Matrix-Vector Nonnegative Tensor Factorization for Blind Unmixing of Hyperspectral Imagery

Abstract

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 (BTD) 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 spectralspatial domain are retreated interdependently. Experiments demonstrate that the proposed method has outperformed several state-of-the-art NMF based unmixing methods.

Hyperspectral Super-Resolution: A Coupled Nonnegative Block-Term Tensor Decomposition Approach

“…But utilizing this model for HSR was never considered, to the best knowledge of the authors. Nevertheless, the interesting results obtained in [14] offers numerical evidence for that real spectral images can be well approximated by the (Lr, Lr, 1) BTD model.…”

“…To circumvent this issue, in this work, we propose to employ the block term decomposition (BTD) [11,12,13] model for the spectral images. Under BTD with rank-(Lr, Lr, 1) terms, the latent factors have explicit physical explanations under the widely adapted LMM-i.e., endmembers and abundances maps-if the abundance maps are approximately low-rank matrices [14]. We show that, under reasonable conditions, the new model also guarantees the recovery of the SRI.…”

“…Before we go to the next subsection, we should mention that using the (Lr, Lr, 1)-BTD to model a spectral image was first considered in [14] for hyperspectral unmixing. But utilizing this model for HSR was never considered, to the best knowledge of the authors.…”

NPSA: Nonorthogonal Principal Skewness Analysis

“…S INCE hyperspectral imagery consists of tens or hundreds of bands with a very high spectral resolution, it has drawn more attention from various applications in the past decades, such as spectral unmixing [1], [2], classification [3], [4], target detection [5], [6] and so on. However, high spectral dimensionality with strong intraband correlations also results in informantion redundancy and computational burden of data processing [7].Therefore, dimensionality reduction (DR) has become one of the most important techniques for addressing these problems.…”

Bilateral Filter Regularized L2 Sparse Nonnegative Matrix Factorization for Hyperspectral Unmixing

“…These kinds of methods have been proven to be able to unmix highly mixed data and obtain a higher decomposition accuracy, despite the relatively high computation complexity. The representative algorithms of the statistical methods include the independent component analysis (ICA) [15,16], nonnegative matrix/tensor factorization (NMF/NTF) [17,18], and Bayesian approaches [19].…”