Sparse Unmixing in the Presence of Mixed Noise Using ℓ0-Norm Constraint and Log-Cosh Loss

Over the past two decades, sparse unmixing (SU) has gained significant attention in the realm of hyperspectral imaging. The aims of SU are to seek a subset of spectral signatures and estimate their fractional abundances to represent each mixed spectral pixel. Conventional SU methods often employ the Frobenius norm and thus cannot work satisfactorily in the presence of non-Gaussian noise. Second, the ideal
$\ell _{0}$
-norm is usually substituted with its convex or nonconvex approximation in most existing algorithms, which may degrade the recovery performance. To address these issues, this article proposes a novel approach, termed sparse unmixing using
$\ell _{0}$
-norm constraint and log-cosh loss (SUNNING). We exploit the
$\log $
-
$\cosh $
function to minimize the fitting errors subject to three constraints, namely, nonnegativity, sum-to-one, and upper bounded
$\ell _{0}$
-norm. Then, we adopt the projected gradient descent (PGD) framework to solve such an optimization problem. SUNNING includes two alternating steps, gradient descent and nonconvex projection, where an optimality of the solution is guaranteed. Also, we prove the convergence of SUNNING, including the objective value and variable sequence. In addition, to attain higher unmixing accuracy, we exploit the spectral library pruning (SLP) strategy to eliminate inactive endmembers, yielding an improved SUNNING. Experimental results on synthetic and real-world datasets exhibit improved robustness and effectiveness of the suggested methods over the state-of-the-art algorithms. MATLAB code is available at:
https://github.com/freeLix-YY/
IEEE_TGRS2024_SparseUnmixing_SUNNING_demo