In this paper, we introduce a novel linear model tailored for semisupervised/library-based unmixing. Our model incorporates considerations for library mismatch while enabling the enforcement of the abundance sum-to-one constraint (ASC). Unlike conventional sparse unmixing methods, this model involves nonconvex optimization, presenting significant computational challenges. We demonstrate the efficacy of Alternating Methods of Multipliers (ADMM) in cyclically solving these intricate problems. We propose two semisupervised unmixing approaches, each relying on distinct priors applied to the new model in addition to the ASC: sparsity prior and convexity constraint. Our experimental results validate that enforcing the convexity constraint outperforms the sparsity prior for the endmember library. These results are corroborated across three simulated datasets (accounting for spectral variability and varying pixel purity levels) and the Cuprite dataset. Additionally, our comparison with conventional sparse unmixing methods showcases considerable advantages of our proposed model, which entails nonconvex optimization. Notably, our implementations of the proposed algorithms-fast semisupervised unmixing (FaSUn) and sparse unmixing using soft-shrinkage (SUnS)-prove considerably more efficient than traditional sparse unmixing methods. SUnS and FaSUn were implemented using PyTorch and provided in a dedicated Python package called Fast Semisupervised Unmixing (FUnmix), which is open-source and available at https://github.com/BehnoodRasti/FUnmix

翻译：本文提出了一种专为半监督/基于光谱库的解混设计的新型线性模型。该模型在考虑光谱库失配的同时，能够强制执行丰度和为一约束（ASC）。与传统稀疏解混方法不同，此模型涉及非凸优化，带来了显著的计算挑战。我们证明了交替方向乘子法（ADMM）在循环求解这些复杂问题中的有效性。我们提出了两种半监督解混方法，除了ASC外，每种方法都对新建模应用了不同的先验：稀疏性先验和凸性约束。实验结果表明，对端元光谱库施加凸性约束优于稀疏性先验。这些结果在三个模拟数据集（考虑光谱变异性和不同像元纯度）和Cuprite数据集上得到了验证。此外，与传统稀疏解混方法的比较显示，我们提出的非凸优化模型具有显著优势。值得注意的是，我们所提算法的实现——快速半监督解混（FaSUn）和使用软阈值收缩的稀疏解混（SUnS）——被证明比传统稀疏解混方法高效得多。SUnS和FaSUN基于PyTorch实现，并集成于名为Fast Semisupervised Unmixing（FUnmix）的专用Python软件包中，该软件包已开源，可通过https://github.com/BehnoodRasti/FUnmix获取。