowman A., Khambay B.S., Ju X., Ayoub A., Ray A.K. Objective evaluation of the latissimus dorsi flap for breast reconstruction using three-dimensional imaging. J. Plastic Reconstr. Aesthetic Surg. 2012;66:1209–1215. PubMed

Hood C., Bock M., Hosey M., Bowman A., Ayoub A. Facial asymmetry — 3D assessment of infants with cleft lip & palate. Int. J. Paediat. Dent. 2003;13:404–410. PubMed

Hulbert A., Poggio T. Synthesizing a color algorithm from examples. Science. 1998;239:482–485. PubMed

Kakumanu P., Makrogiannis S., Bourbakis N. A survey of skin-color modeling and detection methods. Pattern Recognit. 2007;40:1106–1122.

Kass M., Witkin A., Terzopoulos D. Snakes — active contour models. Int. J. Comput. Vis. 1987;1:321–331.

Koenderink J.J. MIT Press; Boston: 1990. Solid Shape.

Krueger, M., Delmas, P., Gimel’farb, G., 2008. Active contour based segmentation of 3D surfaces, in: Computer Vision — ECCV 2008 Proceedings, Pt II 5303, 350–363.

Larsen, R., 2005. Functional 2D procrustes shape analysis, in: Image Analysis, Proceedings 3540, 205–213.

Lopez A.M., Lumbreras F., Serrat J., Villanueva J.J. Evaluation of methods for ridge and valley detection. IEEE Trans. Pattern Anal. Mach. Intell. 1999;21:327–335.

Matthews I., Cootes T.F., Bangham J.A., Cox S., Harvey R. Extraction of visual features for lipreading. IEEE Trans. Pattern Anal. Mach. Intell. 2002;24:198–213.

McNeil K. The University of Glasgow; 2012. Analysis of Three-Dimensional Facial Shape. (Ph.D. thesis)

Müller H. Change-points in nonparametric regression analysis. Ann. Statist. 1992;20:737–761.

Nkenke E., Lehner B., Kramer M., Haeusler G., Benz S., Schuster M., Neukam F., Vairaktaris E., Wurm J. Determination of facial symmetry in unilateral cleft lip and palate patients from three-dimensional data: Technical report and assessment of measurement errors. Cleft Palate Craniofac. J. 2006;43:129–137. PubMed

Ozgur, E., Yilmaz, B., Karabalkan, H., Erdogan, H., Unel, M., 2008. Lip segmentation using adaptive color space training, in: AVSP.

Ramsay J., Silverman B. 2nd ed. Springer; New York: 2005. Functional Data Analysis.

R Foundation for Statistical Computing; Vienna, Austria: 2013. R: A Language and Environment for Statistical Computing. R Development Core Team.

Schyns P.G., Petro L.S., Smith M.L. Transmission of facial expressions of emotion co-evolved with their efficient decoding in the brain: Behavioral and brain evidence. Plos One. 2009;4:e5625. PubMed PMC

Seber G. Wiley; New York: 1977. Linear Regression Analysis.

Stylianou G., Farin G. Crest lines for surface segmentation and flattening. IEEE Trans. Vis. Comput. Graphics. 2004;10:536–544. PubMed

Wood S. Chapman and Hall/CRC; London: 2006. Generalized Additive Models: An Introduction With R.

Wyse J., Friel N., Rue H. Approximate simulation-free bayesian inference for multiple changepoint models with dependence within segments. Bayesian Anal. 2011;6:501–527.

A Hierarchical Curve-Based Approach to the Analysis of Manifold Data