Skip to main content
SpringerLink
Log in

Two Sample Tests for Mean 3D Projective Shapes from Digital Camera Images

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

In this article, we extend mean 3D projective shape change in matched pairs to independent samples. We provide a brief introduction of projective shapes of spatial configurations obtained from their digital camera images, building on previous results of Crane and Patrangenaru (J Multivar Anal 102:225–237, 2011). The manifold of projective shapes of k-ads in 3D containing a projective frame at five given landmark indices has a natural Lie group structure, which is inherited from the quaternion multiplication. Here, given the small sample size, one estimates the mean 3D projective shape change in two populations, based on independent random samples of possibly different sizes using Efron’s nonparametric bootstrap. This methodology is applied in three relevant applications of analysis of 3D scenes from digital images: visual quality control, face recognition, and scene recognition.

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

Similar content being viewed by others

References

  • Beran R, Fisher NI (1998) Nonparametric comparison of mean axes. Ann Stat 26:472–493

    Article  MATH  MathSciNet  Google Scholar 

  • Bhattacharya A (2008) Statistical analysis on manifolds: a nonparametric approach for inference on shape spaces. Sankhya Ser A 70(2):223–266

    MATH  Google Scholar 

  • Bhattacharya RN, Bhattacharya A (2012) Nonparametric statistics on manifolds with applications to shape spaces. In: Institute of mathematical statistics lecture notes–monograph series, vol 3. Cambridge University Press.

  • Bhattacharya RN, Ellingson L, Liu X, Patrangenaru V, Crane M (2012) Extrinsic analysis on manifolds is computationally faster than intrinsic analysis, with applications to quality control by machine vision. Appl Stoch Model Bus Ind 28:222–235

    Article  MathSciNet  Google Scholar 

  • Bhattacharya RN, Ghosh JK (1978) On the validity of the formal Edgeworth expansion. Ann Stat 6(2):434–451

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Bhattacharya RN, Patrangenaru V (2005) Large sample theory of intrinsic and extrinsic sample means on manifolds—part II. Ann Stat 33(3):1211–1245

    Article  MathSciNet  Google Scholar 

  • Billingsley P (1995) Probability and measure, 3rd edn. Eds Wiley

  • Buibas M, Crane M, Ellingson L, Patrangenaru V (2012) A projective frame based shape analysis of a rigid scene from noncalibrated digital camera imaging outputs. In: JSM proceedings, 2011, institute of mathematical statistics. Miami, FL, pp 4730–4744

  • Crane M, Patrangenaru V (2011) Random change on a Lie group and mean glaucomatous projective shape change detection from stereo pair image. J Multivar Anal 102:225–237

    Article  MATH  MathSciNet  Google Scholar 

  • Efron B (1982) The Jackknife, the Bootstrap and other resampling plans. In: CBMS-NSF regional conference series in applied mathematics, vol 38. SIAM

  • Faugeras OD (1992) What can be seen in three dimensions with an uncalibrated stereo rig? In: Proc. European conference on computer vision, LNCS, vol 588, pp 563–578

  • Fisher NI, Hall P, Jing B-Y, Wood ATA (1996) Improved pivotal methods for constructing confidence regions with directional data. J Am Stat Assoc 91:1062–1070

    Article  MATH  MathSciNet  Google Scholar 

  • Fréchet M (1948) Les élements aléatoires de nature quelconque dans un espace distancié. Ann Inst H Poincaré 10:215–310

    Google Scholar 

  • Ma Y, Soatto S, Kosecka J, Sastry SS (2006) An invitation to 3-D vision. Springer, New York

    Google Scholar 

  • Goodall C, Mardia KV (1999) Projective shape analysis. J Graph Comput Stat 8:143–168

    MathSciNet  Google Scholar 

  • Hall P (1997) The bootstrap and Edgeworth expansion. Springer Series in Statistics, New York

  • Hall P, Hart JD (1990) Bootstrap test for difference between means in nonparametric regression. J Am Stat Assoc 85:1039–104

    Article  MATH  MathSciNet  Google Scholar 

  • Hartley RI, Gupta R, Chang T (1992) Stereo from uncalibrated cameras. In: Proc. IEEE conference on computer vision and pattern recognition, pp 761–764

  • Hartley RI, Zisserman A (2004) Multiple view geometry in computer vision, 2nd edn. Cambridge University Press

  • Hendriks H, Landsman Z (1998) Mean location and sample mean location on manifolds: asymptotics, tests, confidence regions. J Multivar Anal 67(2):227–243

    Article  MATH  MathSciNet  Google Scholar 

  • Johnson RA,Wichern DW (2007) Applied multivariate statistical analysis, 6th edn. Prentice Hall

  • Longuet-Higgins C (1981) A computer algorithm for reconstructing a scene from two projections. Nature 293:133–135

    Article  Google Scholar 

  • Mardia KV, Goodall C, Walder A (1996) Distributions of projective invariants and model-based machine vision. Adv Appl Probab 28:641–661

    Article  MATH  MathSciNet  Google Scholar 

  • Mardia KV, Patrangenaru V (2005) Directions and projective shapes. Ann Stat 33:1666–1699

    Article  MATH  MathSciNet  Google Scholar 

  • Mardia KV, Patrangenaru V, Derado G, Patrangenaru VP (2003) Reconstruction of planar scenes from multiple views using affine and projective shape. In: Proceedings of the 2003 workshop on statistical signal processing, pp 285–288

  • Munk A, Paige R, Pang J, Patrangenaru V, Ruymgaart FH (2008) The one and multisample problem for functional data with applications to projective shape analysis. J Multiv Anal 99:815–833

    Article  MATH  MathSciNet  Google Scholar 

  • Osborne D, Patrangenaru V, Ellingson L, Groisser D, Schwartzman A (2013) Nonparametric two-sample tests on homogeneous Riemannian manifolds, Cholesky decompositions and diffusion tensor image analysis. J Multivar Anal 103:163–175

    Article  MathSciNet  Google Scholar 

  • Patrangenaru V (1999) Moving projective frames and spatial scene identification. In: Mardia KV, Aykroyd RG, Dryden IL (eds) Proceedings in spatial-temporal modeling and applications. Leeds University Press, pp 53–57

  • Patrangenaru V (2001) New large sample and bootstrap methods on shape spaces in high level analysis of natural images. Commun Stat Theory Methods 30:1675–1695

    Article  MATH  MathSciNet  Google Scholar 

  • Patrangenaru V, Crane MA, Liu X, Descombes X, Derado G, Liu W, Balan V, Patrangenaru VP, Thompson HW (2012) Methodology for 3D scene reconstruction from digital camera images. In: Proceedings of the international conference of Differential Geometry and Dynamical Systems (DGDS-2011), vol 19, BSG Proceedings, 6–9 October 2011. Bucharest, Romania, pp 110–124

  • Patrangenaru V, Liu X, Sugathadasa S (2010) Nonparametric 3D projective shape estimation from pairs of 2D images—I, in memory of Dayawansa WP. J Multivar Anal 101:11–31

    Article  MATH  MathSciNet  Google Scholar 

  • Spivak M (1979) A comprehensive introduction to differential geometry, vols I–II, 2nd edn. Publish or Perish, Inc., Wilmington, Del

    Google Scholar 

  • Sughatadasa SM (2006) Affine and projective shape analysis with applications. Ph.D. Dissertation, Texas Tech Univesity

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Florida State University, Tallahassee, FL, USA

    Vic Patrangenaru & Mingfei Qiu

  2. Brain Corporation, San Diego, CA, USA

    Marius Buibas

Authors
  1. Vic Patrangenaru
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mingfei Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marius Buibas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vic Patrangenaru.

Additional information

Vic Patrangenaru’s research supported by National Science Foundation Grants DMS-0805977 and DMS-1106935. Mingfei Qiu’s research supported by National Science Foundation Grant DMS-1106935.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Patrangenaru, V., Qiu, M. & Buibas, M. Two Sample Tests for Mean 3D Projective Shapes from Digital Camera Images. Methodol Comput Appl Probab 16, 485–506 (2014). https://doi.org/10.1007/s11009-013-9363-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-013-9363-6

Keywords

AMS 2000 Subject Classifications

Search

Navigation