CSDL Home IEEE Transactions on Pattern Analysis & Machine Intelligence 2010 vol.32 Issue No.04 - April

Subscribe

Issue No.04 - April (2010 vol.32)

pp: 593-603

Thomas Hotz , Georgia Augusta University, Göttingen

Stephan Huckemann , Georgia Augusta University, Göttingen

ABSTRACT

We propose an intrinsic multifactorial model for data on Riemannian manifolds that typically occur in the statistical analysis of shape. Due to the lack of a linear structure, linear models cannot be defined in general; to date only one-way MANOVA is available. For a general multifactorial model, we assume that variation not explained by the model is concentrated near elements defining the effects. By determining the asymptotic distributions of respective sample covariances under parallel transport, we show that they can be compared by standard MANOVA. Often in applications manifolds are only implicitly given as quotients, where the bottom space parallel transport can be expressed through a differential equation. For Kendall's space of planar shapes, we provide an explicit solution. We illustrate our method by an intrinsic two-way MANOVA for a set of leaf shapes. While biologists can identify genotype effects by sight, we can detect height effects that are otherwise not identifiable.

INDEX TERMS

Shape analysis, nonlinear multivariate analysis of variance, Riemannian manifolds, orbifolds, orbit spaces, geodesics, Lie group actions, nonlinear multivariate statistics, covariance, inference, test, intrinsic mean, forest biometry.

CITATION

Thomas Hotz, Stephan Huckemann, "Intrinsic MANOVA for Riemannian Manifolds with an Application to Kendall's Space of Planar Shapes",

*IEEE Transactions on Pattern Analysis & Machine Intelligence*, vol.32, no. 4, pp. 593-603, April 2010, doi:10.1109/TPAMI.2009.117REFERENCES

- [1] G.J.A. Amaral, I.L. Dryden, and A.T.A. Wood, "Pivotal Bootstrap Methods for $k$ -Sample Problems in Directional Statistics and Shape Analysis,"
J. Am. Statistical Assoc., vol. 102, no. 478, pp. 695-707, 2007.- [2] R.V. Ambartzumian,
Factorization, Calculus and Geometric Probability. Cambridge Univ. Press, 1990.- [3] T.W. Anderson,
An Introduction to Multivariate Analysis, second ed. Wiley, 1984.- [4] R.N. Bhattacharya and V. Patrangenaru, "Large Sample Theory of Intrinsic and Extrinsic Sample Means on Manifolds II,"
Ann. Statistics, vol. 33, no. 3, pp. 1225-1259, 2005.- [5] N. Bissantz, L. Dümbgen, H. Holzmann, and A. Munk, "Nonparametric Confidence Bands in Deconvolution Density Estimation,"
J. Royal Statistical Soc. Series B, vol. 69, pp. 483-506, 2007.- [6] N. Bissantz, H. Holzmann, and M. Pawlak, "Testing for Image Symmetries—with Application to Confocal Microscopy,"
IEEE Trans. Information Theory, 2009.- [7] H. Blum and R.N. Nagel, "Shape Description Using Weighted Symmetric Axis Features,"
Pattern Recognition, vol. 10, no. 3, pp. 167-180, 1978.- [8] T.F. Cox,
An Introduction to Multivariate Data Analysis. Hodder Ar nold, 2005.- [9] I.L. Dryden, "Statistical Analysis on High-Dimensional Spheres and Shape Spaces,"
Ann. Statistics, vol. 33, pp. 1643-1665, 2005.- [10] I.L. Dryden and K.V. Mardia,
Statistical Shape Analysis. Wiley, 1998.- [11] J.R. England and P.M. Attiwill, "Changes in Leaf Morphology and Anatomy with Tree Age and Height in the Broadleaved Evergreen Species, Eucalyptus Regnans F. Muell,"
J. Trees—Structure and Function, vol. 20, no. 1, pp. 79-90, 2006.- [12] K. Evans, I.L. Dryden, and H. Le, "Shape Curves and Geodesic Modelling," http://www.maths.nottingham.ac.uk/personal/ ild/paperscurves2.pdf, 2009.
- [13] P.T. Fletcher and S.C. Joshi, "Principal Geodesic Analysis on Symmetric Spaces: Statistics of Diffusion Tensors,"
Proc. European Conf. Computer Vision Workshops Computer Vision Approaches to Medical Image Analysis and Math. Methods in Biomedical Image Analysis, pp. 87-98, 2004.- [14] M. Fuchs and O. Scherzer, "Regularized Reconstruction of Shapes with Statistical A Priori Knowledge,"
Int'l J. Computer Vision, vol. 79, no. 2, pp. 119-135, 2008.- [15] A.K. Gupta and J. Tang, "Distribution of Likelihood Ratio Statistic for Testing Equality of Covariance Matrices of Multivariate Gaussian Models,"
Biometrika, vol. 71, no. 3, pp. 555-559, 1984.- [16] M. Hallin and D. Paindaveine, "Optimal Tests for Homogeneity of Covariance, Scale, and Shape,"
J. Multivariate Analysis, vol. 100, pp. 422-444, 2009.- [17] H. Hendriks and Z. Landsman, "Mean Location and Sample Mean Location on Manifolds: Asymptotics, Tests, Confidence Regions,"
J. Multivariate Analysis, vol. 67, pp. 227-243, 1998.- [18] T. Hotz, S. Huckemann, D. Gaffrey, A. Munk, and B. Sloboda, "Shape Spaces for Pre-Aligned Star-Shaped Objects in Studying the Growth of Plants,"
J. Royal Statistical Soc., Series C, vol. 59, no. 1, pp. 127-143, 2010.- [19] S. Huckemann, "Dynamic Shape Analysis and Comparison of Leaf Growth," submitted, 2010.
- [20] S. Huckemann, T. Hotz, and A. Munk, "Global Models for the Orientation Field of Fingerprints: An Approach Based on Quadratic Differentials,"
IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 30, no. 9, pp. 1507-1519, Sept. 2008.- [21] S. Huckemann and T. Hotz, "Principal Components Geodesics for Planar Shape Spaces,"
J. Multivariate Analysis, vol. 100, pp. 699-714, 2009.- [22] S. Huckemann, T. Hotz, and A. Munk, "Intrinsic Shape Analysis: Geodesic Principal Component Analysis for Riemannian Manifolds Modulo Lie Group Actions (with Discussion),"
Statistica Sinica, vol. 20, no. 1, pp. 1-100, 2010.- [23] S. Huckemann and H. Ziezold, "Principal Component Analysis for Riemannian Manifolds with an Application to Triangular Shape Spaces,"
Advances in Applied Probability, vol. 38, no. 2, pp. 299-319, 2006.- [24] P.E. Jupp and J.T. Kent, "Fitting Smooth Paths to Spherical Data,"
Applied Statistics, vol. 36, no. 1, pp. 34-46, 1987.- [25] H. Karcher, "Riemannian Center of Mass and Mollifier Smoothing,"
Comm. Pure and Applied Math. vol. 30, pp. 509-541, 1977.- [26] D.G. Kendall, D. Barden, T.K. Carne, and H. Le,
Shape and Shape Theory. Wiley, 1999.- [27] P.T. Kim and D.St.P. Richards, "Deconvolution Density Estimation on Compact Lie Groups,"
Contemporary Math., vol. 287, pp. 155-171, 2001.- [28] E. Klassen, A. Srivastava, W. Mio, and S.H. Joshi, "Analysis on Planar Shapes Using Geodesic Paths on Shape Spaces,"
IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 3, pp. 372-383, Mar. 2004.- [29] S. Kobayashi and K. Nomizu,
Foundations of Differential Geometry, vol. I. Wiley, 1963.- [30] S. Kobayashi and K. Nomizu,
Foundations of Differential Geometry, vol. II. Wiley, 1969.- [31] H. Le, I. Dryden, and A. Kume, "Shape-Space Smoothing Splines for Planar Landmark Data,"
Biometrika, vol. 94, no. 3, pp. 513-528, 2007.- [32] H. Le, "Locating Fréchet Means with an Application to Shape Spaces,"
Advances in Applied Probability, vol. 33, pp. 324-338, 2001.- [33] H. Le, "Unrolling Shape Curves,"
J. London Math. Soc., vol. 2, no. 68, pp. 511-526, 2003.- [34] H. Le and A. Kume, "Detection of Shape Changes in Biological Features,"
J. Microscopy, vol. 200, pp. 140-147, 2000.- [35] J.M. Lee,
Riemannian Manifolds: An Introduction to Curvature. Springer, 1997.- [36] E.L. Lehmann,
Testing Statistical Hypotheses. Springer, 1997.- [37] K.V. Mardia, J.T. Kent, and J.M. Bibby,
Mulitvariate Analysis. Academic Press, 1979.- [38] K.V. Mardia and V. Patrangenaru, "On Affine and Projective Shape Data Analysis,"
Proc. 20th LASR Workshop Functional and Spatial Data Analysis, K.V. Mardia and R.G. Aykroyd, eds., pp. 39-45, 2001.- [39] K.V. Mardia and V. Patrangenaru, "Directions and Projective Shapes,"
Ann. Statistics, vol. 33, pp. 1666-1699, 2005.- [40] B. O'Neill, "The Fundamental Equations of a Submersion,"
Michigan Math. J., vol. 13, no. 4, pp. 459-469, 1966.- [41] V. Patrangenaru, "New Large Sample and Bootstrap Methods on Shape Spaces in High Level Analysis of Natural Images,"
Comm. Statistics—Theory and Methods, vol. 30, nos. 8/9, pp. 1675-1693, 2001.- [42] H. Ziezold, "Expected Figures and a Strong Law of Large Numbers for Random Elements in Quasi-Metric Spaces,"
Proc. Seventh Prague Conf. Information Theory, Statistical Decision Function, Random Processes (A), pp. 591-602, 1977. |