  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
