Abstract
Shape analysis is widely used in many application areas such as computer vision, medical and biological studies. One challenge to analyze the shape of an object in an image is its invariant property to shape-preserving transformations. To measure the distance or dissimilarity between two different shapes, we worked with the square-root velocity function (SRVF) representation and the elastic metric. Since shapes are inherently high-dimensional in a nonlinear space, we adopted a tangent space at the mean shape and a few principal components (PCs) on the linearized space. We proposed classification methods based on logistic regression using these PCs and tangent vectors with the elastic net penalty. We then compared its performance with other model-based methods for shape classification in application to shape of algae in watersheds as well as simulated data generated by the mixture of von Mises-Fisher distributions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
The codes and data of the paper are available from the supplementary material.
References
Aguilera, A. M., Escabias, M., & Valderrama, M. J. (2006). Using principal components for estimating logistic regression with high-dimensional multicollinear data. Computational Statistics and Data Analysis, 50(8), 1905–1924.
Cencov, N. N. (2000). Statistical decision rules and optimal inference. Providence: American Mathematical Society.
Cho, M. H., Kurtek, S., & Bharath, K. (2022). Tangent functional canonical correlation analysis for densities and shapes, with applications to multimodal imaging data. Journal of Multivariate Analysis, 189, 104870.
Cho, M. H., Kurtek, S., & MacEachern, S. N. (2021). Aggregated pairwise classification of elastic planar shapes. The Annals of Applied Statistics, 15(2), 619–637.
Cootes, T. F., Taylor, C. J., Cooper, D. H., & Graham, J. (1995). Active shape models: Their training and application. Computer Vision and Image Understanding, 61(1), 38–59.
Dryden, I. L., & Mardia, K. V. (1992). Size and shape analysis of landmark data. Biometrika, 79(1), 57–68.
Dryden, I. L., & Mardia, K. V. (2016). Statistical shape analysis: With applications in R. Wiley.
Escabias, M., Aguilera, A., & Valderrama, M. (2004). Principal component estimation of functional logistic regression: Discussion of two different approaches. Journal of Nonparametric Statistics, 16(3–4), 365–384.
Escabias, M., Aguilera, A., & Valderrama, M. (2005). Modeling environmental data by functional principal component logistic regression. Environmetrics, 16(1), 95–107.
Fisher, N. I., Lewis, T., & Embleton, B. J. (1993). Statistical analysis of spherical data. Cambridge University Press.
Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software, 33(1), 1.
Hilbe, J. M. (2009). Logistic regression models. CRC Press.
Joshi, S.H., Klassen, E., Srivastava, A., & Jermyn, I.H. (2007). A novel representation for Riemannian analysis of elastic curves in \({\mathbb{R}}^n\). In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–7
Kendall, D. G. (1984). Shape manifolds, Procrustean metrics, and complex projective spaces. Bulletin of the London Mathematical Society, 16(2), 81–121.
Kurtek, S., & Bharath, K. (2015). Bayesian sensitivity analysis with the fisher-rao metric. Biometrika, 102(3), 601–616.
Kurtek, S., Srivastava, A., Klassen, E., & Ding, Z. (2012). Statistical modeling of curves using shapes and related features. Journal of the American Statistical Association, 107(499), 1152–1165.
Kurtek, S., Su, J., Grimm, C., Vaughan, M., Sowell, R., & Srivastava, A. (2013). Statistical analysis of manual segmentations of structures in medical images. Computer Vision and Image Understanding, 117(9), 1036–1050.
Laga, H., Kurtek, S., Srivastava, A., Golzarian, M., & Miklavcic, S.J. (2012). A Riemannian elastic metric for shape-based plant leaf classification. In: International Conference on Digital Image Computing Techniques and Applications, pp. 1–7
Lang, S. (1999). Fundamentals of differential geometry. Springer.
Le, H. (2001). Locating Fréchet means with application to shape spaces. Advances in Applied Probability, 33(2), 324–338.
Mardia, K. V., & Jupp, P. E. (2009). Directional statistics. John Wiley and Sons.
Mio, W., Srivastava, A., & Joshi, S. H. (2007). On shape of plane elastic curves. International Journal of Computer Vision, 73(3), 307–324.
Mousavi, S. N., & Sørensen, H. (2018). Functional logistic regression: A comparison of three methods. Journal of Statistical Computation and Simulation, 88(2), 250–268.
Pal, S., Woods, R.P., Panjiyar, S., Sowell, E.R., Narr, K.L., & Joshi, S.H. (2017). A Riemannian framework for linear and quadratic discriminant analysis on the tangent space of shapes. In: Workshop on Differential Geometry in Computer Vision and Machine Learning, pp. 726–734
Park, M. Y., & Hastie, T. (2007). L1-regularization path algorithm for generalized linear models. Journal of the Royal Statistical Society Series B (Statistical Methodology), 69(4), 659–677.
Park, J., Lee, H., Park, C. Y., Hasan, S., Heo, T.-Y., & Lee, W. H. (2019). Algal morphological identification in watersheds for drinking water supply using neural architecture search for convolutional neural network. Water, 11(7), 1338.
Pennec, X. (2006). Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. Journal of Mathematical Imaging and Vision, 25(1), 127.
Robinson, D.T. (2012). Functional data analysis and partial shape matching in the square root velocity framework. PhD thesis, The Florida State University
Srivastava, A., & Klassen, E. P. (2016). Functional and shape data analysis (Vol. 1). Springer.
Srivastava, A., Klassen, E., Joshi, S. H., & Jermyn, I. H. (2011). Shape analysis of elastic curves in Euclidean spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(7), 1415–1428.
Vaillant, M., Miller, M. I., Younes, L., & Trouvé, A. (2004). Statistics on diffeomorphisms via tangent space representations. NeuroImage, 23, 161–169.
Xie, W., Chkrebtii, O., & Kurtek, S. (2019). Visualization and outlier detection for multivariate elastic curve data. IEEE Transactions on Visualization and Computer Graphics, 26(11), 3353–3364.
Younes, L., Michor, P. W., Shah, J., Mumford, D., & Lincei, R. (2008). A metric on shape space with explicit geodesics. Mathematica E Applicazioni, 19(1), 25–57.
Zhu, J., & Hastie, T. (2004). Classification of gene microarrays by penalized logistic regression. Biostatistics, 5(3), 427–443.
Acknowledgements
This work was supported by Inha University Research Grant and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2022-00167077).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Heo, TY., Lee, J.M., Woo, M.H. et al. Logistic regression models for elastic shape of curves based on tangent representations. J. Korean Stat. Soc. (2024). https://doi.org/10.1007/s42952-023-00252-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s42952-023-00252-1