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Articles

Spectral geometry and Riemannian manifold mesh approximations: some autocorrelation lessons from spatial statistics

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Pages 293-313 | Received 17 Apr 2023, Accepted 23 Oct 2023, Published online: 01 Nov 2023

References

  • Aflalo, Y., Brezis, H., & Kimmel, R. (2015). On the optimality of shape and data representation in the spectral domain. SIAM Journal on Imaging Sciences, 8(2), 1141–1160. https://doi.org/10.1137/140977680
  • Alsnayyan, A., & Shanker, B. (2022). Laplace-Beltrami based multi-resolution shape reconstruction on subdivision surfaces. The Journal of the Acoustical Society of America, 151(3), 2207–2222. https://doi.org/10.1121/10.0009851
  • Bérard, P. (1986). Spectral geometry: Direct and inverse problems. Berlin: Springer, Lecture Notes in Mathematics.
  • Besag, J. (1975). Statistical analysis of non-lattice data. Journal of the Royal Statistical Society: Series D (The Statistician), 24(3), 179–195.
  • Boissonnat, J. D., Dyer, R., & Ghosh, A. (2018). Delaunay triangulation of manifolds. Foundations of Computational Mathematics, 18(2), 399–431. https://doi.org/10.1007/s10208-017-9344-1
  • Borcard, D., & Legendre, P. (2002). All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices. Ecological Modelling, 153(1–2), 51–68. https://doi.org/10.1016/S0304-3800(01)00501-4
  • Borovitskiy, V., Terenin, A., & Mostowsky, P. (2020, December 6–12). Matérn Gaussian processes on Riemannian manifolds. Advances in neural information processing systems, proceedings of the 34th conference on neural information processing systems (NeurIPS 2020), Vancouver, Canada (Vol. 33, 12426–12437).
  • Cammarasana, S., & Patané, G. (2021). Localised and shape-aware functions for spectral geometry processing and shape analysis: A survey & perspectives. Computers & Graphics, 97, 1–18. https://doi.org/10.1016/j.cag.2021.03.006
  • Chen, R., Xu, Y., Gotsman, C., & Liu, L. (2010). A spectral characterization of the Delaunay triangulation. Computer Aided Geometric Design, 27(4), 295–300. https://doi.org/10.1016/j.cagd.2010.02.002
  • Chen, W., Zheng, X., Ke, J., Lei, N., Luo, Z., & Gu, X. (2019). Quadrilateral mesh generation I: Metric based method. Computer Methods in Applied Mechanics and Engineering, 356, 652–668. https://doi.org/10.1016/j.cma.2019.07.023
  • Christensen, R. (2007). General prediction theory and the role of R2. Unpublished manuscript. https://www.math.unm.edu/~fletcher/JPG/rsq.pdf; last accessed on 7 June 2023.
  • Cliff, A., & Ord, J. (1972). Spatial Autocorrelation. Pion.
  • Cosmo, L., Panine, M., Rampini, A., Ovsjanikov, M., Bronstein, M., & Rodola, E. (2019, June 15–20). Isospectralization, or how to hear shape, style, and correspondence. In L. O’Connor (editorial producer) (Ed.), Proceedings of the IEEE/CVF conference on computer vision and pattern recognition (CVPR 2019) (pp. 7529–7538). Long Beach, CA, Conference Publishing Services (CPS) & IEEE Computer Society.
  • De Goes, F., Butts, A., & Desbrun, M. (2020). Discrete differential operators on polygonal meshes. ACM Transactions on Graphics, 39(4), 110, 14. https://doi.org/10.1145/3386569.3392389
  • de Jong, P., Sprenger, C., & van Veen, F. (1984). On the extreme values of Moran's I and Geary's c. Geographical Analysis, 16(1), 17–24. https://doi.org/10.1111/j.1538-4632.1984.tb00797.x
  • Frobenius, G. (1912, May 23). Ueber Matrizen aus nicht negativen Elementen [Translation: On matrices of non-negative elements]. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 456–477. https://archive.org/details/mobot31753002089602/page/6/mode/2up
  • Garcia, C. (2012). A simple procedure for the comparison of covariance matrices. BMC Evolutionary Biology, 12(1), 222. https://doi.org/10.1186/1471-2148-12-222
  • Garimella, R. V., Shashkov, M. J., & Knupp, P. M. (2004). Triangular and quadrilateral surface mesh quality optimization using local parametrization. Computer Methods in Applied Mechanics and Engineering, 193(9–11), 913–928. https://doi.org/10.1016/j.cma.2003.08.004
  • Gasparetto, A., & Torsello, A. (2015, June 7–12). A statistical model of riemannian metric variation for deformable shape analysis. In Proceedings of the IEEE conference on computer vision and pattern recognition (CVPR) (pp. 1219–1228). Boston, MA. https://doi.org/10.1109/CVPR.2015.7298726
  • Geary, R. (1954). The contiguity ratio and statistical mapping. The Incorporated Statistician, 5(3), 115–141. https://doi.org/10.2307/2986645
  • Ghosh, K. (2020). Time series analysis: A brief history and its future challenges. Indian Science Cruiser, 34(5), 22–27. https://doi.org/10.24906/isc/2020/v34/i5/206994
  • Griffith, D. (1993). Spatial regression analysis on the PC: Spatial statistics using SAS. Association of American Geographers.
  • Griffith, D. (1996). Spatial autocorrelation and eigenfunctions of the geographic weights matrix accompanying geo-referenced data. Canadian Geographer, 40(4), 351–367. https://doi.org/10.1111/j.1541-0064.1996.tb00462.x
  • Griffith, D. (2000). Eigenfunction properties and approximations of selected incidence matrices employed in spatial analyses. Linear Algebra & Its Applications, 321(1–3), 95–112. https://doi.org/10.1016/S0024-3795(00)00031-8
  • Griffith, D. (2003). Spatial autocorrelation and spatial filtering: Gaining understanding through theory and scientific visualization. Springer-Verlag.
  • Griffith, D. (2016). Spatial autocorrelation and art. Cybergeo: European Journal of Geography, Cybergeo anniversary – Cross section, Online since 22 January 2016, connection on 23 January 2016. https://doi.org/10.4000/cybergeo.27429
  • Griffith, D. (2020). A spatial analysis of selected art: A GIScience-humanities interface. International Journal of Humanities and Arts Computing, 14(1–2), 154–175. https://doi.org/10.3366/ijhac.2020.0250
  • Griffith, D. (2021). Eigenvector visualization and art. Journal of Mathematics and the Arts, 15(2), 170–187. https://doi.org/10.1080/17513472.2021.1922239
  • Griffith, D. (2022). Art, geography/GIScience, and mathematics: A surprising interface. In Annals of the American association of geographers (p. 12). Taylor & Francis Online. https://doi.org/10.1080/24694452.2022.2086101.
  • Griffith, D., & Chun, Y. (2019). Implementing Moran eigenvector spatial filtering for massively large georeferenced datasets. International Journal of Geographical Information Science, 33(9), 1703–1717. https://doi.org/10.1080/13658816.2019.1593421
  • Griffith, D., Chun, Y., & Li, B. (2019). Spatial regression analysis using eigenvector spatial filtering. Academic Press.
  • Griffith, D., & Li, B. (2017, August 2–4). A geocomputation and geovisualization comparison of Moran and Geary eigenvector spatial filtering. In CPGIS publication committee, proceedings of the 25th international conference on geoinformatics, geoinformatics 2017 (p. 4). SUNY/Buffalo, Buffalo, NY.
  • Hua, J., & Zhong, J. (2020). Spectral geometry of shapes. Academic Press.
  • Izyurov, K., & Khristoforov, M. (2022). Asymptotics of the determinant of discrete Laplacians on triangulated and quadrangulated surfaces. Communications in Mathematical Physics, 394(2), 531–572. https://doi.org/10.1007/s00220-022-04437-3
  • Jungnickel, D., & Pott, A. (2017). Correlation property for sequences. Encyclopedia of Mathematics. http://encyclopediaofmath.org/index.php?title=Correlation_property_for_sequences&oldid=46527
  • Kac, M. (1966). Can one hear the shape of a drum? The American Mathematical Monthly, 73(4), 1–23. https://doi.org/10.2307/2313748
  • Kutner, M., Nachtsheim, C., Neter, J., & Li, W. (2005). Applied linear statistical models (5th ed., pp. 55–63). McGraw Hill/Irwin.
  • Leibon, G., & Letscher, D. (2000, June 12–14). Delaunay triangulations and voronoi diagrams for riemannian manifolds. In Proceedings of the sixteenth annual symposium on computational geometry (pp. 341–349). Hong Kong. https://doi.org/10.1145/336154.336221.
  • Lescoat, T., Liu, H.-T., Thiery, J.-M., Jacobson, A., Boubekeur, T., & Ovsjanikov, M. (2020). Spectral mesh simplification. Computer Graphics Forum, 39(2), 315–324. https://doi.org/10.1111/cgf.13932
  • Luo, Q., Griffith, D., & Wu, H. (2018). Spatial autocorrelation for massive spatial data: Verification of efficiency and statistical power asymptotics. Journal of Geographical Systems, 21(2), 237–269. https://doi.org/10.1007/s10109-019-00293-3
  • Maćkiewicz, A., & Ratajczak, W. (1996). Towards a new definition of topological accessibility. Transportation Research Part B: Methodological, 30(1), 47–79. https://doi.org/10.1016/0191-2615(95)00020-8
  • Marin, R., Rampini, A., Castellani, U., Rodolà, E., Ovsjanikov, M., & Melzi, S. (2021). Spectral shape recovery and analysis via data-driven connections. International Journal of Computer Vision, 129(10), 2745–2760. https://doi.org/10.1007/s11263-021-01492-6
  • Martz, R. (2018). The MCNP6 book on unstructured mesh geometry: User's guide for MCNP 6.2.1. Los Alamos Research Report #LA-UR-18-27630. Los Alamos National Laboratory.
  • Minakshisundaram, S., & Pleijel, Å. (1949). Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Canadian Journal of Mathematics, 1(3), 242–256. https://doi.org/10.4153/CJM-1949-021-5
  • Mohar, B. (1997). Some applications of Laplacian eigenvalues of graphs. In G. Hahn & G. Sabidussi (Eds.), Graph symmetry: Algebraic methods and applications (pp. 225–275). Kluwer.
  • Moran, P. (1948). The interpretation of statistical maps. Journal of the Royal Statistical Society, 10B, 243–251.
  • Na, L., Zheng, X., Luo, Z., Luo, F., & Gu, X. (2020). Quadrilateral mesh generation II: Meromorphic quartic differentials and Abel–Jacobi condition. Computer Methods in Applied Mechanics and Engineering, 366, 112980. https://doi.org/10.1016/j.cma.2020.112980
  • O’Neill, B. (2020). The double-constant matrix, centering matrix and equicorrelation matrix: Theory and applications. Working Paper. Australian National University. https://arxiv.org/abs/2109.05814v1
  • Pereira, M., Desassis, N., & Allard, D. (2022). Geostatistics for large datasets on Riemannian manifolds: a matrix-free approach. arXiv preprint arXiv:2208.12501.
  • Perron, O. (1907). Zur theorie der matrices [translation: On the theory of matrices]. Mathematische Annalen, 64(2), 248–263. https://doi.org/10.1007/BF01449896. hdl:10338.dmlcz/104432, S2CID 123460172.
  • Potter, K., Koch, F., Oswalt, C., & Iannone, B. (2016). Data, data everywhere: Detecting spatial patterns in fine-scale ecological information collected across a continent. Landscape Ecology, 31, 67–84.
  • Reuter, M., Wolter, F., & Peinecke, N. (2006). Laplace–Beltrami spectra as ‘Shape-DNA’ of surfaces and solids. Computer-Aided Design, 38(4), 342–366. https://doi.org/10.1016/j.cad.2005.10.011
  • Straffin, P. (1980). Linear algebra in geography: Eigenvectors of networks. Mathematics Magazine, 53(5), 269–276. https://doi.org/10.1080/0025570X.1980.11976869
  • Tait, M., & Tobin, J. (2018). Characterizing graphs of maximum principal ratio. Electronic Journal of Linear Algebra, 34, 61–70. https://doi.org/10.13001/1081-3810.3200
  • Taubin, G. (1995, August 6–11). A signal processing approach to fair surface design. Proceedings of ACM SIGGRAPH, annual conference (pp. 351–358). ACM Press, Los Angeles, CA.
  • Tsay, R. (2000). Time series and forecasting: Brief history and future research. Journal of the American Statistical Association, 95(450), 638–643. https://doi.org/10.1080/01621459.2000.10474241
  • Weyl, H. (1911). Über die asymptotische Verteilung der Eigenwerte, Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-physikalusche. Klasse, 110–117.
  • White, D., Kimerling, J., & Overton, S. (1992). Cartographic and geometric components of a global sampling design for environmental monitoring. Cartography & Geographic Information Systems, 19(1), 5–22. https://doi.org/10.1559/152304092783786636
  • Wu, H., Zha, H., Luo, T., Wang, X., & Ma, S. (2010, June 13–18). Global and local isometry-invariant descriptor for 3D shape comparison and partial matching. In IEEE Staff (Ed.), 23rd IEEE computer society conference on computer vision and pattern recognition (CVPR 2010) (pp. 438–445), IEEE, San Francisco, CA, Piscataway, NJ. https://doi.org/10.1109/CVPR.2010.5540180
  • Xu, G. (2004). Convergence of discrete Laplace-Beltrami operators over surfaces. Computers & Mathematics with Applications, 48(3–4), 347–360. https://doi.org/10.1016/j.camwa.2004.05.001
  • Zelditch, S. (2017). Eigenfunctions of the Laplacian on a Riemannian Manifold. American Mathematical Society.
  • Zhang, H., van Kaick, O., & Dyer, R. (2007). Spectral methods for mesh processing and analysis. In D. Meneveaux & G. Patanè (Eds.), Proceedings of eurographics state-of-the-art report (Vol. 122). Eurographics Association.
  • Zhang, H., van Kaick, O., & Dyer, R. (2010). Spectral mesh processing. Computer Graphics Forum, 29(6), 1865–1894. https://doi.org/10.1111/j.1467-8659.2010.01655.x
  • Zheng, X., Zhu, Y., Chen, W., Lei, N., Luo, Z., & Gu, X. (2021). Quadrilateral mesh generation III: Optimizing singularity configuration based on Abel–Jacobi theory. Computer Methods in Applied Mechanics and Engineering, 387, 114146. https://doi.org/10.1016/j.cma.2021.114146

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