ℓ₀-Norm Sparse Hyperspectral Unmixing Using Arctan Smoothing

The goal of sparse linear hyperspectral unmixing is to determine a scanty subset of spectral signatures of materials contained in each mixed pixel and to estimate their fractional abundances. This turns into an ℓ₀ -norm minimization, which is an NP-hard problem. In this paper, we propose a new iterative method, which starts as an ℓ₁ -norm optimization that is convex, has a unique solution, converges quickly and iteratively tends to be an ℓ₀ -norm problem. More specifically, we employ the arctan function with the parameter σ ≥ 0 in our optimization. This function is Lipschitz continuous and approximates ℓ₁ -norm and ℓ₀ -norm for small and large values of σ, respectively. We prove that the set of local optima of our problem is continuous versus σ. Thus, by a gradual increase of σ in each iteration, we may avoid being trapped in a suboptimal solution. We propose to use the alternating direction method of multipliers (ADMM) for our minimization problem iteratively while increasing σ exponentially. Our evaluations reveal the superiorities and shortcomings of the proposed method compared to several state-of-the-art methods. We consider such evaluations in different experiments over both synthetic and real hyperspectral data, and the results of our proposed methods reveal the sparsest estimated abundances compared to other competitive algorithms for the subimage of AVIRIS cuprite data.