Advertisement

Advances in Data Analysis and Classification

, Volume 10, Issue 1, pp 103–132 | Cite as

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

  • Guillermo Vinué
  • Amelia Simó
  • Sandra Alemany
Regular Article

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.

Keywords

Shape space Statistical shape analysis \(k\)-means algorithm  Procrustes type distances Procrustes mean shape Sizing systems 

Mathematics Subject Classification

62H11 62H30 

References

  1. 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 1–10Google Scholar
  2. Amaral G, Dore L, Lessa R, Stosic B (2010) k-means algorithm in statistical shape analysis. Commun Stat Simul Comput 39(5):1016–1026CrossRefMathSciNetzbMATHGoogle Scholar
  3. Anderberg M (1973) Cluster analysis for applications. Academic Press, New YorkzbMATHGoogle Scholar
  4. Best D, Fisher N (1979) Efficient simulation of the von mises distribution. J R Stat Soc Ser C (Appl Stat) 28(2):152–157zbMATHGoogle Scholar
  5. Bhattacharya R, Patrangenaru V (2002) Nonparametric estimation of location and dispersion on riemannian manifolds. J Stat Plann Inference 108:23–35CrossRefMathSciNetzbMATHGoogle Scholar
  6. Bhattacharya R, Patrangenaru V (2003) Large sample theory of intrinsic and extrinsic sample means on manifolds. Ann Stat 31(1):1–29CrossRefMathSciNetzbMATHGoogle Scholar
  7. 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 161–172CrossRefGoogle Scholar
  8. Bock HH (2008) Origins and extensions of the k-means algorithm in cluster analysis. Electron J Hist Prob Stat 4(2):1–18MathSciNetGoogle Scholar
  9. Cai X, Li Z, Chang CC, Dempsey P (2005) Analysis of alignment influence on 3-D anthropometric statistics. Tsinghua Sci Technol 10(5):623–626CrossRefMathSciNetGoogle Scholar
  10. Chernoff H (1970) Metric considerations in cluster analysis. In: Proc. 6th Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, pp 621–629Google Scholar
  11. Chung M, Lina H, Wang MJJ (2007) The development of sizing systems for taiwanese elementary- and high-school students. Int J Ind Ergon 37:707–716CrossRefGoogle Scholar
  12. Claude J (2008) Morphometrics with R. use R!. Springer, New YorkGoogle Scholar
  13. Dryden IE, Mardia KV (1998) Statistical shape analysis. Wiley, ChichesterzbMATHGoogle Scholar
  14. Dryden IL (2012) Shapes package. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org, contributed package
  15. European Committee for Standardization. European Standard EN 13402–2: Size system of clothing. Primary and secondary dimensions (2002)Google Scholar
  16. Fletcher P, Lu C, Pizer S, Joshi S (2004) Principal geodesic analysis for the study of nonlinear statistics of shape. Med Imaging IEEE Trans 23:995–1005CrossRefGoogle Scholar
  17. Fréchet M (1948) Les éléments aléatoires de nature quelconque dans un espace distancié. Ann Inst Henri Poincare Prob Stat 10(4):215–310Google Scholar
  18. García-Escudero LA, Gordaliza A (1999) Robustness properties of k-means and trimmed k-means. J Am Stat Assoc 94(447):956–969zbMATHGoogle Scholar
  19. 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 1679–1684Google Scholar
  20. Hand DJ, Krzanowski WJ (2005) Optimising k-means clustering results with standard software packages. Comput Stat Data Anal 49:969.973 short communicationCrossRefMathSciNetGoogle Scholar
  21. Hartiga JA, Wong MA (1979) A K-means clustering algorithm. Appl Stat 100–108Google Scholar
  22. Hastie T, Tibshirani R, Friedman J (2008) The elements of statistical learning. Springer, New YorkGoogle Scholar
  23. 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 39:10,512–10,520CrossRefGoogle Scholar
  24. Jain AK (2010) Data clustering: 50 years beyond k-means. Pattern Recognit Lett 31:651–666CrossRefGoogle Scholar
  25. 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):881–892CrossRefGoogle Scholar
  26. Karcher H (1977) Riemannian center of mass and mollifier smoothing. Commun Pure Appl Math 30(5):509–541CrossRefMathSciNetzbMATHGoogle Scholar
  27. Kaufman L, Rousseeuw P (1990) Finding groups in data: an introduction to cluster analysis. Wiley, New YorkGoogle Scholar
  28. Kendall D (1977) The diffusion of shape. Adv Appl Prob 9:428–430CrossRefGoogle Scholar
  29. Kendall DG, Barden D, Carne T, Le H (2009) Shape and shape theory. Wiley, ChichesterGoogle Scholar
  30. Kendall WS (1990) Probability, convexity, and harmonic maps with small image i: uniqueness and fine existence. Proc Lond Math Soc 3(2):371–406CrossRefMathSciNetGoogle Scholar
  31. Kent J, Mardia K (1997) Consistency of procrustes estimators. J R Stat Soc Ser B 59(1):281–290CrossRefMathSciNetzbMATHGoogle Scholar
  32. Kobayashi S, Nomizu K (1969) Foundations of differential geometry, vol 2. Wiley, ChichesterzbMATHGoogle Scholar
  33. Lawing A, Polly P (2010) Geometric morphometrics: recent applications to the study of evolution and development. J Zool 280(1):1–7CrossRefGoogle Scholar
  34. Le H (1998) On the consistency of Procrustean mean shapes. Adv Appl Prob 30(1):53–63CrossRefzbMATHGoogle Scholar
  35. Lloyd SP (1957) Least squares quantization in pcm. bell telephone labs memorandum, murray hill, nj. reprinted. In: IEEE Trans Information Theory IT-28 (1982) 2:129–137Google Scholar
  36. 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 281–297Google Scholar
  37. 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 1–5Google Scholar
  38. Ng R, Ashdown S, Chan A (2007) Intelligent size table generation. Sen’i Gakkaishi (J Soc Fiber Sci Technol Jpn) 63(11):384–387Google Scholar
  39. Pennec X (2006) Intrinsic statistics on riemannian manifolds: basic tools for geometric measurements. J Math Imaging Vis 25(1):127–154Google Scholar
  40. 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
  41. 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
  42. Rohlf JF (1999) Shape statistics: Procrustes superimpositions and tangent spaces. J Classif 16:197–223CrossRefzbMATHGoogle Scholar
  43. 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
  44. Simmons K (2002) Body shape analysis using three-dimensional body scanning technology. PhD thesis, North Carolina State UniversityGoogle Scholar
  45. Small C (1996) The statistical theory of shape. Springer, New YorkCrossRefzbMATHGoogle Scholar
  46. Sokal R, Sneath PH (1963) Principles of numerical taxonomy. Freeman, San FranciscoGoogle Scholar
  47. Steinhaus H (1956) Sur la division des corps matériels en parties. Bull Acad Pol Sci IV(12):801–804MathSciNetGoogle Scholar
  48. Steinley D (2006) K-means clustering: a half-century synthesis. Br J Math Stat Psychol 59:1–34CrossRefMathSciNetGoogle Scholar
  49. Stoyan LA, Stoyan H (1995) Fractals, random shapes and point fields. Wiley, ChichesterGoogle Scholar
  50. Theodoridis S, Koutroumbas K (1999) Pattern recognition. Academic, New YorkGoogle Scholar
  51. Veitch D, Fitzgerald C et al (2013) Sizing up Australia—the next step. Safe Work Australia, CanberraGoogle Scholar
  52. 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
  53. Woods R (2003) Characterizing volume and surface deformations in an atlas framework: theory, applications, and implementation. NeuroImage 18:769–788CrossRefGoogle Scholar
  54. Zheng R, Yu W, Fan J (2007) Development of a new chinese bra sizing system based on breast anthropometric measurements. Int J Ind Ergon 37:697–705CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Guillermo Vinué
    • 1
  • Amelia Simó
    • 2
  • Sandra Alemany
    • 3
  1. 1.Department of StatisticsO.R. University of ValenciaValenciaSpain
  2. 2.Department of Mathematics-IMACUniversitat Jaume ICastellónSpain
  3. 3.Biomechanics Institute of ValenciaUniversidad Politécnica de ValenciaValenciaSpain

Personalised recommendations