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The \(k\)-means algorithm for 3D shapes with an application to apparel design

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Abstract

Clustering of objects according to shapes is of key importance in many scientific fields. In this paper we focus on the case where the shape of an object is represented by a configuration matrix of landmarks. It is well known that this shape space has a finite-dimensional Riemannian manifold structure (non-Euclidean) which makes it difficult to work with. Papers about clustering on this space are scarce in the literature. The basic foundation of the \(k\)-means algorithm is the fact that the sample mean is the value that minimizes the Euclidean distance from each point to the centroid of the cluster to which it belongs, so, our idea is integrating the Procrustes type distances and Procrustes mean into the \(k\)-means algorithm to adapt it to the shape analysis context. As far as we know, there have been just two attempts in that way. In this paper we propose to adapt the classical \(k\)-means Lloyd algorithm to the context of Shape Analysis, focusing on the three dimensional case. We present a study comparing its performance with the Hartigan-Wong \(k\)-means algorithm, one that was previously adapted to the field of Statistical Shape Analysis. We demonstrate the better performance of the Lloyd version and, finally, we propose to add a trimmed procedure. We apply both to a 3D database obtained from an anthropometric survey of the Spanish female population conducted in this country in 2006. The algorithms presented in this paper are available in the Anthropometry R package, whose most current version is always available from the Comprehensive R Archive Network.

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References

  • Alemany S, González JC, Nácher B, Soriano C, Arnáiz C, Heras H (2010) Anthropometric survey of the spanish female population aimed at the apparel industry. In: Proceedings of the 2010 Intl Conference on 3D Body scanning Technologies, Lugano, Switzerland, pp 1–10

  • Amaral G, Dore L, Lessa R, Stosic B (2010) k-means algorithm in statistical shape analysis. Commun Stat Simul Comput 39(5):1016–1026

    Article  MathSciNet  MATH  Google Scholar 

  • Anderberg M (1973) Cluster analysis for applications. Academic Press, New York

    MATH  Google Scholar 

  • Best D, Fisher N (1979) Efficient simulation of the von mises distribution. J R Stat Soc Ser C (Appl Stat) 28(2):152–157

    MATH  Google Scholar 

  • Bhattacharya R, Patrangenaru V (2002) Nonparametric estimation of location and dispersion on riemannian manifolds. J Stat Plann Inference 108:23–35

    Article  MathSciNet  MATH  Google Scholar 

  • Bhattacharya R, Patrangenaru V (2003) Large sample theory of intrinsic and extrinsic sample means on manifolds. Ann Stat 31(1):1–29

    Article  MathSciNet  MATH  Google Scholar 

  • 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 Heidelberg, pp 161–172

    Chapter  Google Scholar 

  • Bock HH (2008) Origins and extensions of the k-means algorithm in cluster analysis. Electron J Hist Prob Stat 4(2):1–18

    MathSciNet  Google Scholar 

  • Cai X, Li Z, Chang CC, Dempsey P (2005) Analysis of alignment influence on 3-D anthropometric statistics. Tsinghua Sci Technol 10(5):623–626

    Article  MathSciNet  Google Scholar 

  • Chernoff H (1970) Metric considerations in cluster analysis. In: Proc. 6th Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, pp 621–629

  • Chung M, Lina H, Wang MJJ (2007) The development of sizing systems for taiwanese elementary- and high-school students. Int J Ind Ergon 37:707–716

    Article  Google Scholar 

  • Claude J (2008) Morphometrics with R. use R!. Springer, New York

    Google Scholar 

  • Dryden IE, Mardia KV (1998) Statistical shape analysis. Wiley, Chichester

    MATH  Google Scholar 

  • Dryden IL (2012) Shapes package. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org, contributed package

  • European Committee for Standardization. European Standard EN 13402–2: Size system of clothing. Primary and secondary dimensions (2002)

  • Fletcher P, Lu C, Pizer S, Joshi S (2004) Principal geodesic analysis for the study of nonlinear statistics of shape. Med Imaging IEEE Trans 23:995–1005

    Article  Google Scholar 

  • Fréchet M (1948) Les éléments aléatoires de nature quelconque dans un espace distancié. Ann Inst Henri Poincare Prob Stat 10(4):215–310

    Google Scholar 

  • García-Escudero LA, Gordaliza A (1999) Robustness properties of k-means and trimmed k-means. J Am Stat Assoc 94(447):956–969

    MATH  Google Scholar 

  • Georgescu V (2009) Clustering of fuzzy shapes by integrating Procrustean metrics and full mean shape estimation into k-means algorithm. In: IFSA-EUSFLAT Conference (Lisbon, Portugal), pp 1679–1684

  • Hand DJ, Krzanowski WJ (2005) Optimising k-means clustering results with standard software packages. Comput Stat Data Anal 49:969.973 short communication

    Article  MathSciNet  Google Scholar 

  • Hartiga JA, Wong MA (1979) A K-means clustering algorithm. Appl Stat 100–108

  • Hastie T, Tibshirani R, Friedman J (2008) The elements of statistical learning. Springer, New York

    Google Scholar 

  • Ibáñez MV, Vinué G, Alemany S, Simó A, Epifanio I, Domingo J, Ayala G (2012) Apparel sizing using trimmed PAM and OWA operators. Expert Syst Appl 39:10,512–10,520

    Article  Google Scholar 

  • Jain AK (2010) Data clustering: 50 years beyond k-means. Pattern Recognit Lett 31:651–666

    Article  Google Scholar 

  • Kanungo T, Mount DM, Netanyahu NS, Piatko C, Silverman R, Wu AY (2002) An efficient k-means clustering algorithm: analysis and implementation. IEEE Trans Pattern Anal Mach Intell 24(7):881–892

    Article  Google Scholar 

  • Karcher H (1977) Riemannian center of mass and mollifier smoothing. Commun Pure Appl Math 30(5):509–541

    Article  MathSciNet  MATH  Google Scholar 

  • Kaufman L, Rousseeuw P (1990) Finding groups in data: an introduction to cluster analysis. Wiley, New York

  • Kendall D (1977) The diffusion of shape. Adv Appl Prob 9:428–430

    Article  Google Scholar 

  • Kendall DG, Barden D, Carne T, Le H (2009) Shape and shape theory. Wiley, Chichester

    Google Scholar 

  • Kendall WS (1990) Probability, convexity, and harmonic maps with small image i: uniqueness and fine existence. Proc Lond Math Soc 3(2):371–406

    Article  MathSciNet  Google Scholar 

  • Kent J, Mardia K (1997) Consistency of procrustes estimators. J R Stat Soc Ser B 59(1):281–290

    Article  MathSciNet  MATH  Google Scholar 

  • Kobayashi S, Nomizu K (1969) Foundations of differential geometry, vol 2. Wiley, Chichester

    MATH  Google Scholar 

  • Lawing A, Polly P (2010) Geometric morphometrics: recent applications to the study of evolution and development. J Zool 280(1):1–7

    Article  Google Scholar 

  • Le H (1998) On the consistency of Procrustean mean shapes. Adv Appl Prob 30(1):53–63

    Article  MATH  Google Scholar 

  • Lloyd SP (1957) Least squares quantization in pcm. bell telephone labs memorandum, murray hill, nj. reprinted. In: IEEE Trans Information Theory IT-28 (1982) 2:129–137

  • MacQueen J (1967) Some methoods for classification and analysis of mulivariate observations. In: Proc 5th Berkely Symp Math Statist Probab. Univ of California Press B (ed) 1965/66, vol 1, pp 281–297

  • 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

  • Ng R, Ashdown S, Chan A (2007) Intelligent size table generation. Sen’i Gakkaishi (J Soc Fiber Sci Technol Jpn) 63(11):384–387

  • Pennec X (2006) Intrinsic statistics on riemannian manifolds: basic tools for geometric measurements. J Math Imaging Vis 25(1):127–154

  • Qiu W, Joe H (2013) ClusterGeneration: random cluster generation (with specified degree of separation. http://CRAN.R-project.org/package=clusterGeneration, R package version 1.3.1

  • R Development Core Team (2014) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org, ISBN 3-900051-07-0

  • Rohlf JF (1999) Shape statistics: Procrustes superimpositions and tangent spaces. J Classif 16:197–223

    Article  MATH  Google Scholar 

  • S-plus original by Ulric Lund and R port by Claudio Agostinelli (2012) CircStats: Circular Statistics, from “Topics in circular Statistics” (2001). http://CRAN.R-project.org/package=CircStats, R package version 0.2–4

  • Simmons K (2002) Body shape analysis using three-dimensional body scanning technology. PhD thesis, North Carolina State University

  • Small C (1996) The statistical theory of shape. Springer, New York

    Book  MATH  Google Scholar 

  • Sokal R, Sneath PH (1963) Principles of numerical taxonomy. Freeman, San Francisco

    Google Scholar 

  • Steinhaus H (1956) Sur la division des corps matériels en parties. Bull Acad Pol Sci IV(12):801–804

    MathSciNet  Google Scholar 

  • Steinley D (2006) K-means clustering: a half-century synthesis. Br J Math Stat Psychol 59:1–34

    Article  MathSciNet  Google Scholar 

  • Stoyan LA, Stoyan H (1995) Fractals, random shapes and point fields. Wiley, Chichester

    Google Scholar 

  • Theodoridis S, Koutroumbas K (1999) Pattern recognition. Academic, New York

    Google Scholar 

  • Veitch D, Fitzgerald C et al (2013) Sizing up Australia—the next step. Safe Work Australia, Canberra

    Google Scholar 

  • Vinué G, Epifanio I, Simó A, Ibáñez MV, Domingo J, Ayala G (2014) Anthropometry: an R Package for analysis of anthropometric data. http://CRAN.R-project.org/package=Anthropometry, R package version 1.0

  • Woods R (2003) Characterizing volume and surface deformations in an atlas framework: theory, applications, and implementation. NeuroImage 18:769–788

    Article  Google Scholar 

  • Zheng R, Yu W, Fan J (2007) Development of a new chinese bra sizing system based on breast anthropometric measurements. Int J Ind Ergon 37:697–705

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Statistics, O.R. University of Valencia, C/ Dr. Moliner, 50, Burjassot, 46100, Valencia, Spain

    Guillermo Vinué

  2. Department of Mathematics-IMAC, Universitat Jaume I, Avda. del Riu Sec s/n, 12071, Castellón, Spain

    Amelia Simó

  3. Biomechanics Institute of Valencia, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022, Valencia, Spain

    Sandra Alemany

Authors
  1. Guillermo Vinué
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  2. Amelia Simó
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  3. Sandra Alemany
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Correspondence to Amelia Simó.

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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). https://doi.org/10.1007/s11634-014-0187-1

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  • DOI: https://doi.org/10.1007/s11634-014-0187-1

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