Blind Spectral Unmixing Based on Sparse Nonnegative Matrix Factorization
Nonnegative matrix factorization (NMF) is a widely used method for blind spectral unmixing (SU), which aims at obtaining the endmembers and corresponding fractional abundances, knowing only the collected mixing spectral data. It is noted that the abundance may be sparse (i.e., the endmembers may be with sparse distributions) and sparse NMF tends to lead to a unique result, so it is intuitive and meaningful to constrain NMF with sparseness for solving SU. However, due to the abundance sum-to-one constraint in SU, the traditional sparseness measured by L0/L1-norm is not an effective constraint any more. A novel measure (termed as S-measure) of sparseness using higher order norms of the signal vector is proposed in this paper. It features the physical significance. By using the S-measure constraint (SMC), a gradient-based sparse NMF algorithm (termed as NMF-SMC) is proposed for solving the SU problem, where the learning rate is adaptively selected, and the endmembers and abundances are simultaneously estimated. In the proposed NMF-SMC, there is no pure index assumption and no need to know the exact sparseness degree of the abundance in prior. Yet, it does not require the preprocessing of dimension reduction in which some useful information may be lost. Experiments based on synthetic mixtures and real-world images collected by AVIRIS and HYDICE sensors are performed to evaluate the validity of the proposed method.
Robust Collaborative Nonnegative Matrix Factorization for Hyperspectral Unmixing
Spectral unmixing is an important technique for remotely sensed hyperspectral data exploitation. It amounts to identifying a set of pure spectral signatures, which are called endmembers, and their corresponding fractional, draftrulesabun-dances in each pixel of the hyperspectral image. Over the last years, different algorithms have been developed for each of the three main steps of the spectral unmixing chain: 1) estimation of the number of endmembers in a scene; 2) identification of the spectral signatures of the endmembers; and 3) estimation of the fractional abundance of each endmember in each pixel of the scene. However, few algorithms can perform all the stages involved in the hyperspectral unmixing process. Such algorithms are highly desirable to avoid the propagation of errors within the chain. In this paper, we develop a new algorithm, which is termed robust collaborative nonnegative matrix factorization (R-CoNMF), that can perform the three steps of the hyperspectral unmixing chain. In comparison with other conventional methods, R-CoNMF starts with an overestimated number of endmembers and removes the redundant endmembers by means of collaborative regularization. Our experimental results indicate that the proposed method provides better or competitive performance when compared with other widely used methods.
Matrix-Vector Nonnegative Tensor Factorization for Blind Unmixing of Hyperspectral Imagery
Many spectral unmixing approaches ranging from geometry, algebra to statistics have been proposed, in which nonnegative matrix factorization (NMF)-based ones form an important family. The original NMF-based unmixing algorithm loses the spectral and spatial information between mixed pixels when stacking the spectral responses of the pixels into an observed matrix. Therefore, various constrained NMF methods are developed to impose spectral structure, spatial structure, and spectral-spatial joint structure into NMF to enforce the estimated endmembers and abundances preserve these structures. Compared with matrix format, the third-order tensor is more natural to represent a hyperspectral data cube as a whole, by which the intrinsic structure of hyperspectral imagery can be losslessly retained. Extended from NMF-based methods, a matrix-vector nonnegative tensor factorization (NTF) model is proposed in this paper for spectral unmixing. Different from widely used tensor factorization models, such as canonical polyadic decomposition CPD) and Tucker decomposition, the proposed method is derived from block term decomposition, which is a combination of CPD and Tucker decomposition. This leads to a more flexible frame to model various application-dependent problems. The matrix-vector NTF decomposes a third-order tensor into the sum of several component tensors, with each component tensor being the outer product of a vector (endmember) and a matrix (corresponding abundances). From a formal perspective, this tensor decomposition is consistent with linear spectral mixture model. From an informative perspective, the structures within spatial domain, within spectral domain, and cross spectral-spatial domain are retreated interdependently. Experiments demonstrate that the proposed method has outperformed several state-of-the-art NMF-based unmixing methods.
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