Archetypal Analysis for Sparse Representation-Based Hyperspectral Sub-pixel Quantification

Abstract

The estimation of land cover fractions from remote sensing images is a frequently used indicator of the environmental quality. This paper focuses on the quantification of land cover fractions in an urban area of Berlin, Germany, using simulated hyperspectral <small>EnMAP</small> data with a spatial resolution of 30 m × 30 m. We use constrained sparse representation, where each pixel with unknown surface characteristics is expressed by a weighted linear combination of elementary spectra with known land cover class. We automatically determine the elementary spectra from image reference data using archetypal analysis by simplex volume maximization, and combine it with reversible jump Markov chain Monte Carlo method. In our experiments, the estimation of the automatically derived elementary spectra is compared to the estimation obtained by a manually designed spectral library by means of reconstruction error, mean absolute error of the fraction estimates, sum of fractions, R2, and the number of used elementary spectra. The experiments show that a collection of archetypes can be an adequate and efficient alternative to the manually designed spectral library with respect to the mentioned criteria.

Parameterized Nonlinear Least Squares for Unsupervised Nonlinear Spectral Unmixing

“…where m p , m p denote the pth endmember in M, M respectively. The root mean square error (RMSE) is also included [32]. It evaluates the errors between all the entries of the true abundance matrix and the estimated abundance matrix and which is defined as follows:…”

Hierarchical Sparse Nonnegative Matrix Factorization for Hyperspectral Unmixing with Spectral Variability

“…If only ANC and ASC are considered, this model is equivalent to the robust constrained matrix factorization [36]. When the sum-to-one constraint is additionally imposed for each column ofB, it yields the conventional AA [31][32][33][34][35][36]. This shows that the proposed model generalizes these two families of method by allowing for pixel-wise bundling coefficients that lead to a pixel-wise description of endmember variability.…”

“…Such a formulation has been, for instance, advocated in the unsupervised unmixing method in [54]. It is also intimately related to AA [29], already advocated as a convenient way to implicitly define the endmember signatures as combinations of the measured pixel spectra [31][32][33][34][35][36]. Satisfying the requirements stated in Section 2.2.3, it has the great advantage of jointly performing bundle extraction, bundle clustering, and unmixing within a unifying framework.…”

“…More precisely, the conventional formulation of AA assumes that observations are convex combinations of so-called archetypes, themselves defined as convex combinations of the observations [29]. AA and some variants have been already instanced in the specific context of hyperspectral image analysis [30,31]. In particular, when conducting unsupervised unmixing, the archetypes have shown to provide physically meaningful estimates of the endmember spectral signatures [32][33][34][35][36].…”

Archetypal analysis as an imputation method and multivariate data augmentation

“…There are applications of AA in various fields of knowledge such as health applications (Reeve et al, 2017) and environmental quality assessment (Drees et al, 2018). Among the several known applications of the archetypes, since they are able to rewrite the sample elements with a minimal error, it was proposed to use them to augment the sample data , generating elements not observed in the original sample.…”