Massive analysis of multi-angular images by inverse regression of reflectance models for the physical characterization of planetary surfaces.

Remote sensing of the Earth and other planets of the Solar System return huge volumes of data. Retrieving information of interest from such data may consist of inverting a direct or forward physical model, which theoretically describes how parameters of interest x ∈ X are translated into observations y ∈ Y. The observations y are high-dimensional (of dimension D) because they represent signals in time, angles or wavelengths. Besides, many such high-dimensional observations are available and the application requires a very large number of inversions (denoted by Nobs in what follows). The parameters x to be predicted (of dimension L) is itself multi-dimensional with correlated dimensions. In [1] a statistical method is put forward with a Bayesian and machine learning framework capable of solving the problem of inverting physical models on multi-dimensional data in order to estimate the value of their parameters. The method addresses: 1) the large number of observations to be analysed, 2) their high dimension, 3) the need to provide predictions for several correlated parameters, 4) the possible existence of multiple solutions and 5) the requirement to provide the latter with a confidence measure (e.g. uncertainty quantification). 
Here our science case consists of analysing hyperspectral images acquired in the visible and infrared at different angles over regions of interest at the surface of Mars. The interpretation of the BRDF (Bidirectional Reflectance Distribution Function) extracted from such multi-angular observations in terms of microtexture of surface materials such as grain size, shape, roughness and internal structure, which can be used as tracers of geological processes, is based on the inversion of physical models of radiative transfer (Hapke and Shkuratov models). The latter link in a nonlinear way physical and observable parameters (functional y = F(x)). In this case y = yobs is a vector of D reflectance values for D geometries (up to a few tens), x is vector of L = 4 or L = 6 photometric parameters, while the number of observations to be inverted Nobs can be of the order of a few millions. This great number of observations Nobs, organized in data cubes, results from the combination of spectral and spatial sampling of the scene. From an experimental vector yobs of reflectance values, the objective is to estimate the mean or the most probable value(s) for each of the L parameters (the components of vector x) of the physical model F and to provide a measurement of uncertainty about the estimations. Our pipeline performs the estimation in four steps:
1. The generation of a database of N couples DN = {(xn,yn), n = 1 : N} using the direct physical model, y = F(x). More specifically, xn values are simulated from a chosen prior distribution, e.g. uniform over the parameter range, and the direct model F is applied to produce F(xn) to which a typically Gaussian noise is added to provide yn. F is supposed to be available as a closed-form expression. 


2. The learning phase in which the pipeline constructs from the database a direct and an inverse parametric statistical model of the functional y = F(x). The learned model is a Gaussian mixture model with a structured parameterization referred to as GLLiM for Gaussian Locally Linear Mapping (see [1] for details). Because it is trained on the data from step 1, it depends on the physical direct model. The inverse model is then approximated by a surrogate probability distribution function (PDF) expression pGLLiM which is learned once, for all possible yobs to be inverted, p(xy) ≈ pGLLiM(xy) .


3. This surrogate model is then used for all vectors yobs in order to build a set of a PDFs pGLLiM(xyobs).


4. The PDFs are then exploited each independently by different techniques to estimate the solution x̂ corresponding to each vector yobs: estimation by the mean/mode of the PDF, fusion of the components of the Gaussian mixture model into a small number of centroids (usually two or three) to identify possible multiple modes (we consider up to 2 modes in the following), importance sampling of the true target PDF p(xyobs) with the proposal distribution set to pGLLiM(xyobs) around the mean or around the centroids/modes, [1] to improve the quality of the predictions (eg. posterior mean, modes and variance).

The pipeline is implemented into a high-performance, documented, and open-source software PlanetGLLiM* that is specifically tailored to handle inversions of reflectance models given cubes of multi-spectral data and geometries typically produced by planetary remote sensing. PlanetGLLiM offers an easy to use graphical user interface and implements the Hapke and the Shkuratov models. Custom models can be added by the user in the form of a single Python class. The PlanetGLLiM application played a crucial role in conducting the first large-scale inversion of a dataset derived from multi-angular observations across various type localities on Mars using the CRISM sensor [2], enabling their spectrophotometric and physical characterization. The method was also applied to perform a massive Hapke model inversion on Pleiades satellite data to map the physical properties of the Moon surface [3].

* https://kernelo-mistis.gitlabpages.inria.fr/planet-gllim-front-end/

[1] B. Kugler, F. Forbes, and S. Douté. Fast Bayesian inversion for high dimensional inverse problems. Statistics and Computing, 32(2):31.

[2] S. Douté. In LPI Contributions, volume 3040 of LPI Contributions, page 2149, March 2024

[3] D. T. Nguyen, S. Jacquemoud, A. Lucas, S. Douté, C. Ferrari, S. Coustance, S. Marcq, and A. Meygret. In LPI Contributions, volume 3040 of LPI Contributions, page 1998, March 2024.