Caroline Bernard-Michel

Retrieval of Mars surface physical properties from OMEGA hyperspectral images using Regularized Sliced Inverse Regression

Hyperspectral remote sensing, also known as imaging spectroscopy, is a promising space technology regularly selected by agencies with regard to the exploration and observation of planets, to earths geology or to the monitoring of the environment. It allows to collect for each pixel of a scene, the intensity of light energy reflected from planets as it varies across different wavelengths. More than one hundred spectels in the visible and near infra-red are typically recorded, making it possible to observe a continuous spectrum for each image cell. Usually, in space exploration, the analysis of these spectral signatures allows to retrieve the physical, chemical or mineralogical properties of surfaces and of atmospheres that may help to understand the geological and climatological history of planets. We propose in this paper a statistical method to evaluate the physical properties of surface materials on Mars from hyperspectral images collected by the OMEGA instrument aboard the Mars express spacecraft. The approach we develop is based on the estimation of the functional relationship F between some physical parameters and observed spectra. For this purpose, a database of synthetic spectra is generated by a physical radiative transfer model and used to estimate F. The high dimension of spectra is reduced by using Gaussian regularized sliced inverse regression (GRSIR) to overcome the curse of dimensionality and consequently the sensitivity of the inversion to noise (ill-conditioned problems). Compared with a naive spectrum matching approach such as the k-nearest neighbors algorithm, estimates are more accurate and realistic.

Gaussian Regularized Sliced Inverse Regression

Sliced Inverse Regression (SIR) is an effective method for dimension reduction in high-dimensional regression problems. The original method, however, requires the inversion of the predictors covariance matrix. In case of collinearity between these predictors or small sample sizes compared to the dimension, the inversion is not possible and a regularization technique has to be used. Our approach is based on a Fisher Lecture given by R.D. Cook where it is shown that SIR axes can be interpreted as solutions of an inverse regression problem. We propose to introduce a Gaussian prior distribution on the unknown parameters of the inverse regression problem in order to regularize their estimation. We show that some existing SIR regularizations can enter our framework, which permits a global understanding of these methods. Three new priors are proposed leading to new regularizations of the SIR method. A comparison on simulated data as well as an application to the estimation of Mars surface physical properties from hyperspectral images are provided.

A Note on Sliced Inverse Regression with Regularizations

In Li and Yin (2008, Biometrics 64, 124-131), a ridge SIR estimator is introduced as the solution of a minimization problem and computed thanks to an alternating least-squares algorithm. This methodology reveals good performance in practice. In this note, we focus on the theoretical properties of the estimator. It is shown that the minimization problem is degenerated in the sense that only two situations can occur: Either the ridge SIR estimator does not exist or it is zero.

Machine learning techniques for the inversion of planetary hyperspectral images

In this paper, the physical analysis of planetary hyperspectral images is addressed. To deal with high dimensional spaces (image cubes present 256 bands), two methods are proposed. The first method is the support vectors machines regression (SVM-R) which applies the structural risk minimization to perform a non-linear regression. Several kernels are investigated in this work. The second method is the Gaussian regularized sliced inverse regression (GRSIR). It is a two step strategy; the data are map onto a lower dimensional vector space where the regression is performed. Experimental results on simulated data sets have showed that the SVM-R is the most accurate method. However, when dealing with real data sets, the GRSIR gives the most interpretable results.