Spatial Multivariate Morphing Transformation

Abstract

Earth science phenomena, in particular mineralization of ore deposits, can be characterized by the spatial and statistical features of multivariate information. The relationships among these variables are often complex, encountering non-linear features, compositional constraints, and heteroscedasticity. Capturing and reproducing their statistical and spatial distributions is essential for uncertainty management, allowing for better decision-making and process control. In this work, we present a novel spatial multivariate morphing transformation that maps the initial multivariate space into a spatially and statistically decorrelated multi-Gaussian space. The spatial structures of the Gaussian random variables are modeled independently, and values are simulated at unsampled locations using a conventional univariate geostatistical simulation algorithm. Multivariate features and relationships are reintroduced by mapping from the multi-Gaussian distribution into the initial space. The spaces are paired following the fundamentals of point cloud morphing using discrete optimal transport to minimize the distance between landmark points between spaces. New simulated values are mapped from the anchored multi-Gaussian space into the multivariate space via thin-plate spline interpolation conditioned to the k-spatially known closest samples. The effectiveness of the method is demonstrated in a 6-dimensional dataset with strong non-linear relationships and spatial continuity. The resulting multivariate statistical and spatial metrics have been compared with simulations obtained by projection pursuit multivariate transformation.

Appendices

Appendix A: Thin-Plate Spline Interpolation

Physically inspired by the bending energy function of a two-dimensional steel sheet under small deformations, the integral bending norm or integral quadratic variation of the energy E(f) has the form

$$\begin{aligned} E(f) = \iint _{{\mathbb {R}}^2} \left( \frac{\partial ^2 f}{\partial x_1^2}\right) ^2 + 2\left( \frac{\partial ^2 f}{\partial x_1 \partial x_2}\right) ^2 + \left( \frac{\partial ^2 f}{\partial x_2^2}\right) ^2 \text {d} x_1 \, \text {d}x_2. \end{aligned}$$

(8)

It has been often applied in computer vision applications (Donato and Belongie 2002; Agbolade et al. 2020) on three-dimensional object surfaces under the form

$$\begin{aligned} \begin{aligned} E(f)&= \iiint _{{\mathbb {R}}^3} \left( \frac{\partial ^2 f}{\partial x_1^2}\right) ^2 + \left( \frac{\partial ^2 f}{\partial x_2^2}\right) ^2 + \left( \frac{\partial ^2 f}{\partial x_3^2}\right) ^2 \\&\quad + 2\left( \frac{\partial ^2 f}{\partial x_1 \partial x_2}\right) ^2 + 2\left( \frac{\partial ^2 f}{\partial x_1 \partial x_3}\right) ^2 + 2\left( \frac{\partial ^2 f}{\partial x_2 \partial x_3}\right) ^2 \text {d} x_1 \, \text {d}x_2 \, \text {d}x_3, \end{aligned} \end{aligned}$$

(9)

with a natural extension to d-dimensions and for an arbitrary order m of derivatives (Wood 2003; Rohr et al. 2001) as

$$\begin{aligned} E(f) = \int \dots \int _{{\mathbb {R}}^d} \sum \limits _{v_1+ \cdots + v_d = m} \frac{m!}{v_1! \dots v_d!} \left( \frac{\partial ^m f}{\partial x_1^{v_1} \ldots \partial x_d^{v_d}} \right) ^2 \text {d} x_1 \, \ldots \, \text {d}x_d, \end{aligned}$$

(10)

with $v_1, \ldots , v_d$ being positive integers (Wahba 1990). Note that with $d=m=2$, we recover Eq. 8, and with $d=3, m=2$ we recover Eq. 9.

The minimum number of landmark points must be greater than $\dfrac{(d+m-1)!}{d!(m-1)!}$ (Rohr et al. 2001). Thus, for $m=2$ and $d \ge 1$, the number of landmark points must be greater than $d+1$. In this work, we consider the case where $m=2$ and $d \ge 1$.

For a detailed explanation on bending energy function and minimum landmark point selection, we refer readers to Wahba (1990) and Rohr et al. (2001).

Appendix B: Bivariate Comparison Between PPMT and the Morphing Transformation

Full comparison between samples and a single realization from PPMT and Morphing transformation are provided on Figs. 26, 27, and 28, complementing results obtained in Sect. 5.5.

Fig. 26

Scatter-plots with kernel density estimation comparison between PPMT and Morphing. Samples (left) and a single realization from PPMT (center) and Morphing (right). Part 1 of 3

Fig. 27

Scatter-plots with kernel density estimation comparison between PPMT and Morphing. Samples (left) and a single realization from PPMT (center) and Morphing (right). Part 2 of 3

Fig. 28

Scatter-plots with kernel density estimation comparison between PPMT and Morphing. Samples (left) and a single realization from PPMT (center) and Morphing (right). Part 3 of 3