A new extended linear mixing model to address spectral variability

Abstract

Spectral variability is a phenomenon due, to a grand extend, to variations in the illumination and atmospheric conditions within a hyperspectral image, causing the spectral signature of a material to vary within a image. Data spectral fluctuation due to spectral variability compromises the linear mixing model (LMM) sum-to-one constraint, and is an important source of error in hyperspectral image analysis. Recently, spectral variability has raised more attention and some techniques have been proposed to address this issue, i.e. spectral bundles. Here, we propose the definition of an extended LMM (ELMM) to model spectral variability and we show that the use of spectral bundles models the ELMM implicitly. We also show that the constrained least squares (CLS) is an explicit modelling of the ELMM when the spectral variability is due to scaling effects. We give experimental validation that spectral bundles (and sparsity) and CLS are complementary techniques addressing spectral variability. We finally discuss on future research avenues to fully exploit the proposed ELMM.

Estimating the Number of Endmembers to Use in Spectral Unmixing of Hyperspectral Data with Collaborative Sparsity

“…For example, significant SV for a material with a small abundance will be very hard to recover. We mixed the data using the (full) ELMM of [17,8] and finally added white Gaussian noise, such that the SNR was 25 dB. This resulted in a 40 × 40 × 300 hyperspectral image.…”

“…In the usual linear SU setting, the abundances are estimated by nonnegative linear least squares, with the additional abundance sum to one constraint (ASC). However, here, following [17], we change the mixing model to incorporate SV into the unmixing at a negligible cost. We consider the following mixing model:…”

“…locally changing illumination conditions in the image, due to the topography of the imaged scene and to the photometric properties of the materials. This model is a simplified version of the Extended Linear Mixing Model (ELMM) [17,8] (which considers distinct scaling factors for each material). With this model, S 0 is then a reference endmember matrix, and we can define local endmember matrices in each pixel by computing S k ψ k S 0 .…”

“…With this model, S 0 is then a reference endmember matrix, and we can define local endmember matrices in each pixel by computing S k ψ k S 0 . If this model holds, [17] shows that the quantity estimated by nonnegative least squares in each pixel and for each material actually incorporates SV information, via the product φ kp ψ k a kp . The model reduces to the LMM when all scaling factors are equal to 1.…”

A perturbed linear mixing model accounting for spectral variability

“…In [3], the variability is assumed to only result from scaling factors. Conversely, in this paper, inspired by a model designed in [4], each endmember is represented by a "pure" spectral signature corrupted by an additive perturbation accounting for spectral variability.…”