×

A nonparametric approach to 3D shape analysis from digital camera images. I. (English) Zbl 1177.62059

Summary: This article, for the first time, develops a nonparametric methodology for the analysis of projective shapes of configurations of landmarks on real 3D objects from their regular camera pictures. A fundamental result in computer vision, emulating the principle of human vision in space, claims that, generically, a finite 3D configuration of points can be retrieved from corresponding configurations in a pair of camera images, up to a projective transformation. Consequently, the projective shape of a 3D configuration can be retrieved from two of its planar views, and a projective shape analysis can be pursued from a sample of images.
Projective shapes are here regarded as points on projective shape manifolds. Using large sample and nonparametric bootstrap methodology for extrinsic means on manifolds, one gives confidence regions and tests for the mean projective shape of a 3D configuration from its 2D camera images. Two examples are given: an example of testing for accuracy of a simple manufactured object using mean projective shape analysis, and a face identification example. Both examples are data driven based on landmark registration in digital images.

MSC:

62G09 Nonparametric statistical resampling methods
62H35 Image analysis in multivariate analysis
62H11 Directional data; spatial statistics
68U10 Computing methodologies for image processing
62H10 Multivariate distribution of statistics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Faugeras, O. D., What can be seen in three dimensions with an uncalibrated stereo rig?, (Proc. European Conference on Computer Vision. Proc. European Conference on Computer Vision, LNCS, vol. 588 (1992)), 563-578
[2] R.I. Hartley, R. Gupta, T. Chang, Stereo from uncalibrated cameras, in: Proc. IEEE Conference on Computer Vision and Pattern Recognition, 1992; R.I. Hartley, R. Gupta, T. Chang, Stereo from uncalibrated cameras, in: Proc. IEEE Conference on Computer Vision and Pattern Recognition, 1992
[3] S. Sugathadasa, Affine and projective shape analysis with applications, Ph.D. Thesis, Texas Tech University, 2006; S. Sugathadasa, Affine and projective shape analysis with applications, Ph.D. Thesis, Texas Tech University, 2006
[4] Hartley, R. I.; Zisserman, A., Multiple View Geometry in Computer Vision (2004), Cambridge University Press · Zbl 1072.68104
[5] X. Liu, V. Patrangenaru, S. Sugathadasa, Projective Shape Analysis for Noncalibrated Pinhole Camera Views. To the Memory of W.P. Dayawansa, Technical Report M983, Florida State University Department of Statistics, 2007; X. Liu, V. Patrangenaru, S. Sugathadasa, Projective Shape Analysis for Noncalibrated Pinhole Camera Views. To the Memory of W.P. Dayawansa, Technical Report M983, Florida State University Department of Statistics, 2007
[6] Maybank, S. J., Classification based on the cross ratio, (Mundy, J. L.; Zisserman, A.; Forsyth, D., Applications of Invariance in Computer Vision. Applications of Invariance in Computer Vision, Lecture Notes in Comput. Sci., vol. 825 (1994), Springer: Springer Berlin), 433-472
[7] Mardia, K. V.; Goodall, C.; Walder, A. N., Distributions of projective invariants and model based machine vision, Adv. Appl. Probab., 28, 641-661 (1996) · Zbl 0866.62028
[8] Goodall, C.; Mardia, K. V., Projective shape analysis, J. Graph. Comput. Statist., 8, 143-168 (1999)
[9] Patrangenaru, V., New large sample and bootstrap methods on shape spaces in high level analysis of natural images, Comm. Statist., 30, 1675-1693 (2001) · Zbl 1008.62616
[10] Lee, J. L.; Paige, R.; Patrangenaru, V.; Ruymgaart, F., Nonparametric density estimation on homogeneous spaces in high level image analysis, (Aykroyd, R. G.; Barber, S.; Mardia, K. V., Bioinformatics, Images, and Wavelets (2004), Department of Statistics, University of Leeds), 37-40
[11] Paige, R.; Patrangenaru, V.; Ruymgaart, F.; Wang, W., Analysis of projective shapes of curves using projective frames, (Barber, S.; Baxter, P. D.; Mardia, K. V.; Walls, R. E., Quantitative Biology, Shape Analysis, and Wavelets (2005), Leeds University Press: Leeds University Press Leeds), 71-74
[12] Mardia, K. V.; Patrangenaru, V., Directions and projective shapes, Ann. Statist., 33, 4, 1666-1699 (2005) · Zbl 1078.62068
[13] Kent, J. T.; Mardia, K. V., A new representation for projective shape, (Barber, S.; Baxter, P. D.; Mardia, K. V.; Walls, R. E., Interdisciplinary Statistics and Bioinformatics (2006), Leeds University Press: Leeds University Press Leeds), 75-78 · Zbl 0543.62043
[14] Munk, A.; Paige, R.; Pang, J.; Patrangenaru, V.; Ruymgaart, F., The one and multisample problem for functional data with applications to projective shape analysis, J. Multivar. Anal., 99, 815-833 (2008) · Zbl 1286.62050
[15] Bhattacharya, R. N.; Patrangenaru, V., Large sample theory of intrinsic and extrinsic sample means on manifolds — Part II, Ann. Statist., 33, 3, 1211-1245 (2005)
[16] Ma, Y.; Soatto, S.; Kosecka, J.; Sastry, S. S., An Invitation to 3-D Vision (2006), Springer: Springer New York
[17] R.I. Hartley, Projective reconstruction and invariants from multiple images, Preprint, 1993; R.I. Hartley, Projective reconstruction and invariants from multiple images, Preprint, 1993
[18] Patrangenaru, V., Moving projective frames and spatial scene identification, (Mardia, K. V.; Aykroyd, R. G.; Dryden, I. L., Proceedings in Spatial-Temporal Modeling and Applications (1999), Leeds University Press), 53-57
[19] Bhattacharya, R. N.; Patrangenaru, V., Large sample theory of intrinsic and extrinsic sample means on manifolds—I, Ann. Statist., 31, 1, 1-29 (2003) · Zbl 1020.62026
[20] Spivak, M., A Comprehensive Introduction to Differential Geometry. vol. I (1979), Publish or Perish, Inc.: Publish or Perish, Inc. Wilmington, Del. · Zbl 0439.53001
[21] Dimitric, I., A note on equivariant embeddings of Grassmannians, Publ. Inst. Math. (Beograd) (N.S.), 59, 131-137 (1996) · Zbl 0965.53038
[22] Watson, G. S., (Statistics on Spheres. Statistics on Spheres, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 6 (1983), A Wiley-Interscience Publication, John Wiley and Sons Inc.: A Wiley-Interscience Publication, John Wiley and Sons Inc. New York)
[23] Prentice, M. J., A distribution-free method of interval estimation for unsigned directional data, Biometrika, 71, 147-154 (1984) · Zbl 0549.62033
[24] Bhattacharya, R. N.; Ghosh, J. K., On the validity of the formal Edgeworth expansion, Ann. Statist., 6, 434-451 (1978) · Zbl 0396.62010
[25] Longuet-Higgins, C., computer algorithm for reconstructing a scene from two projections, Nature, 293, 133-135 (1981)
[26] Bhattacharya, A.; Bhattacharya, R., Nonparametric statistics on manifolds with applications to shape spaces, (Ghoshal, S.; Clarke, B., Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh. Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, IMS Collections, vol. 3 (2008)), 282-301
[27] Hall, P., The Bootstrap and Edgeworth Expansion (1997), Springer Series in Statistics: Springer Series in Statistics New York
[28] Ferguson, T. S., A Course in Large Sample Statistics (1996), Chapmann and Hall
[29] V. Balan, M. Crane, V. Patrangenaru, X. Liu, Projective shape manifolds and coplanarity of landmark configurations. A nonparametric approach (2009) (in press); V. Balan, M. Crane, V. Patrangenaru, X. Liu, Projective shape manifolds and coplanarity of landmark configurations. A nonparametric approach (2009) (in press) · Zbl 1180.62067
[30] Efron, B., Bootstrap methods: Another look at the jackknife, Ann. Statist., 7, 1, 1-26 (1979) · Zbl 0406.62024
[31] J.T. Kent, Projective shape analysis, in: Summer 2007 Program on the Geometry and Statistics of Shape Spaces, July 7-13, 2007, Speaker Abstracts; J.T. Kent, Projective shape analysis, in: Summer 2007 Program on the Geometry and Statistics of Shape Spaces, July 7-13, 2007, Speaker Abstracts
[32] K.V. Mardia, V. Patrangenaru, Directions and projective shapes, Technical Report No. 02/04, Dept. Statistics, Univ. Leeds, 2002; K.V. Mardia, V. Patrangenaru, Directions and projective shapes, Technical Report No. 02/04, Dept. Statistics, Univ. Leeds, 2002 · Zbl 1078.62068
[33] Maybank, S. J., Probabilistic analysis of the application of the cross ratio to model based vision: Misclassification, Int. J. Comput. Vis., 14, 3 (1995)
[34] S. Birchfield, http://www.ces.clemson.edu/ stb/projective/node19.html; S. Birchfield, http://www.ces.clemson.edu/ stb/projective/node19.html
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.