Shahram Hosseini

Linear–Quadratic Mixing Model for Reflectances in Urban Environments

In the field of remote sensing, the unmixing of hyperspectral images is usually based on the use of a mixing model. Most existing spectral unmixing methods, used in the reflective range (0.4-2.5 μm), rely on a linear model of endmember reflectances. Nevertheless, such a model supposes the pixels at the ground level to be uniformly irradiated and the scene to be flat. When considering a 3-D landscape, such a model is no longer valid as irradiated and shadowed areas are present, as well as radiative interactions between facing surfaces. This paper introduces a new mixing model adapted to urban environments and which aims to overcome these limitations. This model is derived from physical equations based on radiative transfer theory, and its analytic expression is linear-quadratic. Similar models have already been used in the literature for unmixing purposes but without being justified by physical analysis. Our proposed model is validated using a synthetic but realistic European 3-D urban scene. Then, simplifications are introduced, based on a study of the different radiative components contributing to the signal in a way to make the model easy to use for spectral unmixing. This paper also shows that the quadratic term cannot be neglected in many cases in urban environments since it can, e.g., range from 15% to 20% of the reflectances in canyons.

Recurrent networks for separating extractable-target nonlinear mixtures. Part II. Blind configurations

While most reported blind source separation methods concern linear mixtures, we here address the nonlinear case. In the first part of this paper, we introduced a general class of nonlinear mixtures which can be inverted using recurrent networks. That part was focused on separating structures themselves and therefore on the non-blind configuration, whereas the current paper addresses the estimation of the parameters of this large class of structures in a blind context. We propose a maximum likelihood approach to this end. The main advantage of this approach is that it exploits the knowledge of the parametric model of mixing transformation in the separation procedure while its implementation does not require the knowledge of the explicit inverse of the model because the separating structure can be designed using a recurrent network. In particular, we illustrate in detail the proposed approach for a linear-quadratic mixture by using an extended recurrent network with self-feedback parameters which guarantee its local stability. Simulation results show the very good performance of the proposed algorithm.

Blind spatial unmixing of multispectral images: New methods combining sparse component analysis, clustering and non-negativity constraints

Remote sensing has become an unavoidable tool for better managing our environment, generally by realizing maps of land cover using classification techniques. Traditional classification techniques assign only one class (e.g., water, soil, grass) to each pixel of remote sensing images. However, the area covered by one pixel contains more than one surface component and results in the mixture of these surface components. In such situations, classical classification is not acceptable for many major applications, such as environmental monitoring, agriculture, mineral exploration and mining, etc. Most methods proposed for treating this problem have been developed for hyperspectral images. On the contrary, there are very few automatic techniques suited to multispectral images. In this paper, we propose new unsupervised spatial methods (called 2D-Corr-NLS and 2D-Corr-NMF) in order to unmix each pixel of a multispectral image for better recognizing the surface components constituting the observed scene. These methods are related to the blind source separation (BSS) problem, and are based on sparse component analysis (SCA), clustering and non-negativity constraints. Our approach consists in first identifying the mixing matrix involved in this BSS problem, by using the first stage of a spatial correlation-based SCA method with very limited source sparsity constraints, combined with clustering. Non-negative least squares (NLS) or non-negative matrix factorization (NMF) methods are then used to extract spatial sources. An important advantage of our proposed methods is their applicability to the possibly globally underdetermined, but locally (over)determined BSS model in multispectral remote sensing images. Experiments based on realistic synthetic mixtures and real multispectral images collected by the Landsat ETM+ and the Formosat-2 sensors are performed to evaluate the performance of the proposed approach. We also show that our methods significantly outperform the sequential maximum angle convex cone (SMACC) method.

Blind separation of piecewise stationary non-Gaussian sources

We address independent component analysis (ICA) of piecewise stationary and non-Gaussian signals and propose a novel ICA algorithm called Block EFICA that is based on this generalized model of signals. The method is a further extension of the popular non-Gaussianity-based FastICA algorithm and of its recently optimized variant called EFICA. In contrast to these methods, Block EFICA is developed to effectively exploit varying distribution of signals, thus, also their varying variance in time (nonstationarity) or, more precisely, in time-intervals (piecewise stationarity). In theory, the accuracy of the method asymptotically approaches Cramer-Rao lower bound (CRLB) under common assumptions when variance of the signals is constant. On the other hand, the performance is practically close to the CRLB even when variance of the signals is changing. This is demonstrated by comparing our algorithm with various methods that are asymptotically efficient within ICA models based either on the non-Gaussianity or the nonstationarity. The benefit of our algorithm is demonstrated by examples with real-world audio signals.

Blind separation of linear instantaneous mixtures of non-stationary signals in the frequency domain

Blind source separation (BSS) methods aim at restoring source signals from their mixtures. For linear instantaneous mixtures of stationary random sources, a natural and widely used approach consists in using some statistics associated to the temporal representation of the signals. On the contrary, we here consider non-stationary real sources and we show that they have interesting frequency-domain properties which motivate the introduction of two new frequency-domain BSS methods. The first method works by diagonalizing a zero-lag, second-order statistics matrix, created using both covariance and pseudo-covariance matrices of Fourier transforms of real-valued observations. In practice, this method is specially suitable for separating cyclo-stationary sources. The second method is particularly important because it allows the existing time-domain algorithms developed for stationary, temporally correlated sources (like AMUSE or SOBI) to be extended to non-stationary, temporally uncorrelated sources just by mapping the mixtures into the frequency domain. Both methods set no constraint on the piecewise stationarity of the sources, unlike most previously reported BSS methods exploiting source non-stationarity. The experimental results using artificial and real-world sources confirm the good performance of the proposed methods for non-stationary sources.

Physical modelling and non-linear unmixing method for urban hyperspectral images

Hyperspectral unmixing consists in decomposing pixel spectra into constituent spectra corresponding to pure materials (endmembers). Most existing methods are based on linear mixtures of the reflectance endmembers, which is only valid when the surface is flat and homogeneous. In the case of urban area sensing, due to the 3D structure of the buildings, such assumptions are no more valid. In this paper, we first derive the mixing model faced in urban spectral unmixing. Starting from physical equations based on radiative transfer theory, we deduce a linear-quadratic model. Then, a suitable unmixing method is proposed to extract the spectra of the materials composing the pixels. Our study led us to an original use of NMF (Non-negative Matrix Factorization) methods. The proposed algorithm is tested with synthetic mixtures of real endmember spectra, corresponding to cases with simple geometrical shapes that can be encountered in urban areas. First results are encouraging and show good agreement with respect to spectral signature.

Improvement of remote sensing multispectral image classification by using Independent Component Analysis

This paper deals with the application of Independent Component Analysis (ICA) as a solution to Blind Source Separation (BSS), in order to pre-process remote sensing multispectral images before we classify them. We analyze the structure of the considered data, and especially show that each recorded image corresponding to a spectral band may be seen as an observation consisting of a mixture (linear combination) of source images. The latter images correspond to the abundances of the pure elements (endmembers) in the pixels. Using BSS methods, one can hope to reduce the mixing effect in these observations, which then allows better recognition of the classes constituting the observed scene. Based on this approach, we create new images (i.e. at least partly separated images) by using ICA, starting from HRV SPOT images. These images are then used as inputs of a supervised classifier integrating textural information. The separated image classification results show a clear improvement compared to classification of initial images. This show the contribution of ICA as an attractive pre-processing for classification of multispectral remote sensing imagery.

Blind Separation of Parametric Nonlinear Mixtures of Possibly Autocorrelated and Non-Stationary Sources

In this paper, we present a new method, formulated in a maximum-likelihood framework, for blindly separating nonlinear mixtures of statistically independent signals. Our method exploits, on the one hand, the knowledge of the parametric model of the mixing transformation (with unknown parameter values), and on the other hand, the possible structure of source signals, i.e., their autocorrelation and/or nonstationarity. One of the main advantages of the proposed method is that it can be implemented even if the analytical expression of the inverse model is unknown. The method is first addressed in a general configuration, then detailed for two special cases, i.e., a simple bijective “toy” model and a linear-quadratic model. The study of the toy model is interesting because of its simplicity and its global bijectivity, which allows us to focus our efforts on parameter estimation. The linear-quadratic model is chosen due to its capacity to describe real-world mixing phenomena. Simulation results, using the toy model and using a subclass of the linear-quadratic model (i.e., the bilinear model), show that taking into account the nonlinearity of the mixing transformations and the structure of signals considerably improves separation performance.

A new unsupervised method for hyperspectral image unmixing using a linear-quadratic model

This paper deals with pixel unmixing in hyperspectral remote sensing images. The surface covered by one pixel of these images may contain more than one material component. Thus, the spectrum obtained from such a pixel may result from different contributions of several materials. Using a blind source separation (BSS) method, we aim at decomposing each mixed pixel spectrum into several pure material spectra and their relative contributions. Due to the non-flat landscape, for example in urban areas, the resulting spectrum is obtained from reflections between the various surfaces. As a result, the classical linear mixing model, generally used in BSS methods, is no longer valid. In this work, we propose a new non-linear unmixing method for hyperspectral remote sensing images using a linear-quadratic model. Our approach starts by estimating the number of pure materials called endmembers. Then, supposing the existence of at least two pure pixels per material in the image, it achieves source separation by going through the following stages: endmember spectra estimation, classification into clusters, and estimation of abundance and reflection fractions. Experimental results on artificial mixtures of real spectra confirm the effectiveness of the proposed method.

Hyperspectral image unmixing by non-negative matrix factorization initialized with modified independent component analysis

In this paper, we propose an unsupervised unmixing approach for hyperspectral images, consisting of a modified version of ICA, followed by NMF. In the ideal case of a hyperspectral image combining (C-1) statistically independent source images, and a Cth image which is dependent on them due to the sum-to-one constraint, our modified ICA first estimates these (C-1) sources and associated mixing coefficients, and then derives the remaining source and coefficients, while also removing the BSS scale indeterminacy. In real conditions, the above (C-1) sources may be somewhat dependent. Our modified ICA method then only yields approximate data. These are then used as the initial values of an NMF method, which refines them. Our tests show that this joint modifICA-NMF approach significantly outperforms the considered classical methods.