Chapter 2.6 - Nonlinear spectral unmixing

Abstract

Abstract In hyperspectral imaging, no matter the spatial resolution, the spectral signatures collected in natural environments are invariably a mixture of the signatures of the various materials found within the spatial extent of the ground instantaneous field view of the imaging instrument. Spectral unmixing consists in the identification of the spectrally pure signatures of the materials present in the scene (called endmembers in hyperspectral imaging terminology) and the estimation of the fractional abundances for each pixel associated with the endmembers or, in other words, the contribution of each endmember on each mixed pixel. Thus a crucial issue is the definition of endmember, which may depend on the target application or, at least, the considered problem. To perform the spectral unmixing process, one of the simplest and most widely used approaches to characterize mixed pixels in hyperspectral imagery is the linear mixture model (LMM). In the LMM it is possible to represent each mixed pixel as a linear combination of the endmembers times the fractional abundances associated with it. Let us define a hyperspectral image with l bands as a collection of n spectral signatures, yi ∈ R l for i ∈ 1 … n. These spectral signatures are mixtures of p pure spectral signatures or endmembers mj ∈ R l for j ∈ 1 … p with abundances s i j . The LMM can be represented in vector form as yi = Msi, where M ≡ [m1, m2, …, mp] and s i ∈ R p ≡ [ s i 1 , s i 2 , … , s i p ] T . It may be also represented in matrix form as Y = MST with Y ≡ [y1, y2, …, yn] and S ≡ [ s 1 , s 2 , … , s n ] T . Typically, two constraints on the abundance fractions are associated with this model; these are the so-called abundance nonnegativity constraint (ANC) and the abundance sum-to-1 constraint (ASC). The ANC enforces that all the abundance fractions are nonnegative s i j ≥ 0 , ∀ i ∈ 1 , … , n and ∀j ∈ 1, …, p. The ASC enforces that the abundances of a given pixel sum to 1, that is, ∑ j = 1 p s i j = 1 , ∀ i ∈ 1 , … , n . A large variety of unmixing methods are based on the LMM.