Fast Semisupervised Unmixing Using Nonconvex Optimization

TL;DR

This work tackles library-based semisupervised hyperspectral unmixing under endmember library mismatch while enforcing the abundance sum-to-one constraint (ASC). It introduces a nonconvex linear model $Y=DBA+N$ with endmember contributions $B$ and abundances $A$, and develops two ADMM-based methods: FaSUn, which imposes a convexity prior on endmember combinations, and SUnS, which applies a sparsity prior on $B$. Empirical results on synthetic datasets with spectral variability and three real datasets (Houston and Cuprite) show that FaSUn consistently outperforms competing methods, including SUnS, with substantial speedups from GPU implementations in the FUnmix package. The methods demonstrate strong scalability and practical utility, offering open-source PyTorch implementations and clear advantages over traditional sparse unmixing approaches in terms of accuracy and efficiency.

Abstract

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

Figures (6)

• Figure 1: Abundance SRE ($\uparrow$) in dB for the selected semi-supervised methods on two simulated datasets (DC1 and DC2) for three noise levels.
• Figure 2: Abundance SRE ($\uparrow$) in dB for the selected semi-supervised methods on three different pixel purity levels ($\rho$) for a given input SNR (30 dB)
• Figure 3: Visual comparisons of abundance maps estimated by using different semi-supervised unmixing methods applied to DC1 (20 dB).
• Figure 4: Visual comparisons of abundance maps estimated by using different semi-supervised unmixing methods applied to DC2 (20 dB).
• Figure 5: Estimated abundances obtained by different semi-supervised methods on the Houston dataset.