Fermat Active Laplace Learning for Semi-Supervised Hyperspectral Image Classification

2026-08-03

AI Summary

This study addresses the challenges of scarce labeled samples and underutilized manifold structure in semi-supervised hyperspectral image classification by proposing a Fermat distance–based active learning framework, FALL, along with its scalable variant, A-FALL. The method introduces density-aware Fermat distance into active learning for the first time, integrating Poisson reweighted harmonic label propagation with uncertainty-guided farthest-point sampling to jointly model data density and geometric structure. The distance exponent \( p \) is adaptively selected via landmark multidimensional scaling combined with leave-one-out cross-validation. Experimental results demonstrate that FALL achieves high classification accuracy on the Salinas A and Pavia datasets, while A-FALL exhibits excellent scalability in large-scale scenarios.

Abstract

Two active learning algorithms for hyperspectral image (HSI) classification are proposed that combine density-aware Fermat distances with Poisson-reweighted harmonic label propagation. Our methods actively query points using an uncertainty-based acquisition function, extending Poisson ReWeighted Laplace Learning (PWLL). Our first algorithm, Fermat Active Laplace Learning (FALL), builds an affinity matrix using Fermat distances between all data points. Then, PWLL is run with a diagonal perturbation using the minimum-norm acquisition function. In contrast, Approximate FALL (A-FALL) computes Fermat distances between each data point and landmark pixels selected via farthest-point sampling and constructs the affinity matrix using landmark multidimensional scaling. After several query rounds, A-FALL selects the Fermat exponent $p$ using a leave-one-out cross-validation variant. FALL and A-FALL leverage Fermat distances and subsequent harmonic label propagation to provide a density-aware estimation of the data manifold, improving labeling accuracy. Experiments on Salinas A and Pavia show the effectiveness of FALL and the scalability of A-FALL to large HSI scenes.

Problem

• hyperspectral image classification
• semi-supervised learning
• active learning
• label propagation
• manifold estimation

Innovation

• Fermat distance
• active learning
• semi-supervised classification
• hyperspectral image
• harmonic label propagation