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It has been proved that the determination of independent components (ICs) in the independent component analysis (ICA) can be attributed to calculating the eigenpairs of high-order statistical tensors of the data. However, previous works can only obtain approximate solutions, which may affect the accuracy of the ICs. In addition, the number of ICs would need to be set manually. Recently, an algorithm based on semidefinite programming (SDP) has been proposed, which utilizes the first-order gradient information of the Lagrangian function and can obtain all the accurate real eigenpairs. In this article, for the first time, we introduce this into the ICA field, which tends to further improve the accuracy of the ICs. Note that the number of eigenpairs of symmetric tensors is usually larger than the number of ICs, indicating that the results directly obtained by SDP are redundant. Thus, in practice, it is necessary to introduce second-order derivative information to identify local extremum solutions. Therefore, originating from the SDP method, we present a new modified version, called modified SDP (MSDP), which incorporates the concept of the projected Hessian matrix into SDP and, thus, can intellectually exclude redundant ICs and select true ICs. Some cases that have been tested in the experiments demonstrate its effectiveness. Experiments on the image/sound blind separation and real multi/hyperspectral image also show its superiority in improving the accuracy of ICs and automatically determining the number of ICs. In addition, the results on hyperspectral simulation and real data also demonstrate that MSDP is also capable of dealing with cases, where the number of features is less than the number of ICs.

Principal skewness analysis (PSA) has been introduced for feature extraction in hyperspectral imagery. As a thirdorder generalization of principal component analysis (PCA), its solution of searching for the local maximum skewness direction is transformed into the problem of calculating the eigenpairs (the eigenvalues and the corresponding eigenvectors) of a coskewness tensor. By combining a fixed-point method with an orthogonal constraint, the new eigenpairs are prevented from converging to the same previously determined maxima. However, in general, the eigenvectors of the supersymmetric tensor are not inherently orthogonal, which implies that the results obtained by the search strategy used in PSA may unavoidably deviate from the actual eigenpairs. In this paper, we propose a new nonorthogonal search strategy to so lve this problem and the new algorithm is named nonorthogonal principal skewness analysis (NPSA). The contribution of NPSA lies in the finding that the search space of the eigenvector to be determined can be enlarged by using the orthogonal complement of the Kronecker product of the previous eigenvector with itself, instead of its orthogonal complement space. We also give a detailed theoretical proof on why we can obtain the more accurate eigenpairs through the new search strategy by comparison with PSA. In addition, after some algebraic derivations, the complexity of the presented algorithm is also greatly reduced. Experiments with both simulated data and real multi/hyperspectral imagery demonstrate its validity in feature extraction.

The linear mixture model (LMM) plays a crucial role in the spectral unmixing of hyperspectral data. Under the assumption of LMM, the solution with the minimum reconstruction error is considered to be the ideal endmember. However, for practical hyperspectral data sets, endmembers that enclose all the pixels are physically meaningless due to the effect of noise. Therefore, in many cases, it is not sufficient to consider only the reconstruction error, some constraints (for instance, volume constraint) need to be added to the endmembers. The two terms can be considered as serving two forces: minimizing the reconstruction error forces the endmembers to move outward and thus enlarges the volume of the simplex while the endmember constraint acts in the opposite direction by driving the endmembers to move inward so as to constrain the volume to be smaller. Many existing methods obtain their solution just by balancing the two contradictory forces. The solution acquired in this way can not only minimize the reconstruction error but also be physically meaningful. Interestingly, we find, in this paper, that the two forces are not completely contradictory with each other, and the reconstruction error can be further reduced without changing the volume of the simplex. And more interestingly, our method can further optimize the solution provided by all the endmember extraction methods (both endmember selection methods and endmember generation methods). After optimization, the final endmembers outperform the initial solution in terms of reconstruction error as well as accuracy. The experiments on simulated and real hyperspectral data verify the validation of our method.

Few band selection methods are specially designed for small target detection. It is well known that the information of small targets is most likely contained in non-Gaussian bands, where small targets are more easily separated from the background. On the other hand, correlation of band set also plays an important role in the small target detection. When the selected bands are highly correlated, it will be unbeneficial for the subsequent detection. However, the existing non-Gaussianity-based band selection methods have not taken the correlation of bands into account, which generally result in high correlation of obtained bands. In this paper, combining the third-order (third-order tensor) and second-order (correlation) statistics of bands, we define a new concept, named joint skewness, for multivariate data. Moreover, we also propose an easy-to-implement approach to estimate this index based on high-order singular value decomposition (HOSVD). Based on the definition of joint skewness, we present an unsupervised band selection for small target detection for hyperspectral data, named joint skewness band selection (JSBS). The evaluation results demonstrate that the bands selected by JSBS are very effective in terms of small target detection.

Recently, high-order statistics have received more and more interest in the field of hyperspectral anomaly detection. However, most of the existing high-order statistics based anomaly detection methods require stepwise iterations since they are the direct applications of blind source separation. Moreover, these methods usually produce multiple detection maps rather than a single anomaly distribution image. In this study, we exploit the concept of coskewness tensor and propose a new anomaly detection method, which is called COSD (coskewness detector). COSD does not need iteration and can produce single detection map. The experiments based on both simulated and real hyperspectral data sets verify the effectiveness of our algorithm.