Improved Hyperspectral Anomaly Detection via Unsupervised Subspace Modeling in the Signed Cumulative Distribution Transform Domain

TL;DR

The paper addresses hyperspectral anomaly detection by modeling background pixels as transport-induced deformations of a template within the signed cumulative distribution transform (SCDT) domain. By performing unsupervised subspace learning in the SCDT space, anomalies are identified as deviations from a learned background subspace, with the anomaly score defined as $\\varepsilon_a = \\| \\widehat{s} - B B^T \\widehat{s} \\|^2_{L_2}$. The approach is non-iterative and parameter-light, demonstrated to outperform several state-of-the-art HAD methods across five datasets, particularly at low false-positive rates, despite occasional limitations when background inhomogeneity (e.g., shadows) complicates the model. Overall, the method leverages convexity and composition properties of the SCDT to linearize nonlinear deformations, enabling robust background modeling and reliable anomaly detection in real-world hyperspectral data.

Abstract

Hyperspectral anomaly detection (HAD), a crucial approach for many civilian and military applications, seeks to identify pixels with spectral signatures that are anomalous relative to a preponderance of background signatures. Significant effort has been made to improve HAD techniques, but challenges arise due to complex real-world environments and, by definition, limited prior knowledge of potential signatures of interest. This paper introduces a novel HAD method by proposing a transport-based mathematical model to describe the pixels comprising a given hyperspectral image. In this approach, hyperspectral pixels are viewed as observations of a template pattern undergoing unknown deformations that enables their representation in the signed cumulative distribution transform (SCDT) domain. An unsupervised subspace modeling technique is then used to construct a model of abundant background signals in this domain, whereupon anomalous signals are detected as deviations from the learned model. Comprehensive evaluations across five distinct datasets illustrate the superiority of our approach compared to state-of-the-art methods.

Figures (8)

• Figure 1: The proposed approach begins by estimating the background subspace $\widehat{\mathbb{V}}^{(b)}$ in the SCDT domain, which spans the set $\widehat{\mathbb{S}}^{(b)}$. The anomaly score $\varepsilon_a$ for a test data point is then computed based on its subspace distance from $\widehat{\mathbb{V}}^{(b)}$.
• Figure 2: Approximate RGB images of the datasets.
• Figure 3: Experimental results on AVIRIS-I dataset: (left) full ROC curves up to $\text{FPR}= 10^{0}$, and (right) partial ROC curves restricted to $\text{FPR}\leq 10^{-2}$.
• Figure 4: Experimental results on AVIRIS-II dataset: (left) full ROC curves up to $\text{FPR}= 10^{0}$, and (right) partial ROC curves restricted to $\text{FPR}\leq 10^{-2}$.
• Figure 5: Experimental results on Urban dataset: (left) full ROC curves up to $\text{FPR}= 10^{0}$, and (right) partial ROC curves restricted to $\text{FPR}\leq 10^{-2}$.