Lucas Drumetz

A new extended linear mixing model to address spectral variability

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.

Blind Hyperspectral Unmixing Using an Extended Linear Mixing Model to Address Spectral Variability

Spectral unmixing is one of the main research topics in hyperspectral imaging. It can be formulated as a source separation problem, whose goal is to recover the spectral signatures of the materials present in the observed scene (called endmembers) as well as their relative proportions (called fractional abundances), and this for every pixel in the image. A linear mixture model (LMM) is often used for its simplicity and ease of use, but it implicitly assumes that a single spectrum can be completely representative of a material. However, in many scenarios, this assumption does not hold, since many factors, such as illumination conditions and intrinsic variability of the endmembers, induce modifications on the spectral signatures of the materials. In this paper, we propose an algorithm to unmix hyperspectral data using a recently proposed extended LMM. The proposed approach allows a pixelwise spatially coherent local variation of the endmembers, leading to scaled versions of reference endmembers. We also show that the classic nonnegative least squares, as well as other approaches to tackle spectral variability can be interpreted in the framework of this model. The results of the proposed algorithm on two different synthetic datasets, including one simulating the effect of topography on the measured reflectance through physical modelling, and on two real data sets, show that the proposed technique outperforms other methods aimed at addressing spectral variability, and can provide an accurate estimation of endmember variability along the scene because of the scaling factors estimation.

Hyperspectral Local Intrinsic Dimensionality

The intrinsic dimensionality (ID) of multivariate data is a very important concept in spectral unmixing of hyperspectral images. A good estimation of the ID is crucial for a correct retrieval of the number of endmembers (the spectral signatures of macroscopic materials) in the image, for dimensionality reduction or for subspace learning, among others. Recently, some approaches to perform spectral unmixing and superresolution locally have been proposed, which require a local estimation of the number of endmembers to use. However, the role of ID in local regions of hyperspectral images has not been properly addressed. Some important issues when dealing with small regions of hyperspectral data can seriously affect the performance of conventional hyperspectral ID estimators. We show that three factors mainly affect local ID estimation: the number of pixels in the local regions, which has to be high enough for the estimations to be relevant, the number of hyperspectral bands which complicates the estimations if the ambient space has a high dimensionality, and the noise, which can be misinterpreted as a signal when its power is important. Here, we review the hyperspectral ID estimators on the literature for local ID estimation, we show how they behave in a local setting on synthetic and real data sets, and we provide some guidelines to make proper use of these estimators in local approaches.

Blind hyperspectral unmixing using an extended linear mixing model to address spectral variability

The Linear Mixing Model is often used to perform Hyperspectral Unmixing because of its simplicity, but it assumes that a single spectral signature can be completely representative of an endmember. However, in many scenarios, this assumption does not hold since many factors such as illumination conditions and intrinsic variability of the endmembers have consequences on the spectral signatures of the materials. In this paper, we propose a simple yet flexible algorithm to unmix hyperspectral data using a recently proposed Extended Linear Mixing Model. This model allows a pixelwise variation of the endmembers, which leads to consider scaled versions of reference endmember spectra. The results on synthetic data show that the proposed technique outperforms other methods aimed at tackling spectral variability, and provides an accurate estimation of endmember variability along the observed scene thanks to the scaling factors estimation, provided the abundance of the corresponding material is sufficient.

Binary partition tree-based local spectral unmixing

The linear mixing model (LMM) is a widely used methodology for the spectral unmixing (SU) of hyperspectral data. In this model, hyperspectral data is formed as a linear combination of spectral signatures corresponding to macroscopically pure materials (endmembers), weighted by their fractional abundances. Some of the drawbacks of the LMM are the presence of multiple mixtures and the spectral variability of the endmembers due to illumination and atmospheric effects. These issues appear as variations of the spectral conditions of the image along its spatial domain. However, these effects are not so severe locally and could be at least mitigated by working in smaller regions of the image. The proposed local SU works over a partition of the image, performing the spectral unmixing locally in each region of the partition. In this work, we first introduce the general local SU methodology, then we propose an implementation of the local SU based on a binary partition tree representation of the hyperspectral image and finally we give an experimental validation of the approach using real data.

Hyperspectral unmixing with material variability using social sparsity

We apply social ℓ-norms for the first time to the problem of hyperspectral unmixing while modeling spectral variability. These norms are built with inter-group penalties which are combined in a global intra-group penalization that can enforce selection of entire endmember bundles; this results in the selection of a few representative materials even in the presence of large endmembers bundles capturing each materials variability. We demonstrate improvements quantitatively on synthetic data and qualitatively on real data for three cases of social norms: group, elitist, and a fractional social norm, respectively. We find that the greatest improvements arise from using either the group or fractional flavor.

Variability of the endmembers in spectral unmixing: Recent advances

Endmember variability has been identified as one of the main limitations of the usual Linear Mixing Model, conventionally used to perform spectral unmixing of hyperspectral data. The topic is currently receiving a lot of attention from the community, and many new algorithms have recently been developed to model this variability and take it into account. In this paper, we review state of the art methods dealing with this problem and classify them into three categories: the algorithms based on endmember bundles, the ones based on computational models, and the ones based on parametric physics-based models. We discuss the advantages and drawbacks of each category of methods and list some open problems and current challenges.

Improved Local Spectral Unmixing of hyperspectral data using an algorithmic regularization path for collaborative sparse regression

Local Spectral Unmixing (LSU) methods perform the unmixing of hyperspectral data locally in regions of the image. The endmembers and their abundances in each pixel are extracted region-wise, instead of globally to mitigate spectral variability effects, which are less severe locally. However, it requires the local estimation of the number of endmembers to use. Algorithms for intrinsic dimensionality (ID) estimation tend to overestimate the local ID, especially in small regions. The ID only provides an upper bound of the application and scale dependent number of endmembers, which leads to extract irrelevant signatures as local endmembers, associated with meaningless local abundances. We propose a method to select in each region the best subset of the locally extracted endmembers. Collaborative sparsity is used to detect spurious endmembers in each region and only keep the most influent ones. We compute an algorithmic regularization path for this problem, giving access to the sequence of successive active sets of endmembers when the regularization parameter is increased. Finally, we select the optimal set in the sense of the Bayesian Information Criterion (BIC), favoring models with a high likelihood, while penalizing those with too many endmembers. Results on real data show the interest of the proposed approach.