Skip to main content
SpringerLink
Log in

A robust tangent PCA via shape restoration for shape variability analysis

  • Theoretical advances
  • Published:
Pattern Analysis and Applications Aims and scope Submit manuscript

Abstract

This paper presents a novel method for handling the effects of shape outliers in statistical shape analysis. Usually performed by a variant of classical principal component analysis (PCA), variability analysis may be highly affected by erroneous shapes. Principal components may thus imply aberrant modes, while eigenshapes may not accurately describe variability in a given set of shapes. Our robust analysis is performed using an elastic metric associated with the square-root velocity representation of shapes. This elastic shape analysis allows shape variability to be described with natural and intuitive deformations. The proposed method based on shape outlier detection applies the shape restoration procedure to rectify aberrant shapes. The resultant components are thus obtained from a tangent PCA on the restored database. By performing experiments based on MPEG-7 and HAND databases, we demonstrate that the proposed scheme is effective for shape variability analysis in the presence of outlying shapes. Our method is then compared with two existing schemes for robust data variability analysis: minimum covariance determinant-based PCA and projection pursuit-based PCA.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24

Similar content being viewed by others

References

  1. Abboud M, Benzinou A, Nasreddine K, Jazar M (2014) Geodesics-based statistical shape analysis. In: International conference on image processing (ICIP). IEEE, pp 4747–4751

  2. Arribas-Gil A, Romo J (2014) Shape outlier detection and visualization for functional data: the outliergram. Biostatistics 15(4):603–619

    Article  Google Scholar 

  3. Cardot H, Godichon-Baggioni A (2017) Fast estimation of the median covariation matrix with application to online robust principal components analysis. Test 26(3):461–480

    Article  MathSciNet  MATH  Google Scholar 

  4. Chiang LH, Pell RJ, Seasholtz MB (2003) Exploring process data with the use of robust outlier detection algorithms. J Process Control 13(5):437–449

    Article  Google Scholar 

  5. Croux C, Filzmoser P, Oliveira M (2007) Algorithms for projection-pursuit robust principal component analysis. Chemom Intell Lab Syst 87(2):218–225

    Article  Google Scholar 

  6. Croux C, Haesbroeck G (2000) Principal component analysis based on robust estimators of the covariance or correlation matrix: influence functions and efficiencies. Biometrika 87(3):603–618

    Article  MathSciNet  MATH  Google Scholar 

  7. Daszykowski M, Kaczmarek K, Vander Heyden Y, Walczak B (2007) Robust statistics in data analysis, a review: basic concepts. Chemom Intell Lab Syst 85(2):203–219

    Article  Google Scholar 

  8. Drira H, Amor BB, Srivastava A, Daoudi M, Slama R (2013) 3d face recognition under expressions, occlusions, and pose variations. IEEE Trans Pattern Anal Mach Intell 35(9):2270–2283

    Article  Google Scholar 

  9. Dryden I, Mardia K (1998) Statistical analysis of shape. Wiley, New York

    MATH  Google Scholar 

  10. Engelen S, Hubert M, Branden KV (2016) A comparison of three procedures for robust PCA in high dimensions. Austrian J Stat 34(2):117–126

    Article  Google Scholar 

  11. Fawcett T, Provost F (1997) Adaptive fraud detection. Data Min Knowl Discov 1(3):291–316

    Article  Google Scholar 

  12. Fletcher PT, Lu C, Pizer SM, Joshi S (2004) Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Trans Med Imaging 23(8):995–1005

    Article  Google Scholar 

  13. Fletcher PT, Venkatasubramanian S, Joshi S (2008) Robust statistics on Riemannian manifolds via the geometric median. In: International conference on computer vision and pattern recognition CVPR. IEEE, pp 1–8

  14. Friedman JH, Tukey JW (1974) A projection pursuit algorithm for exploratory data analysis. IEEE Trans Comput 100(9):881–890

    Article  MATH  Google Scholar 

  15. Hawkins DM (1980) Identification of outliers. Springer, Berlin

    Book  MATH  Google Scholar 

  16. Hodge VJ, Austin J (2004) A survey of outlier detection methodologies. Artif Intell Rev 22(2):85–126

    Article  MATH  Google Scholar 

  17. Huang W, Gallivan KA, Srivastava A, Absil PA (2016) Riemannian optimization for registration of curves in elastic shape analysis. J Math Imaging Vis 54(3):320–343

    Article  MathSciNet  MATH  Google Scholar 

  18. Huckemann S, Hotz T (2009) Principal component geodesics for planar shape spaces. J Multivar Anal 100(4):699–714

    Article  MathSciNet  MATH  Google Scholar 

  19. Huckemann S, Ziezold H (2006) Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces. Adv Appl Probab 38(2):299–319

    Article  MathSciNet  MATH  Google Scholar 

  20. Jeannin S, Bober M (1999) Description of core experiments for MPEG-7 motion/shape. MPEG7, ISO/IEC JTC1/SC29/WG11 N2690, document N2690, Seoul

  21. Jermyn IH, Kurtek S, Laga H, Srivastava A (2017) Elastic shape analysis of three-dimensional objects. Synth Lect Comput Vis 12(1):1–185

    Google Scholar 

  22. Jiang T, Jia H, Yuan H, Zhou N, Li F (2016) Projection pursuit: a general methodology of wide-area coherency detection in bulk power grid. IEEE Trans Power Syst 31(4):2776–2786

    Article  Google Scholar 

  23. Jolicoeur P, Mosimann JE (1960) Size and shape variation in the painted turtle. A principal component analysis. Growth 24(4):339–354

    Google Scholar 

  24. Joshi SH, Klassen E, Srivastava A, Jermyn I (2007) A novel representation for Riemannian analysis of elastic curves in RN. In: International conference on computer vision and pattern recognition CVPR. IEEE, pp 1–7

  25. Knorr EM, Ng RT, Tucakov V (2000) Distance-based outliers: algorithms and applications. VLDB J Int J Very Large Data Bases 8(3–4):237–253

    Article  Google Scholar 

  26. Knox EM, Ng RT (1998) Algorithms for mining distance-based outliers in large datasets. In: Proceedings of the international conference on very large data bases. Citeseer, pp 392–403

  27. Lee JG, Han J, Li X (2008) Trajectory outlier detection: a partition-and-detect framework. In: 24th international conference on data engineering ICDE. IEEE, pp 140–149

  28. Lekadir K, Merrifield R, Yang GZ (2007) Outlier detection and handling for robust 3-d active shape models search. IEEE Trans Med Imaging 26(2):212–222

    Article  Google Scholar 

  29. Matsuda T, Morita T, Kudo T, Takine T (2017) Traffic anomaly detection based on robust principal component analysis using periodic traffic behavior. IEICE Trans Commun 100(5):749–761

    Article  Google Scholar 

  30. Nasreddine K, Benzinou A, Fablet R (2010) Variational shape matching for shape classification and retrieval. Pattern Recognit Lett 31:1650–1657

    Article  Google Scholar 

  31. Prastawa M, Bullitt E, Ho S, Gerig G (2004) A brain tumor segmentation framework based on outlier detection. Med Image Anal 8(3):275–283

    Article  Google Scholar 

  32. Ramaswamy S, Rastogi R, Shim K (2000) Efficient algorithms for mining outliers from large data sets. In: ACM SIGMOD Record, vol 29. ACM, pp 427–438

  33. Robinson D, Duncan A, Srivastava A, Klassen E (2017) Exact function alignment under elastic Riemannian metric. In: Graphs in biomedical image analysis, computational anatomy and imaging genetics. Springer, Berlin, pp 137–151

  34. Rousseeuw PJ (1985) Multivariate estimation with high breakdown point. Math Stat Appl 8:283–297

    Article  MathSciNet  MATH  Google Scholar 

  35. Rousseeuw PJ, Van Zomeren BC (1990) Unmasking multivariate outliers and leverage points. J Am Stat Assoc 85(411):633–639

    Article  Google Scholar 

  36. Sepulveda E, Dowd P, Xu C, Addo E (2017) Multivariate modelling of geometallurgical variables by projection pursuit. Math Geosci 49(1):121–143

    Article  MathSciNet  Google Scholar 

  37. Shekhar S, Lu CT, Zhang P (2002) Detecting graph-based spatial outliers. Intell Data Anal 6(5):451–468

    Article  MATH  Google Scholar 

  38. Srivastava A, Jain A, Joshi S, Kaziska D (2006) Statistical shape models using elastic-string representations. In: ACCV Computer Vision. Springer, Berlin, pp 612–621

  39. Srivastava A, Klassen E, Joshi SH, Jermyn IH (2011) Shape analysis of elastic curves in Euclidean spaces. Trans Pattern Anal Mach Intell 33(7):1415–1428

    Article  Google Scholar 

  40. Srivastava A, Klassen EP (2016) Functional and shape data analysis, Chapter 5. Springer, Berlin

  41. Stegmann MB, Gomez DD (2002) A brief introduction to statistical shape analysis. Informatics and mathematical modelling, Technical University of Denmark, DTU 15(11)

  42. Su Z, Klassen E, Bauer M (2017) The square root velocity framework for curves in a homogeneous space. arXiv preprint arXiv:1706.03095

  43. Sugiyama M, Borgwardt K (2013) Rapid distance-based outlier detection via sampling. In: Burges CJC, Bottou L, Welling M, Ghahramani Z, Weinberger KQ (eds) Advances in neural information processing systems. Curran Associates Inc, USA, pp 467–475

    Google Scholar 

  44. Younes L (1998) Computable elastic distances between shapes. SIAM J Appl Math 58(2):565–586

    Article  MathSciNet  MATH  Google Scholar 

  45. Younes L (2000) Optimal matching between shapes via elastic deformations. Image Vis Comput 17(5):381–389

    Google Scholar 

  46. Younes L (2012) Spaces and manifolds of shapes in computer vision: an overview. Image Vis Comput 30(6):389–397

    Article  MathSciNet  Google Scholar 

  47. Younes L, Michor PW, Shah J, Mumford D (2008) A metric on shape space with explicit geodesics. Rendiconti Lincei - Matematica e Applicazioni 9:25–57

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Univ Bretagne Loire, ENIB, UMR CNRS 6285 LabSTICC, 29238, Brest, France

    Michel Abboud, Abdesslam Benzinou & Kamal Nasreddine

  2. LaMA-Liban, Lebanese University, P.O. Box 37, Tripoli, Lebanon

    Michel Abboud

Authors
  1. Michel Abboud
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdesslam Benzinou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kamal Nasreddine
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abdesslam Benzinou.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abboud, M., Benzinou, A. & Nasreddine, K. A robust tangent PCA via shape restoration for shape variability analysis. Pattern Anal Applic 23, 653–671 (2020). https://doi.org/10.1007/s10044-019-00822-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10044-019-00822-2

Keywords

Search

Navigation