Unsupervised classification of children’s bodies using currents. (English) Zbl 1414.62431

Summary: Object classification according to their shape and size is of key importance in many scientific fields. This work focuses on the case where the size and shape of an object is characterized by a current. A current is a mathematical object which has been proved relevant to the modeling of geometrical data, like submanifolds, through integration of vector fields along them. As a consequence of the choice of a vector-valued reproducing kernel Hilbert space (RKHS) as a test space for integrating manifolds, it is possible to consider that shapes are embedded in this Hilbert Space. A vector-valued RKHS is a Hilbert space of vector fields; therefore, it is possible to compute a mean of shapes, or to calculate a distance between two manifolds. This embedding enables us to consider size-and-shape clustering algorithms. These algorithms are applied to a 3D database obtained from an anthropometric survey of the Spanish child population with a potential application to online sales of children’s wear.


62P10 Applications of statistics to biology and medical sciences; meta analysis
62H30 Classification and discrimination; cluster analysis (statistical aspects)
68T10 Pattern recognition, speech recognition


Matlab; clusfind
Full Text: DOI arXiv


[1] Allen, B.; Curless, B.; Popović, Z., The space of human body shapes: reconstruction and parameterization from range scans, ACM Trans Graph TOG, 22, 587-594, (2003)
[2] Aronszajn, N., Theory of reproducing kernels, Trans Am Math Soc, 68, 337-404, (1950) · Zbl 0037.20701
[3] Baek, SY; Lee, K., Parametric human body shape modeling framework for human-centered product design, Comput Aided Des, 44, 56-67, (2012)
[4] Bauer, M.; Harms, P.; Michor, PW, Sobolev metrics on shape space of surfaces, J Geom Mech, 3, 389-438, (2011) · Zbl 1262.58004
[5] Bock HH (2007) Clustering methods: a history of k-means algorithms. In: Brito P, Bertrand P, Cucumel G, de Carvalho F (eds) Selected contributions in data analysis and classification. Springer, Berlin, pp 161-172
[6] Caponnetto, A.; Micchelli, CA; Pontil, M.; Ying, Y., Universal multi-task kernels, J Mach Learn Res, 9, 1615-1646, (2008) · Zbl 1225.68155
[7] Carmeli, C.; Vito, E.; Toigo, A., Vector valued reproducing kernel hilbert spaces of integrable functions and mercer theorem, Anal Appl, 4, 377-408, (2006) · Zbl 1116.46019
[8] Chung, M.; Lina, H.; Wang, MJJ, The development of sizing systems for taiwanese elementary- and high-school students, Int J Ind Ergon, 37, 707-716, (2007)
[9] Conway JB (2013) A course in functional analysis, vol 96. Springer, Science & Business Media
[10] Cox TF, Cox MA (2000) Multidimensional scaling. CRC press
[11] Do Carmo MP (2012) Differential forms and applications. Springer, Science & Business Media
[12] Durrleman S (2010) Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution. PhD thesis, Université Nice Sophia Antipolis
[13] Durrleman, S.; Pennec, X.; Trouvé, A.; Ayache, N., Statistical models of sets of curves and surfaces based on currents, Med Image Anal, 13, 793-808, (2009)
[14] European Committee for Standardization (2002) European Standard EN 13402-2: Size system of clothing. Primary and secondary dimensions. http://esearch.cen.eu/esearch/Details.aspx?id=5430955
[15] Glaunès J (2005) Transport par difféomorphismes de points, de mesures et de courants pour la comparaison de formes et l’anatomie numérique. PhD thesis, Université Paris 13. http://cis.jhu.edu/joan/TheseGlaunes.pdf. Accessed Sept 2005
[16] Glaunes JA, Joshi S (2006) Template estimation form unlabeled point set data and surfaces for computational anatomy. In: 1st MICCAI workshop on mathematical foundations of computational anatomy: geometrical, statistical and registration methods for modeling biological shape variability
[17] Gual-Arnau, X.; Herold-García, S.; Simó, A., Geometric analysis of planar shapes with applications to cell deformations, Image Anal Stereol, 34, 171-182, (2015) · Zbl 1379.94003
[18] Hsing T, Eubank R (2015) Theoretical foundations of functional data analysis, with an introduction to linear operators. Wiley · Zbl 1338.62009
[19] Huang, H.; Wang, F.; Guibas, L., Functional map networks for analyzing and exploring large shape collections, ACM Trans Graph, 33, 1-11, (2014) · Zbl 1396.65046
[20] Ibáñez, MV; Vinué, G.; Alemany, S.; Simó, A.; Epifanio, I.; Domingo, J.; Ayala, G., Apparel sizing using trimmed PAM and OWA operators, Expert Syst Appl, 29, 10512-10520, (2012)
[21] Jain, A.; Thormählen, T.; Seidel, HP; Theobalt, C., Moviereshape: tracking and reshaping of humans in videos, ACM Trans Graph TOG, 29, 148, (2010)
[22] Jain, AK, Data clustering: 50 years beyond k-means, Pattern Recognit Lett, 31, 651-666, (2010)
[23] Kanungo, T.; Mount, DM; Netanyahu, NS; Piatko, C.; Silverman, R.; Wu, AY, An efficient k-means clustering algorithm: analysis and implementation, IEEE Trans Pattern Anal Mach Intell, 24, 881-892, (2002) · Zbl 1373.68466
[24] Kaufman L, Rousseeuw P (1990) Finding groups in data: an introduction to cluster analysis. Wiley, New York · Zbl 1345.62009
[25] Lang S (1995) Differential and Riemannian manifolds. Springer, New York · Zbl 0824.58003
[26] Lloyd SP (1957) Least squares quantization in pcm. Bell telephone labs memorandum, Murray Hill, nj. Reprinted in IEEE trans information theory IT-28 (1982) vol 2. pp 129-137
[27] MATLAB (2014) version 8.4.0 (R2014b). The MathWorks Inc., Natick
[28] Micchelli, C.; Pontil, M., On learning vector-valued functions, Neural Comput, 17, 177-204, (2005) · Zbl 1092.93045
[29] Morgan F (2008) Geometric measure theory: a beginner’s guide. Academic Press, Cambridge · Zbl 0974.49025
[30] Nazeer KAA, Sebastian MP (2009) Improving the accuracy and efficiency of the k-means clustering algorithm. In: Proceedings of the World Congress on Engineering (London, UK), pp 1-5
[31] Ovsjanikov, M.; Ben-Chen, M.; Solomon, J.; Butscher, A.; Guibas, L., Functional maps: a flexible representation of maps between shapes, ACM Trans Graph, 31, 1-11, (2012)
[32] Pennec, X., Intrinsic statistics on riemannian manifolds: basic tools for geometric measurements, J Math Imaging Vis, 25, 127-154, (2006)
[33] Pishchulin L, Wuhrer S, Helten T, Theobalt C, Schiele B (2015) Building statistical shape spaces for 3d human modeling. arXiv:1503.05860
[34] Quang, MH; Kang, SH; Le, TM, Image and video colorization using vector-valued reproducing kernel hilbert spaces, J Math Imaging Vis, 37, 49-65, (2010) · Zbl 1392.94042
[35] Steinhaus H (1956) Sur la division des corps matériels en parties. Bull Acad Pol Sci IV(12):801-804 · Zbl 0079.16403
[36] Vaillant, M.; Glaunès, J., Surface matching via currents, 381-392, (2005), Berlin, Heidelberg
[37] Vinué, G.; Simó, A.; Alemany, S., The k-means algorithm for 3d shapes with an application to apparel design, Adv Data Anal Classif, 10, 103-132, (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.