Bilinear Mixture Research Articles

Geometrical projection improved multi-objective particle swarm optimization for unsupervised nonlinear hyperspectral unmixing

ABSTRACT In recent years, a series of bilinear mixture models (BMMs) have been well studied, based on which nonlinear unmixing algorithms for hyperspectral images have been proposed. However, the performance of unsupervised nonlinear spectral unmixing can be largely affected by the collinearity and the nonconvex unmixing problem’s inherent complexity. To solve the problems, this paper proposes a geometrical projection improved multi-objective particle swarm optimization (GPMOPSO) method for nonlinear unmixing, which consists of two dynamically updating procedures. First, under the assumption of the BMMs, observed pixels are geometrically projected to their approximate linear mixture components to alleviate the collinearity. Second, the linear mixture components are efficiently decomposed to refined endmembers and abundances in an improved linear unmixing framework using multi-objective particle swarm optimization. Two advanced multi-objective optimization strategies are adopted to balance the influence of an endmember distance constraint and an abundance sparsity constraint on the achieved unsupervised linear unmixing process, respectively. Then, accurate unmixing variables can be fed back to the geometrical projection procedure to produce more reliable linear mixture components which facilitate the execution of the former in turn. Experimental results of both simulated data and real hyperspectral remote sensing data verify the superior nonlinear unmixing performance of the proposed method.

Deep Autoencoders With Multitask Learning for Bilinear Hyperspectral Unmixing

Hyperspectral unmixing is an important problem for remotely sensed data interpretation. It amounts at estimating the spectral signatures of the pure spectral constituents in the scene (endmembers) and their corresponding subpixel fractional abundances. Although the unmixing problem is inherently nonlinear (due to multiple scattering), the nonlinear unmixing of hyperspectral data has been a very challenging problem. This is because nonlinear models require detailed knowledge about the physical interactions between the sunlight scattered by multiple materials. In turn, bilinear mixture models (BMMs) can reach good accuracy with a relatively simple model for scattering. In this article, we develop a new BMM and a corresponding unsupervised unmixing approach which consists of two main steps. In the first step, a deep autoencoder is used to linearly estimate the endmember signatures and their associated abundance fractions. The second step refines the initial (linear) estimates using a bilinear model, in which another deep autoencoder (with a low-rank assumption) is adapted to model second-order scattering interactions. It should be noted that in our developed BMM model, the two deep autoencoders are trained in a mutually interdependent manner under the multitask learning framework, and the relative reconstruction error is used as the stopping criterion. The effectiveness of the proposed method is evaluated using both synthetic and real hyperspectral data sets. Our experimental results indicate that the proposed approach can reasonably estimate the nature of nonlinear interactions in real scenarios. Compared with other state-of-the-art unmixing algorithms, the proposed approach demonstrates very competitive performance.

Multiobjective Endmember Extraction Based on Bilinear Mixture Model

Hyperspectral imagery is always composed of mixed pixels because of the limited spatial resolution of a sensor and the macroscopic/microscopic mixture of distinct substances. The linear mixing model (LMM) is proven to be simple and effective in extensive literature when the macroscopic mixture dominates the mixing process. But when the photons undergo multiple reflections before reaching the sensor, the LMM becomes invalid. In this circumstance, the bilinear mixture model (Bi-LMM), which considers secondary reflections with a bilinear term, is a viable alternative. However, the bilinear term in most existing Bi-LMMs is constructed based on the pre-estimated endmembers, and thus, most Bi-LMMs focus mainly on the abundance estimation. This may lead to inaccurate estimation of endmembers and abundances for a given hyperspectral image. In this article, we propose a multiobjective endmember extraction (Bi-MoEE) method within the bilinear mixture paradigm, which considers each secondary reflection as a virtual endmember. Then, Bi-MoEE selects real and virtual endmembers from an extended spectral library consisting of a standard spectral library and their virtual products. By imposing some intuitive constraints, the solution space is greatly reduced, and the multipoint crossover and restricted bit-flip mutation operators are specially designed. Finally, Bi-MoEE can efficiently obtain a set of tradeoff solutions by minimizing the unmixing residuals and the number of selected endmembers, and automatically determine the optimal solution with multiobjective decision-making techniques. Compared with some advanced endmember extraction methods, the proposed Bi-MoEE does not need to know the number of real endmembers. In addition, the time efficiency of Bi-MoEE is mainly related to the image size and the algorithmic parameters, and has little to do with the size of spectral library, thus facilitating the practical implementation of Bi-MoEE with regard to the oversized spectral library. The experiments on synthetic and real data sets demonstrated the excellent performance of Bi-MoEE.

Unsupervised Nonlinear Hyperspectral Unmixing Based on Bilinear Mixture Models via Geometric Projection and Constrained Nonnegative Matrix Factorization

Bilinear mixture model-based methods have recently shown promising capability in nonlinear spectral unmixing. However, relying on the endmembers extracted in advance, their unmixing accuracies decrease, especially when the data is highly mixed. In this paper, a strategy of geometric projection has been provided and combined with constrained nonnegative matrix factorization for unsupervised nonlinear spectral unmixing. According to the characteristics of bilinear mixture models, a set of facets are determined, each of which represents the partial nonlinearity neglecting one endmember. Then, pixels’ barycentric coordinates with respect to every endmember are calculated in several newly constructed simplices using a distance measure. In this way, pixels can be projected into their approximate linear mixture components, which reduces greatly the impact of collinearity. Different from relevant nonlinear unmixing methods in the literature, this procedure effectively facilitates a more accurate estimation of endmembers and abundances in constrained nonnegative matrix factorization. The updated endmembers are further used to reconstruct the facets and get pixels’ new projections. Finally, endmembers, abundances, and pixels’ projections are updated alternately until a satisfactory result is obtained. The superior performance of the proposed algorithm in nonlinear spectral unmixing has been validated through experiments with both synthetic and real hyperspectral data, where traditional and state-of-the-art algorithms are compared.

Nonlinear Hyperspectral Unmixing Based on Geometric Characteristics of Bilinear Mixture Models

Recently, many nonlinear spectral unmixing algorithms that use various bilinear mixture models (BMMs) have been proposed. However, the high computational complexity and intrinsic collinearity between true endmembers and virtual endmembers considerably decrease these algorithms’ unmixing performances. In this paper, we come up with a novel abundance estimation algorithm based on the BMMs. Motivated by BMMs’ geometric characteristics that are related to collinearity, we conduct a unique nonlinear vertex <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${p}$ </tex-math></inline-formula> to replace all the virtual endmembers. Unlike the virtual endmembers, this vertex <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${p}$ </tex-math></inline-formula> actually works as an additional true endmember that gives affine representations of pixels with other true endmembers. When the pixels’ normalized barycentric coordinates with respect to true endmembers are obtained, they will be directly projected to be their approximate linear mixture components, which removes the collinearity effectively and enables further linear spectral unmixing. After that, based on the analysis of projection bias, two strategies using the projected gradient algorithm and a traditional linear spectral unmixing algorithm, respectively, are provided to correct the bias and estimate more accurate abundances. The experimental results on simulated and real hyperspectral data show that the proposed algorithm performs better compared with both traditional and state-of-the-art spectral unmixing algorithms. Both the unmixing accuracy and speed have been improved.

A second-order blind source separation method for bilinear mixtures

In this paper, we are interested in the problem of Blind Source Separation using a Second-order Statistics (SOS) method in order to separate autocorrelated and mutually independent sources mixed according to a bilinear (BL) model. In this context, we propose a new approach called Bilinear Second-order Blind Source Separation, which is an extension of linear SOS methods, devoted to separate sources present in BL mixtures. These sources, called extended sources, include the actual sources and their products. We first study the statistical properties of the different extended sources, in order to verify the assumption of identifiability when the actual sources are zero-mean and when they are not. Then, we present the different steps performed in order to estimate these actual centred sources and to extract the actual mixing parameters. The obtained results using artificial mixtures of synthetic and real sources confirm the effectiveness of the new proposed approach.

A New Algorithm for Bilinear Spectral Unmixing of Hyperspectral Images Using Particle Swarm Optimization

Spectral unmixing is an important technique for exploiting hyperspectral data. The presence of nonlinear mixing effects poses an important problem when attempting to provide accurate estimates of the abundance fractions of pure spectral components (endmembers) in a scene. This problem complicates the development of algorithms that can address all types of nonlinear mixtures in the scene. In this paper, we develop a new strategy to simultaneously estimate both the endmember signatures and their corresponding abundances using a biswarm particle swarm optimization (BiPSO) bilinear unmixing technique based on Fan's model. Our main motivation in this paper is to explore the potential of the newly proposed bilinear mixture model based on particle swarm optimization (PSO) for nonlinear spectral unmixing purposes. By taking advantage of the learning mechanism provided by PSO, we embed a multiobjective optimization technique into the algorithm to handle the more complex constraints in simplex volume minimization algorithms for spectral unmixing, thus avoiding limitations due to penalty factors. Our experimental results, conducted using both synthetic and real hyperspectral data, demonstrate that the proposed BiPSO algorithm can outperform other traditional spectral unmixing techniques by accounting for nonlinearities in the mixtures present in the scene.

Abundance Estimation for Bilinear Mixture Models via Joint Sparse and Low-Rank Representation

Sparsity-based unmixing algorithms, exploiting the sparseness property of the abundances, have recently been proposed with promising performances. However, these algorithms are developed for the linear mixture model (LMM), which cannot effectively handle the nonlinear effects. In this paper, we extend the current sparse regression methods for the LMM to bilinear mixture models (BMMs), where the BMMs introduce additional bilinear terms in the LMM in order to model second-order photon scattering effects. To solve the abundance estimation problem for the BMMs, we propose to perform a sparsity-based abundance estimation by using two dictionaries: a linear dictionary containing all the pure endmembers and a bilinear dictionary consisting of all the possible second-order endmember interaction components. Then, the abundance values can be estimated from the sparse codes associated with the linear dictionary. Moreover, to exploit the spatial data structure where the adjacent pixels are usually homogeneous and are often mixtures of the same materials, we first employ the joint-sparsity (row-sparsity) model to enforce structured sparsity on the abundance coefficients. However, the joint-sparsity model is often a strict assumption, which might cause some aliasing artifacts for the pixels that lie on the boundaries of different materials. To deal with this problem, the low-rank-representation model, which seeks the lowest rank representation of the data, is further introduced to better capture the spatial data structure. Our simulation results demonstrate that the proposed algorithms provide much enhanced performance compared with state-of-the-art algorithms.