Linear spectral mixture analysis (LSMA) is a widely used technique in remote sensing to estimate abundance fractions of materials present in an image pixel. In order for an LSMA-based estimator to produce accurate amounts of material abundance, it generally requires two constraints imposed on the linear mixture model used in LSMA, which are the abundance sum-to-one constraint and the abundance nonnegativity constraint. The first constraint requires the sum of the abundance fractions of materials present in an image pixel to be one and the second imposes a constraint that these abundance fractions be nonnegative. While the first constraint is easy to deal with, the second constraint is difficult to implement since it results in a set of inequalities and can only be solved by numerical methods. Consequently, most LSMA-based methods are unconstrained and produce solutions that do not necessarily reflect the true abundance fractions of materials. In this case, they can only be used for the purposes of material detection, discrimination, and classification, but not for material quantification. The authors present a fully constrained least squares (FCLS) linear spectral mixture analysis method for material quantification. Since no closed form can be derived for this method, an efficient algorithm is developed to yield optimal solutions. In order to further apply the designed algorithm to unknown image scenes, an unsupervised least squares error (LSE)-based method is also proposed to extend the FCLS method in an unsupervised manner.

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Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery

With very high spectral resolution, hyperspectral sensors can now uncover many unknown signal sources which cannot be identified by visual inspection or a priori. In order to account for such unknown signal sources, we introduce a new definition, referred to as virtual dimensionality (VD) in this paper. It is defined as the minimum number of spectrally distinct signal sources that characterize the hyperspectral data from the perspective view of target detection and classification. It is different from the commonly used intrinsic dimensionality (ID) in the sense that the signal sources are determined by the proposed VD based only on their distinct spectral properties. These signal sources may include unknown interfering sources, which cannot be identified by prior knowledge. With this new definition, three Neyman-Pearson detection theory-based thresholding methods are developed to determine the VD of hyperspectral imagery, where eigenvalues are used to measure signal energies in a detection model. In order to evaluate the performance of the proposed methods, two information criteria, an information criterion (AIC) and minimum description length (MDL), and the factor analysis-based method proposed by Malinowski, are considered for comparative analysis. As demonstrated in computer simulations, all the methods and criteria studied in this paper may work effectively when noise is independent identically distributed. This is, unfortunately, not true when some of them are applied to real image data. Experiments show that all the three eigenthresholding based methods (i.e., the Harsanyi-Farrand-Chang (HFC), the noise-whitened HFC (NWHFC), and the noise subspace projection (NSP) methods) produce more reliable estimates of VD compared to the AIC, MDL, and Malinowski's empirical indicator function, which generally overestimate VD significantly. In summary, three contributions are made in this paper, 1) an introduction of the new definition of VD, 2) three Neyman-Pearson detection theory-based thresholding methods, HFC, NWHFC, and NSP derived for VD estimation, and 3) experiments that show the AIC and MDL commonly used in passive array processing and the second-order statistic-based Malinowski's method are not effective measures in VD estimation.

Estimation of number of spectrally distinct signal sources in hyperspectral imagery

Endmember extraction is a process to identify the hidden pure source signals from the mixture. In the past decade, numerous algorithms have been proposed to perform this estimation. One commonly used assumption is the presence of pure pixels in the given image scene, which are detected to serve as endmembers. When such pixels are absent, the image is referred to as the highly mixed data, for which these algorithms at best can only return certain data points that are close to the real endmembers. To overcome this problem, we present a novel method without the pure-pixel assumption, referred to as the minimum volume constrained nonnegative matrix factorization (MVC-NMF), for unsupervised endmember extraction from highly mixed image data. Two important facts are exploited: First, the spectral data are nonnegative; second, the simplex volume determined by the endmembers is the minimum among all possible simplexes that circumscribe the data scatter space. The proposed method takes advantage of the fast convergence of NMF schemes, and at the same time eliminates the pure-pixel assumption. The experimental results based on a set of synthetic mixtures and a real image scene demonstrate that the proposed method outperforms several other advanced endmember detection approaches

Endmember Extraction From Highly Mixed Data Using Minimum Volume Constrained Nonnegative Matrix Factorization

Spectral unmixing aims at estimating the fractional abundances of pure spectral signatures (also called endmembers) in each mixed pixel collected by a remote sensing hyperspectral imaging instrument. In recent work, the linear spectral unmixing problem has been approached in semisupervised fashion as a sparse regression one, under the assumption that the observed image signatures can be expressed as linear combinations of pure spectra, known a priori and available in a library. It happens, however, that sparse unmixing focuses on analyzing the hyperspectral data without incorporating spatial information. In this paper, we include the total variation (TV) regularization to the classical sparse regression formulation, thus exploiting the spatial-contextual information present in the hyperspectral images and developing a new algorithm called sparse unmixing via variable splitting augmented Lagrangian and TV. Our experimental results, conducted with both simulated and real hyperspectral data sets, indicate the potential of including spatial information (through the TV term) on sparse unmixing formulations for improved characterization of mixed pixels in hyperspectral imagery.

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Total Variation Spatial Regularization for Sparse Hyperspectral Unmixing

Endmember variability in Spectral Mixture Analysis: A review

Target detection in remotely sensed images can be conducted spatially, spectrally or both. The difficulty of detecting targets in remotely sensed images with spatial image analysis arises from the fact that the ground sampling distance is generally larger than the size of targets of interest in which case targets are embedded in a single pixel and cannot be detected spatially. Under this circumstance target detection must be carried out at subpixel level and spectral analysis offers a valuable alternative. In this paper, the problem of subpixel spectral detection of targets in remote sensing images is considered, where two constrained target detection approaches are studied and compared. One is a target abundance-constrained approach, referred to as nonnegatively constrained least squares (NCLS) method. It is a constrained least squares spectral mixture analysis method which implements a nonnegativity constraint on the abundance fractions of targets of interest. Another is a target signature-constrained approach, called constrained energy minimization (CEM) method. It constrains the desired target signature with a specific gain while minimizing effects caused by other unknown signatures. A quantitative study is conducted to analyze the advantages and disadvantages of both methods. Some suggestions are further proposed to mitigate their disadvantages.

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D.C. Heinz

Papers

Constrained subpixel target detection for remotely sensed imagery