Sparse Hyperspectral Unmixing via Heuristic $\ell _p$ -Norm Approach
In this paper, we propose a new approach to approximate the <inline-formula><tex-math notation="LaTeX">$\ell _0$ </tex-math></inline-formula>-norm for linear sparse hyperspectral unmixing of images. We approximate the <inline-formula><tex-math notation="LaTeX">$\ell _0$</tex-math></inline-formula>-norm with the <inline-formula> <tex-math notation="LaTeX">$\ell _p$</tex-math></inline-formula>-norm and iteratively reduce <inline-formula> <tex-math notation="LaTeX">$p$</tex-math></inline-formula> to enhance the results. The <inline-formula> <tex-math notation="LaTeX">$p$</tex-math></inline-formula>-changing heuristic scheme that reduces the value of <inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula> smoothly and iteratively, results in an enhanced sparse solution. We introduce an iteratively reweighted <inline-formula><tex-math notation="LaTeX">$\ell _2$ </tex-math></inline-formula>-norm to approximate the <inline-formula><tex-math notation="LaTeX">$\ell _p$</tex-math> </inline-formula>-norm. In this approach, a parameter <inline-formula><tex-math notation="LaTeX">$\epsilon$</tex-math> </inline-formula> is involved to deal with the fact that <inline-formula><tex-math notation="LaTeX">$\ell _p$</tex-math> </inline-formula>-norm problem is not Lipschitz continuous for the region of <inline-formula><tex-math notation="LaTeX"> $p<1$</tex-math></inline-formula>. We propose two different methods to update the pair <inline-formula> <tex-math notation="LaTeX">$(p,\epsilon)$</tex-math></inline-formula>. Finally, we evaluate our proposed heuristic <inline-formula><tex-math notation="LaTeX">$\ell _p$</tex-math></inline-formula>-norm method over the synthetic data as well as real hyperspectral dataset. Experimental results show that our algorithm outperforms several state-of-the-art algorithms in terms of the reconstruction errors and their probability of success.