Regression Approaches to Small Sample Inverse Covariance Matrix Estimation for Hyperspectral Image Classification

Abstract

Abstract-A key component in most parametric classifiers is the estimation of an inverse covariance matrix. In hyperspectral images the number of bands can be in the hundreds leading to covariance matrices having tens of thousands of elements. Lately, the use of general linear regression models in estimating the inverse covariance matrix have been introduced in the time-series literature. This paper adopts and expands these ideas to ill-posed hyperspectral image classification problems. The results indicate that at least some of the approaches can give a lower classification error than traditional methods such as the linear discriminant analysis (LDA) and the regularized discriminant analysis (RDA). Furthermore, the results show that contrary to earlier beliefs, long-range correlation coefficients appear necessary to build an effective hyperspectral classifier, and that the high correlations between neighboring bands seem to allow differing sparsity configurations of the covariance matrix to obtain similar classification results.

Spatially Adaptive Classification of Land Cover With Remote Sensing Data

“…It is impractical and too demanding, however, to model hyperspectral data directly by cokriging [28], since cokriging requires solving (n + 1) · d linear equations for n data points with d dimensions, and the system becomes sensitive to noise due to the greatly increased number of parameters. There is also a broad literature on estimating the covariance matrix when faced with inadequate amounts of training data [47], [48] that can be applied if one desires to learn full covariance matrices. Estimating nonstationary covariance functions using local estimation methods as in [27] could be also considered, but it would significantly increase the complexity of the proposed framework.…”

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“…Another class of approaches directly produce a regularized estimate of the precision matrix from training samples. Examples in this class include methods based on shrinkage [21]- [23], factor models-based methods [1], [24], regression analysis-based column-by-column methods [25]- [27], and penalized likelihood [28], [29].…”

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“…While shrinkage is the most common regularization scheme, sparse 15,16 and sparse transform [17][18][19][20] methods have also been proposed. To deal with the non-Gaussian nature of most data, both robust 21 and anti-robust 22 estimators have been proposed.…”