Hyperspectral unmixing (HU), an essential procedure for various environmental applications, has garnered significant attention within remote sensing communities. Among different groups of HU methods, nonnegative matrix factorization (NMF)-based ones have gained widespread popularity due to their high capability of simultaneously extracting endmembers and their corresponding abundances. However, converting a 3-D hyperspectral data cube into a 2-D matrix format leads to the loss of spatial and potential correlation information. Consequently, in recent years, nonnegative tensor factorization (NTF) methods, which preserve the 3-D nature of hyperspectral data cube, have been extensively embraced by numerous researchers. Nevertheless, incorporating prior information into NTF-based problems faces limitations owing to the inconsistency of such information, particularly concerning \ell {1} norm sparsity and the abundance sum-to-one constraint (ASC). To address this limitation, our study introduces a novel general regularization term. This term leverages sparsity and ASC simultaneously, integrating it into a matrix-tensor factorization framework. Our proposed method, named a matrix-tensor-based HU method with general \ell {q} norm regularization (MTUHL{q}), is established on the block term decomposition (BTD) paradigm, which ensures physical interpretability and simple implementation. To investigate the performance of the proposed MTUHL{q}, a series of experiments on both synthetic and real hyperspectral datasets were conducted. The results of the implemented experiments indicated that the proposed method outperformed other state-of-the-art HU methods.

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The weighted total least squares (WTLS) has been widely used in many geodetic problems to solve the error-in-variable (EIV) models in which both the observation vector and the design matrix contain random errors. This method is widely applied in its univariate form, where the observations and unknown coefficients appear in vector forms. However, in some geodetic problems, data sets appear in more than one dimension, and the vector representation of the univariate model may not be suitable to efficiently solve the problem. The observation and unknown parameter vectors can then be replaced with their counterparts in matrix representations in a multivariate model. In this paper, we propose a simple, fast, and flexible procedure for solving the multivariate WTLS (MWTLS) problem using the standard least squares theory. The method has the capability of applying to large-size and high-dimensional data sets. Our numerical experiments on both simulated and real datasets demonstrate the high performance of the proposed method for solving multivariate WTLS problems. In terms of computational complexity, our method outperforms the existing state-of-the-art methods, both numerically and analytically.

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Due to the existence of a large number of highly correlated parameters in the rational function model (RFM), known as rational polynomial coefficients (RPCs), RFM suffers from both ill-posedness and overparameterization problems. To tackle these problems, recently, ℓ₁-regularized least-squares, in which ℓ₂-norm of residuals and ℓ₁-norm of RPCs are minimized, has been introduced. Nonetheless, RFM still suffers from these two phenomena, especially in the presence of a limited number of ground control points (GCPs). This study proposes a new parameter-free linear framework for RFM optimization. In this framework, called linearly programed ℓ₁-regularized RFM framework (LPRFM), the ℓ₁-norms of RPCs and residuals are simultaneously minimized through a linear objective function. To solve LPRFM as a linear optimization problem, the commonly used dual-simplex method is applied. LPRFM can be implemented for both RPC estimation and correction. Our experimental results indicated the superiority of the LPRFM against well-known competing methods in both estimation and correction of RPCs. The results showed that, in the estimation of RPCs, the proposed framework has led to accurate and robust results with only five GCPs. Moreover, accurate coordinates were obtained after correcting vendor-provided RPCs with only one GCP.@en