A Convex Optimization-Based Coupled Nonnegative Matrix Factorization Algorithm for Hyperspectral and Multispectral Data Fusion

Fusing a low-spatial-resolution hyperspectral data with a high-spatial-resolution (HSR) multispectral data has been recognized as an economical approach for obtaining HSR hyperspectral data, which is important to accurate identification and classification of the underlying materials. A natural and promising fusion criterion, called coupled nonnegative matrix factorization (CNMF), has been reported that can yield high-quality fused data. However, the CNMF criterion amounts to an ill-posed inverse problem, and hence, advisable regularization can be considered for further upgrading its fusion performance. Besides the commonly used sparsity-promoting regularization, we also incorporate the well-known sum-of-squared-distances regularizer, which serves as a convex surrogate of the volume of the simplex of materials' spectral signature vectors (i.e., endmembers), into the CNMF criterion, thereby leading to a convex formulation of the fusion problem. Then, thanks to the biconvexity of the problem nature, we decouple it into two convex subproblems, which are then, respectively, solved by two carefully designed alternating direction method of multipliers (ADMM) algorithms. Closed-form expressions for all the ADMM iterates are derived via convex optimization theories (e.g., Karush-Kuhn-Tucker conditions), and furthermore, some matrix structures are employed to obtain alternative expressions with much lower computational complexities, thus suitable for practical applications. Some experimental results are provided to demonstrate the superior fusion performance of the proposed algorithm over state-of-the-art methods.