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

Abstract

A spectral geometry utility awareness, with specific reference to isospectralisation and art painting analytics, is permeating the academy today, with special interest in its ability to foster interfaces between a range of analytical quantitative disciplines and art, exhibiting popularity in, for example, computer engineering/image processing and GIScience/spatial statistics, among other subject areas. This paper contributes to the emerging literature about such mathematized interdisciplinarities and synergies. It more specifically extends the matrix algebra based 2-D Graph Moranian operator that dominates spatial statistics/econometrics to the 3-D Riemannian manifold sphere whose analysis the general Graph Laplacian (i.e. Laplace-Beltrami) operator monopolizes today. One conclusion is that harmonizing the use of these two operators offers a way to expand knowledge and comprehension. Another is a continuing demonstration that the understanding and analysis of art sculptures dovetails with mathematics-art studies.

GRAPHICAL ABSTRACT

KEYWORDS:

• Geary ratio
• Moran coefficient
• Riemannian manifold
• spatial autocorrelation
• spectral geometry

Acknowledgements

Daniel A. Griffith is an Ashbel Smith Professor of Geospatial Information Sciences.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Statements and declarations

The author did not receive support from any organization for this work. The author further certifies that he has no affiliations with or involvement in any organization or entity with any financial or non-financial interest in the subject matter or materials discussed in this paper. Finally, the datasets generated and/or analyzed during the study summarized in this paper are available from the corresponding author by reasonable request; a number of the source datasets also are retrievable from online depositories cited in this paper.

Notes

1 The spatial statistics/econometrics literature notation almost universally symbolizes this matrix with C, and its row-standardized Laplacian companion with W.

2 The Laplace-Beltrami operator based upon an unbounded equilateral triangle mesh essentially is a scalar multiple of this Laplacian matrix (Xu, 2004; Wu et al., 2010).

3 They disagree about the spatial autocorrelation portrayal of numerous regular square tessellation eigenvectors, based upon either a rook or a queen definition of adjacency, both of which extract exactly the same eigenvectors.

4 See https://github.com/alecjacobson/common-3d-test-models, https://github.com/MPI-IS/mesh/blob/master/data/unittest/sphere.obj, https://cims.nyu.edu/gcl/datasets.html, https://www.cs.cornell.edu/courses/cs4620/2015fa/assignments/a1/a1mesh.html, https://people.sc.fsu.edu/~jburkardt/data/obj/obj.html, http://graphics.stanford.edu/data/3Dscanrep/, https://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/, https://github.com/yig/graphics101-meshes/tree/master/examples, and https://archive.lib.msu.edu/crcmath/math/math/t/t200.htm.

5 Many digitized surfaces are too large, with many tens- or hundreds-of-thousands, or millions of vertices (e.g. bunny, n = 35,947; https://github.com/alecjacobson/common-3d-test-models/blob/master/data/stanford-bunny.obj), for convenient exploratory work. Others do not constitute a single complete graph, being composed of several subset graphs; e.g. the following 3-D digitized teapot retrieved from the internet consists of a handle-pot-spout (6,072 vertices and 12,151 triangles) and a disconnected lid (1,778 vertices and 3,552triangles):

A merged operator for the two distinct complete graphs interlaces their eigenfunctions, compromising Riemannian manifold reconstruction using eigenvectors.

6 GR = α – [(n–1)/n] MC (Griffith, 1993, p. 23) $\Rightarrow$ E(GR) = E(α) – [(n–1)/n E(MC) $\Rightarrow$ [substituting the well-known quantities for E(GR) and E(MC)] 1 = E(α) – [(n–1)/n][–1/(n–1)] $\Rightarrow$ E(α) = (n–1)/n, where E denotes the expected value operator. E(α), the same initial coefficient of MC, asymptotically converges on 1.

7 For example, Figure 4 entries allude to the following curve-fitting approximations, calculated with nonlinear regression, where the pseudo-R² is the squared correlation between predicted and observed values (e.g. Christensen, 2007):

• Figure 4d: λ_GR ≈ 0.68 + 1.19[LN(1.31 – λ_MC)/ (1.01 + λ_MC)^1.78][1 + 1.51λ_MC – 0.98${\lambda }_{\text{MC}}^3$] , pseudo-R² = 0.9977, where the noninteger exponent 1.78 merits further study to determine whether or not it has some specific meaning (it seems to seek to account for the two plotted outlier values positioned in the upper left corner of Quadrant II),

• Figure 4e: λ_GR ≈ 0.94 – 3.80λ_M + 0.51${\lambda }_{\text{MC}}^2$ – 0.37${\lambda }_{\text{MC}}^3$, R² = 0.9920,

• Figure 4f: λ_GR ≈ 1.32 + 0.36LN(1.52 – λ_MC) – 1.19LN(1.52 + λ_MC), pseudo-R² = 0.9990,

• Figure 4g: λ_GR ≈ 1.542 + 0.63LN(2.02 – λ_MC) – 1.41LN(2.02 + λ_MC), pseudo-R² = 0.9996,

• Figure 4h: λ_GR ≈ 1.65 + 0.76LN(2.30 – λ_MC) – 1.56LN(2.30 + λ_MC), pseudo-R² = 0.9996,

• Figure 4i: λ_GR ≈ –3.83 + 2.91LN(4.90 – λ_MC) – 0.24LN(0.55 + λ_MC), pseudo-R² = 0.9999,

where λ_GR, λ_MC, and LN respectively denote the Graphs Laplacian and Moranian eigenvalues, and natural logarithm.

8 Source: https://commons.wikimedia.org/wiki/File:Buddha_detail,_Japan_-_Taima_Temple_Mandala-_Amida_Welcomes_Ch%C3%BBj%C3%B4hime_to_the_Western_Paradise_-_Google_Art_Project_(cropped).jpg.