Abstract

We demonstrate applications of the Gaussian process-based landmarking algorithm proposed in [T. Gao, S. Z. Kovalsky, and I. Daubechies, SIAM J. Math. Data Sci., 1 (2019), pp. 208--236] to geometric morphometrics, a branch of evolutionary biology centered at the analysis and comparisons of anatomical shapes, and compare the automatically sampled landmarks with the “ground truth” landmarks manually placed by evolutionary anthropologists; the results suggest that Gaussian process landmarks perform equally well or better, in terms of both spatial coverage and downstream statistical analysis. We provide a detailed exposition of numerical procedures and feature filtering algorithms for computing high-quality and semantically meaningful diffeomorphisms between disk-type anatomical surfaces.

Keywords

  1. Gaussian process
  2. experimental design
  3. active learning
  4. manifold learning
  5. geometric morphometrics

MSC codes

  1. 60G15
  2. 62K05
  3. 65D18

Get full access to this article

View all available purchase options and get full access to this article.

Supplementary Material


PLEASE NOTE: These supplementary files have not been peer-reviewed.


Index of Supplementary Materials

Title of paper: Gaussian Process Landmarking for Three-Dimensional Geometric Morphometrics

Authors: Tingran Gao, Shahar Kovalsky, Doug Boyer, and Ingrid Daubechies

File: supplement.pdf

Type: PDF

Contents: Additional figures, proof of a theorem in the main text, and comparative biological discussions.

References

1.
D. C. Adams, F. J. Rohlf, and D. E. Slice, A field comes of age: Geometric morphometrics in the 21st century, Hystrix, 24 (2013), pp. 7--14.
2.
R. Al-Aifari, I. Daubechies, and Y. Lipman, Continuous Procrustes distance between two surfaces, Comm. Pure Appl. Math., 66 (2013), pp. 934--964, https://doi.org/10.1002/cpa.21444.
3.
P. Alliez, D. Cohen-Steiner, O. Devillers, B. Lévy, and M. Desbrun, Anisotropic polygonal remeshing, ACM Trans. Graphics, 22 (2003), pp. 485--493, https://doi.org/10.1145/882262.882296.
4.
M. J. Anderson, A new method for non-parametric multivariate analysis of variance, Austral Ecology, 26 (2001), pp. 32--46.
5.
M. Aubry, U. Schlickewei, and D. Cremers, The wave kernel signature: A quantum mechanical approach to shape analysis, in Proceedings of the 2011 IEEE International Conference on Computer Vision (ICCV Workshops), IEEE, 2011, pp. 1626--1633.
6.
T. F. Banchoff, Critical points and curvature for embedded polyhedral surfaces, Amer. Math. Monthly, 77 (1970), pp. 475--485.
7.
C. Beard, Teilhardina, in The International Encyclopedia of Primatology. 1--2, John Wiley, New York, 2017, https://doi.org/10.1002/9781119179313.wbprim0444.
8.
M. Belkin and P. Niyogi, Laplacian eigenmaps for dimensionality reduction and data representation, Neural Comput., 15 (2003), pp. 1373--1396, https://doi.org/10.1162/089976603321780317.
9.
T. Berry and J. Harlim, Variable bandwidth diffusion kernels, Appl. Comput. Harmon. Anal., 40 (2016), pp. 68--96, https://doi.org/10.1016/j.acha.2015.01.001.
10.
P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods, SIAM J. Math. Anal., 43 (2011), pp. 1457--1472, https://doi.org/10.1137/100795772.
11.
F. L. Bookstein, Morphometric Tools for Landmark Data: Geometry and Biology, Cambridge University Press, Cambridge, UK, 1991.
12.
D. M. Boyer, L. Costeur, and Y. Lipman, Earliest record of Platychoerops (primates, Plesiadapidae), a new species from Mouras Quarry, Mont de Berru, France, Amer. J. Phys. Anthropol., 149 (2012), pp. 329--346, https://doi.org/10.1002/ajpa.22119.
13.
D. M. Boyer, Y. Lipman, E. St. Clair, J. Puente, B. A. Patel, T. Funkhouser, J. Jernvall, and I. Daubechies, Algorithms to automatically quantify the geometric similarity of anatomical surfaces, Proc. Natl. Acad. Sci. USA, 108 (2011), pp. 18221--18226, https://doi.org/10.1073/pnas.1112822108.
14.
D. M. Boyer, J. Puente, J. T. Gladman, C. Glynn, S. Mukherjee, G. S. Yapuncich, and I. Daubechies, A new fully automated approach for aligning and comparing shapes, Anatom. Rec., 298 (2015), pp. 249--276.
15.
A. M. Bronstein, Spectral Descriptors for Deformable Shapes, preprint, https://arxiv.org/abs/1110.5015, 2011.
16.
U. Castellani, M. Cristani, S. Fantoni, and V. Murino, Sparse points matching by combining 3D mesh saliency with statistical descriptors, Computer Graphics Forum, 27 (2008), pp. 643--652.
17.
K. N. Chaudhury, Y. Khoo, and A. Singer, Global registration of multiple point clouds using semidefinite programming, SIAM J. Optim., 25 (2015), pp. 468--501, https://doi.org/10.1137/130935458.
18.
D. Cohen-Steiner and J.-M. Morvan, Restricted Delaunay triangulations and normal cycle, in Proceedings of the 19th Annual Symposium on Computational Geometry, ACM, 2003, pp. 312--321.
19.
R. R. Coifman and S. Lafon, Diffusion maps, Appl. Comput. Harmon. Anal., 21 (2006), pp. 5--30, https://doi.org/10.1016/j.acha.2006.04.006.
20.
W. Czaja and M. Ehler, Schroedinger eigenmaps for the analysis of biomedical data, IEEE Trans. Pattern Anal. Mach. Intell., 35 (2013), pp. 1274--1280.
21.
R. DeVore, G. Petrova, and P. Wojtaszczyk, Greedy algorithms for reduced bases in Banach spaces, Constr. Approx., 37 (2013), pp. 455--466.
22.
I. L. Dryden and K. V. Mardia, Statistical Shape Analysis, Wiley Ser. Probab. Statis. 4, John Wiley & Sons, New York, 1998.
23.
D. S. Fry, Shape Recognition Using Metrics on the Space of Shapes, Ph.D. thesis, Harvard University, Cambridge, MA, 1993. UMI Order No. GAX94-12337.
24.
T. Gao, Hypoelliptic Diffusion Maps and Their Applications in Automated Geometric Morphometrics, Ph.D. thesis, Duke University, Durham, NC, 2015.
25.
T. Gao, The Diffusion Geometry of Fibre Bundles: Horizontal Diffusion Maps, preprint, https://arxiv.org/abs/1602.02330, 2016; Appl. Comput. Harmon. Anal., submitted.
26.
T. Gao, J. Brodzki, and S. Mukherjee, The Geometry of Synchronization Problems and Learning Group Actions, preprint, https://arxiv.org/abs/1610.09051, 2016.
27.
T. Gao, S. Z. Kovalsky, and I. Daubechies, Gaussian process landmarking on manifolds, SIAM J. Math. Data Sci., 1 (2019), pp. 208--236, https://doi.org/10.1137/18M1184035.
28.
T. Gao, G. S. Yapuncich, I. Daubechies, S. Mukherjee, and D. M. Boyer, Development and assessment of fully automated and globally transitive geometric morphometric methods, with application to a biological comparative dataset with high interspecific variation, Anatom. Rec., 301 (2018), pp. 636--658, https://doi.org/10.1002/ar.23700.
29.
L. Z. Garamszegi, Modern Phylogenetic Comparative Methods and Their Application in Evolutionary Biology: Concepts and Practice, Springer, Berlin, 2014.
30.
T. F. Gonzalez, Clustering to minimize the maximum intercluster distance, Theoret. Comput. Sci., 38 (1985), pp. 293--306.
31.
P. I. Good, Permutation, Parametric, and Bootstrap Tests of Hypotheses, Springer Ser. Statist., Springer-Verlag, New York, 2004.
32.
J. C. Gower, Generalized Procrustes analysis, Psychometrika, 40 (1975), pp. 33--51, https://doi.org/10.1007/BF02291478.
33.
J. C. Gower and G. B. Dijksterhuis, Procrustes Problems, Oxford Statist. Sci. Ser. 3, Oxford University Press, Oxford, UK, 2004.
34.
B. Güneysu, The Feynman-Kac formula for Schrödinger operators on vector bundles over complete manifolds, J. Geom. Phys., 60 (2010), pp. 1997--2010.
35.
E. Harjunmaa, A. Kallonen, M. Voutilainen, K. Hämäläinen, M. L. Mikkola, and J. Jernvall, On the difficulty of increasing dental complexity, Nature, 483 (2012), pp. 324--327.
36.
E. Harjunmaa, K. Seidel, T. Häkkinen, E. Renvoisé, I. J. Corfe, A. Kallonen, Z.-Q. Zhang, A. R. Evans, M. L. Mikkola, I. Salazar-Ciudad, and O. D. Klein, Replaying evolutionary transitions from the dental fossil record, Nature, 512 (2014), pp. 44--48.
37.
B. R. Hassett and T. Lewis-Bale, Comparison of 3D landmark and 3D dense cloud approaches to hominin mandible morphometrics using structure-from-motion, Archaeometry, 59 (2017), pp. 191--203, https://doi.org/10.1111/arcm.12229.
38.
B. Helffer and J. Sjöstrand, Puits Multiples en Mecanique Semi-Classique iv Etude du Complexe de Witten, Comm. Partial Differential Equations, 10 (1985), pp. 245--340.
39.
S. C. Joshi and M. I. Miller, Landmark matching via large deformation diffeomorphisms, IEEE Trans. Image Process., 9 (2000), pp. 1357--1370, https://doi.org/10.1109/83.855431.
40.
D. G. Kendall, Shape manifolds, procrustean metrics, and complex projective spaces, Bull. London Math. Soc., 16 (1984), pp. 81--121.
41.
V. Kim, Y. Lipman, and T. Funkhouser, Blended intrinsic maps, ACM Trans. Graphics, 30 (2011), 79.
42.
C.-W. Ko, J. Lee, and M. Queyranne, An exact algorithm for maximum entropy sampling, Oper. Res., 43 (1995), pp. 684--691.
43.
P. Koehl and J. Hass, Landmark-free geometric methods in biological shape analysis, J. Roy. Soc. Interface, 12 (2015), 2015.0795.
44.
S. Z. Kovalsky, M. Galun, and Y. Lipman, Accelerated quadratic proxy for geometric optimization, ACM Trans. Graphics, 35 (2016), 134.
45.
A. Krause, A. Singh, and C. Guestrin, Near-optimal sensor placements in Gaussian processes: Theory, efficient algorithms and empirical studies, J. Mach. Learn. Res., 9 (2008), pp. 235--284.
46.
D. Le Peutrec, F. Nier, and C. Viterbo, Precise Arrhenius law for p-forms: The Witten Laplacian and Morse-Barannikov complex, Ann. Henri Poincaré, 14 (2013), pp. 567--610.
47.
Y. Lipman, Bounded distortion mapping spaces for triangular meshes, ACM Trans. Graphics, 31 (2012), 108.
48.
Y. Lipman and I. Daubechies, Conformal Wasserstein distances: Comparing surfaces in polynomial time, Adv. Math., 227 (2011), pp. 1047--1077, https://doi.org/10.1016/j.aim.2011.01.020.
49.
Y. Lipman, J. Puente, and I. Daubechies, Conformal Wasserstein distance: II. Computational aspects and extensions, Math. Comp., 82 (2013), pp. 331--381, http://dblp.uni-trier.de/db/journals/moc/moc82.html#LipmanPD13.
50.
Y. Lipman, S. Yagev, R. Poranne, D. W. Jacobs, and R. Basri, Feature matching with bounded distortion, ACM Trans. Graphics, 33 (2014), 26.
51.
Y.-S. Liu, M. Liu, D. Kihara, and K. Ramani, Salient critical points for meshes, in Proceedings of the 2007 ACM Symposium on Solid and Physical Modeling, ACM, 2007, pp. 277--282.
52.
D. G. Lowe, Object recognition from local scale-invariant features, in Proceedings of the 7th IEEE International Conference on Computer Vision, Vol. 2, IEEE, 1999, pp. 1150--1157.
53.
N. Mantel, The detection of disease clustering and a generalized regression approach, Cancer Res., 27 (1967), pp. 209--220.
54.
S. Melzi, E. Rodolà, U. Castellani, and M. M. Bronstein, Localized manifold harmonics for spectral shape analysis, Computer Graphics Forum, 37 (2018), pp. 20--34.
55.
P. Mitteroecker and P. Gunz, Advances in geometric morphometrics, Evolutionary Biol., 36 (2009), pp. 235--247.
56.
P. Mitteroecker and S. M. Huttegger, The concept of morphospaces in evolutionary and developmental biology: Mathematics and metaphors, Biological Theory, 4 (2009), pp. 54--67.
57.
C. Moenning and N. A. Dodgson, Fast Marching Farthest Point Sampling, Tech. report, University of Cambridge, Computer Laboratory, Cambridge, UK, 2003.
58.
J. L. Moigne, Introduction to remote sensing image registration, in Proceedings of the 2017 IEEE International Geoscience and Remote Sensing Symposium (IGARSS), IEEE, 2017, pp. 2565--2568, https://doi.org/10.1109/IGARSS.2017.8127519.
59.
B. Nadler, S. Lafon, R. R. Coifman, and I. G. Kevrekidis, Diffusion maps, spectral clustering and reaction coordinates of dynamical systems, Appl. Comput. Harmon. Anal., 21 (2006), pp. 113--127.
60.
A. Naor, O. Regev, and T. Vidick, Efficient rounding for the noncommutative Grothendieck inequality, in Proceedings of the 45th Annual ACM Symposium on Theory of Computing, ACM, 2013, pp. 71--80.
61.
G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher, An analysis of approximations for maximizing submodular set functions I, Math. Program., 14 (1978), pp. 265--294.
62.
A. Nemirovski, Sums of random symmetric matrices and quadratic optimization under orthogonality constraints, Math. Program., 109 (2007), pp. 283--317.
63.
T. E. Nichols and A. P. Holmes, Nonparametric permutation tests for functional neuroimaging: A primer with examples, Human Brain Mapping, 15 (2002), pp. 1--25.
64.
S. Niranjan, A. Krause, S. M. Kakade, and M. Seeger, Gaussian process optimization in the bandit setting: No regret and experimental design, in Proceedings of the 27th International Conference on Machine Learning, 2010, pp. 1015--1022.
65.
M. Ovsjanikov, Q.-X. Huang, and L. Guibas, A condition number for non-rigid shape matching, Computer Graphics Forum, 30 (2011), pp. 1503--1512.
66.
O. Özyeşil, N. Sharon, and A. Singer, Synchronization over Cartan motion groups via contraction, SIAM J. Appl. Algebra Geom., 2 (2018), pp. 207--241, https://doi.org/10.1137/16M1106055.
67.
S. Pandey, W. Voorsluys, M. Rahman, R. Buyya, J. Dobson, and K. Chiu, Brain image registration analysis workflow for fMRI studies on global grids, in Proceedings of the 2009 International Conference on Advanced Information Networking and Applications, 2009, pp. 435--442, https://doi.org/10.1109/AINA.2009.13.
68.
E. Paradis, Analysis of Phylogenetics and Evolution with R, Springer, New York, 2011.
69.
F. Pesarin, Multivariate Permutation Tests: With Applications in Biostatistics, Wiley, Chichester, UK, 2001.
70.
I. Plyusnin, A. R. Evans, A. Karme, A. Gionis, and J. Jernvall, Automated $3$D phenotype analysis using data mining, PLoS One, 3 (2008), e1742.
71.
J. Puente, Distances and Algorithms to Compare Sets of Shapes for Automated Biological Morphometrics, Ph.D. thesis, Princeton University, Princeton, NJ, 2013.
72.
J. Ramsay and B. Silverman, Functional Data Analysis, 2nd ed., Springer Series in Statistics, Springer, New York, 2005.
73.
J. O. Ramsay and B. W. Silverman, Applied Functional Data Analysis: Methods and Case Studies, Springer Series in Statistics 77, Springer, New York, 2002.
74.
F. J. Rohlf and F. L. Bookstein, Proceedings of the Michigan Morphometrics Workshop, University of Michigan Museum of Zoology, Ann Arbor, MI, 1990.
75.
V. L. Roth, On three-dimensional morphometrics, and on the identification of landmark points, in Contributions to Morphometrics, L. F. Marcus, E. Bello, and G.-V. A., eds., Museo Nacional de Ciencias Naturales, Madrid, Spain, 1993, pp. 41--61.
76.
T. J. Santner, B. J. Williams, and W. I. Notz, The Design and Analysis of Computer Experiments, Springer Series in Statistics, Springer, Springer, New York, 2013.
77.
A. Singer, From graph to manifold Laplacian: The convergence rate, Appl. Comput. Harmon. Anal., 21 (2006), pp. 128--134.
78.
A. Singer and H.-T. Wu, Vector diffusion maps and the connection Laplacian, Comm. Pure Appl. Math., 65 (2012), pp. 1067--1144, https://doi.org/10.1002/cpa.21395.
79.
J. Smith and S. Schaefer, Bijective parameterization with free boundaries, ACM Trans. Graphics, 34 (2015), 70.
80.
O. Smolyanov, H. Weizsäcker, and O. Wittich, Brownian motion on a manifold as limit of stepwise conditioned standard Brownian motions, in Stochastic Processes, Physics and Geometry: New Interplays, II (Leipzig, 1999), CMS Conf. Proc. 29, AMS, Providence, RI, 2000, pp. 589--602.
81.
O. G. Smolyanov, H. v. Weizsäcker, and O. Wittich, Chernoff's theorem and discrete time approximations of Brownian motion on manifolds, Potential Anal., 26 (2007), pp. 1--29.
82.
A. M.-C. So, Moment inequalities for sums of random matrices and their applications in optimization, Math. Program., 130 (2011), pp. 125--151.
83.
M. L. Stein, Interpolation of Spatial Data: Some Theory for Kriging, Springer, New York, 2012.
84.
G. K. Tam, Z.-Q. Cheng, Y.-K. Lai, F. C. Langbein, Y. Liu, D. Marshall, R. R. Martin, X.-F. Sun, and P. L. Rosin, Registration of 3D point clouds and meshes: A survey from rigid to nonrigid, IEEE Trans. Visualization Comput. Graphics, 19 (2013), pp. 1199--1217.
85.
O. Van Kaick, H. Zhang, G. Hamarneh, and D. Cohen-Or, A survey on shape correspondence, Computer Graphics Forum, 30 (2011), pp. 1681--1707.
86.
N. S. Vitek, C. L. Manz, T. Gao, J. I. Bloch, S. G. Strait, and D. M. Boyer, Semi-supervised determination of pseudocryptic morphotypes using observer-free characterizations of anatomical alignment and shape, Ecol. Evol., 7 (2017), pp. 5041--5055, https://doi.org/10.1002/ece3.3058.
87.
S. K. Wärmländer, H. Garvin, P. Guyomarc'h, A. Petaros, and S. B. Sholts, Landmark typology in applied morphometrics studies: What's the point?, Anatom. Rec., (2018), https://doi.org/10.1002/ar.24005.
88.
A. Watanabe, How many landmarks are enough to characterize shape and size variation?, PloS One, 13 (2018), e0198341.
89.
J. White, Geometric morphometric investigation of molar shape diversity in modern lemurs and lorises, Anatom. Rec., 292 (2009), pp. 701--719.
90.
E. Witten, Supersymmetry and Morse theory, J. Differential Geom., 17 (1982), pp. 661--692, https://doi.org/10.4310/jdg/1214437492.
91.
M. L. Zelditch, D. L. Swiderski, and H. D. Sheets, Geometric Morphometrics for Biologists: A Primer, 2nd ed., Academic Press, San Diego, CA, 2012.

Information & Authors

Information

Published In

cover image SIAM Journal on Mathematics of Data Science
SIAM Journal on Mathematics of Data Science
Pages: 237 - 267
ISSN (online): 2577-0187

History

Submitted: 27 July 2018
Accepted: 27 December 2018
Published online: 12 February 2019

Keywords

  1. Gaussian process
  2. experimental design
  3. active learning
  4. manifold learning
  5. geometric morphometrics

MSC codes

  1. 60G15
  2. 62K05
  3. 65D18

Authors

Affiliations

Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : BCS-1552848
Simons Foundation https://doi.org/10.13039/100000893 : 400837

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media