Variability of the endmembers in spectral unmixing: recent advances

International audienceEndmember 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

Residual Networks as Flows of Diffeomorphisms

International audienceThis paper addresses the understanding and characterization of residual networks (ResNet), which are among the state-of-the-art deep learning architectures for a variety of supervised learning problems. We focus on the mapping component of ResNets, which map the embedding space towards a new unknown space where the prediction or classification can be stated according to linear criteria. We show that this mapping component can be regarded as the numerical implementation of continuous flows of diffeomorphisms governed by ordinary differential equations. Especially, ResNets with shared weights are fully characterized as numerical approximation of exponential diffeomorphic operators. We stress both theoretically and numerically the relevance of the enforcement of diffeormorphic properties and the importance of numerical issues to make consistent the continuous formulation and the discretized ResNet implementation. We further discuss the resulting theoretical and computational insights on ResNet architectures

From local to global unmixing of hyperspectral images to reveal spectral variability

International audienceThe linear mixing model is widely assumed when unmixing hyperspectral images, but it cannot account for endmembers spectral variability. Thus, several workarounds have arisen in the hyperspectral unmixing literature, such as the extended linear mixing model (ELMM), which authorizes endmembers to vary pixelwise according to scaling factors, or local spectral unmixing (LSU) where the unmixing process is conducted locally within the image. In the latter case however, results are difficult to interpret at the whole image scale. In this work, we propose to analyze the local results of LSU within the ELMM framework, and show that it not only allows to reconstruct global endmembers and fractional abundances from the local ones, but it also gives access to the scaling factors advocated by the ELMM. Results obtained on a real hyperspectral image confirm the soundness of the proposed methodology

Semi-Automatic Classification of Cementitious Materials using Scanning Electron Microscope Images

International audienceSegmentation and classification are prolific research topics in the image processing community, which have been more and more used in the context of analysis of cementitious materials, on images acquired with Scanning Electron Microscopes (SEM). Indeed, there is a need to be able to detect and to quantify the materials present in a cement paste in order to follow the chemical reactions occurring in the material even days after the solidification. In this paper, we propose a new approach for segmentation and classification of cementitious materials based on the denoising of the data with the Block Matching 3D (BM3D) algorithm, Binary Partition Tree (BPT) segmentation, Support Vector Machines (SVM) classification, and the interactivity with the user. The BPT provides a hierarchical representation of the spatial regions of the data, allowing a segmentation to be selected among the admissible partitions of the image. SVMs are used to obtain a classification map of the image. This approach combines state-of-the-art image processing tools with the interactivity with the user to allow a better segmentation to be performed, or to help the classifier discriminate the classes better. We show that the proposed approach outperforms a previous method on synthetic data and several real datasets coming from cement samples, both qualitatively with visual examination and quantitatively with the comparison of experimental results with theoretical ones

Blind hyperspectral unmixing using an Extended Linear Mixing Model to address spectral variability

International audienceThe Linear Mixing Model is often used to perform Hyperspec-tral 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

Spatial Graph Signal Interpolation with an Application for Merging BCI Datasets with Various Dimensionalities

BCI Motor Imagery datasets usually are small and have different electrodes setups. When training a Deep Neural Network, one may want to capitalize on all these datasets to increase the amount of data available and hence obtain good generalization results. To this end, we introduce a spatial graph signal interpolation technique, that allows to interpolate efficiently multiple electrodes. We conduct a set of experiments with five BCI Motor Imagery datasets comparing the proposed interpolation with spherical splines interpolation. We believe that this work provides novel ideas on how to leverage graphs to interpolate electrodes and on how to homogenize multiple datasets.Comment: Submitted to the 2023 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2023

Blind hyperspectral unmixing using an Extended Linear Mixing Model to address spectral variability

International audienceSpectral 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 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 Linear Mixing Model. 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 datasets, 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 thanks to the scaling factors estimation

Neural Koopman prior for data assimilation

With the increasing availability of large scale datasets, computational power and tools like automatic differentiation and expressive neural network architectures, sequential data are now often treated in a data-driven way, with a dynamical model trained from the observation data. While neural networks are often seen as uninterpretable black-box architectures, they can still benefit from physical priors on the data and from mathematical knowledge. In this paper, we use a neural network architecture which leverages the long-known Koopman operator theory to embed dynamical systems in latent spaces where their dynamics can be described linearly, enabling a number of appealing features. We introduce methods that enable to train such a model for long-term continuous reconstruction, even in difficult contexts where the data comes in irregularly-sampled time series. The potential for self-supervised learning is also demonstrated, as we show the promising use of trained dynamical models as priors for variational data assimilation techniques, with applications to e.g. time series interpolation and forecasting