Anis Fradi
ABSTRACT
This paper introduces a shrinkage statistical model designed for analyzing, registering and clustering multi-dimensional curves. The model utilizes reparametrization functions that act as local distributions on curves. Given the intricate nature of the model, we establish a connection with well-understood Riemannian manifolds. This connection enables us to simplify the reparametrization space and enhance the manageability of the optimization task. Moreover, we provide empirical evidence of the practical usefulness of our proposed method by applying it to a potential application involving the clustering of hominin cochlear shapes. Looking ahead, our research interests lie in the development of theoretical extensions that can accommodate more complex spaces. By exploring new aspects of manifold learning and inference on high-dimensional manifolds, we aim to further advance the field.
- S-I. Amari. 1983. Differential geometry of statistical inference. Springer, Berlin, Heidelberg.Google Scholar
- Jos M. F. Ten Berge. 1977. Orthogonal procrustes rotation for two or more matrices. Psychometrika 42 (1977), 267–276.Google ScholarCross Ref
- A. Bhattacharya and R. Bhattacharya. 2012. Nonparametric inference on manifolds: with applications to shape spaces (1st ed.). Cambridge University Press, New York, USA.Google Scholar
- J. Braga, C. Samir, Anis Fradi, Y. Feunteun, K. Jakata, V. Zimmer, B. Zipfel, J. Thackeray, M. Macé, B. Wood, and F. Grine. 2021. Cochlear shape distinguishes southern African early hominin taxa with unique auditory ecologies. Scientific Reports 11 (2021), 17018.Google ScholarCross Ref
- D. W. Bryner and A. Srivastava. 2022. Shape analysis of functional data with Elastic partial matching. IEEE Trans. Pattern Anal. Mach. Intell. 44 (2022), 9589–9602.Google ScholarCross Ref
- C. Chen and A. Srivastava. 2021. SrvfRegNet: Elastic function registration using deep neural networks. In 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW). IEEE.Google Scholar
- X. Chen, N. Ravikumar, Y. Xia, R. Attar, A. Diaz-Pinto, S.K. Piechnik, S. Neubauer, S.E. Petersen, and A.F. Frangi. 2021. Shape registration with learned deformations for 3D shape reconstruction from sparse and incomplete point clouds. Medical Image Analysis 74 (2021), 102228.Google ScholarCross Ref
- I. L. Dryden and K. V. Mardia. 2016. Statistical shape analysis, with applications in R (2nd ed.). John Wiley and Sons, Chichester.Google Scholar
- A. Fradi, C. Samir, J. Braga, S. H. Joshi, and J-M. Loubes. 2022. Nonparametric Bayesian regression and classification on manifolds, with applications to 3D cochlear shapes. IEEE Transactions on Image Processing 31 (2022), 2598–2607.Google ScholarCross Ref
- K. Fritzsche and H. Grauert. 2002. From holomorphic functions to complex manifolds. Springer, New York, USA.Google Scholar
- C. Goodall. 1991. Procrustes methods in the statistical analysis of shape. Journal of the Royal Statistical Society 53 (1991), 285–339.Google Scholar
- Z. He, J. Zhang, M. Kan, S. Shan, and X. Chen. 2017. Robust FEC-CNN: a high accuracy facial landmark detection system. In 2017 IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW). IEEE, Honolulu, HI, USA, 2044–2050.Google Scholar
- A. E. Hoerl and R. W. Kennard. 1970. Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics 12 (1970), 55–67.Google Scholar
- A. Holbrook, S. Lan, J. Streets, and B. Shahbaba. 2020. Nonparametric Fisher Geometry with application to density estimation. In Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI)(Proceedings of Machine Learning Research, Vol. 124). PMLR, 101–110.Google Scholar
- X. Huang, N. Paragios, and D.N. Metaxas. 2006. Shape registration in implicit spaces using information theory and free form deformations. IEEE Transactions on Pattern Analysis and Machine Intelligence 28 (2006), 1303–1318.Google ScholarDigital Library
- I.T. Jolliffe. 1986. Principal component analysis. Springer Verlag, New York, USA.Google Scholar
- Shantanu H. Joshi, Ryan P. Cabeen, Anand A. Joshi, Bo Sun, Ivo Dinov, Katherine L. Narr, Arthur W. Toga, and Roger P. Woods. 2012. Diffeomorphic sulcal shape analysis on the cortex. IEEE Transactions on Medical Imaging 31 (2012), 1195–1212.Google ScholarCross Ref
- S. H. Joshi, E. Klassen, A. Srivastava, and I. Jermyn. 2007. A novel representation for Riemannian analysis of elastic curves in . In Conference on Computer Vision and Pattern Recognition. IEEE, Piscataway, New Jersey, 1–7.Google Scholar
- D. G. Kendall. 1984. Shape manifolds, procrustean metrics, and complex projective spaces. Bulletin of The London Mathematical Society 16 (1984), 81–121.Google ScholarCross Ref
- J.B. Kruskal and M. Wish. 1978. Multidimensional Scaling. Sage Publications, Newbury Park, California.Google Scholar
- S. J. Maybank. 2005. The Fisher-Rao metric for projective transformations of the line. International Journal of Computer Vision 63 (2005), 191–206.Google ScholarDigital Library
- A. Peter and A. Rangarajan. 2006. Shape analysis using the Fisher-Rao Riemannian metric: unifying shape representation and deformation. In 3rd IEEE International Symposium on Biomedical Imaging. Arlington, VA, USA, 1164–1167.Google Scholar
- A. M. Peter, A. Rangarajan, and M. Moyou. 2017. The geometry of orthogonal-series, square-root density estimators: applications in computer vision and model selection. In Computational Information Geometry. Springer, Cham, 175–215.Google ScholarCross Ref
- A. Srivastava, S.H. Joshi, W. Mio, and Xiuwen Liu. 2005. Statistical shape analysis: clustering, learning, and testing. IEEE Transactions on Pattern Analysis and Machine Intelligence 27 (2005), 590–602.Google ScholarDigital Library
- A. Srivastava and E. Klassen. 2016. Functional and shape data analysis. Springer-Verlag, New York, USA.Google Scholar
- A. Srivastava, E. Klassen, S. H. Joshi, and I. H. Jermyn. 2011. Shape analysis of Elastic curves in Euclidean spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 33 (2011), 1415–1428.Google ScholarDigital Library
- G. Trappolini, L. Cosmo, L. Moschella, R. Marin, S. Melzi, and E. Rodolà. 2021. Shape registration in the time of transformers. In Advances in Neural Information Processing Systems. 1–14.Google Scholar
- R. Vandaele, J. Aceto, M. Muller, F. Péronnet, V. Debat, C-W. Wang, C-T. Huang, S. Jodogne, P. Martinive, P. Geurts, and R. Marée. 2018. Landmark detection in 2D bioimages for geometric morphometrics: a multi-resolution tree-based approach. Scientific Reports 8 (2018), 1–14.Google ScholarCross Ref
- H.D. Vinod. 1995. Double bootstrap for shrinkage estimators. Journal of Econometrics 68 (1995), 287–302.Google ScholarCross Ref
- X. Wan, Y. Wu, and X. Li. 2022. Learning robust shape-indexed features for facial landmark detection. Applied Sciences 12, 12 (2022).Google Scholar
Index Terms
- A Shrinkage Method for Learning, Registering and Clustering Shapes of Curves
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