Principal Component Pursuit and Applications in Small Target Detection

High-dimensional data in real-world applications often possess intrinsic low-dimensional structures. While classical principal component analysis (PCA) provides a powerful foundation for dimension reduction, it suffers from significant robustness limitations when confronted with sparse corruptions or gross errors. This chapter provides an overview of robust PCA, which handle sparse corruptions. We focus on principal component pursuit (PCP) and its extensions, which address these fundamental limitations through convex optimisation. We emphasise the impact on a real-world application: infrared small target detection, where extensions of PCP and tensor methods are introduced and compared using extensive public datasets and benchmarks. A critical research gap in a unified detection framework with robust tensor methods that could generalize multimodal imagery is identified. Beginning with the theoretical foundations, we examine how PCP formulates the decomposition of low-rank and sparse components. We then explore stable principal component pursuit (SPCP), which extends the framework to handle sparse corruptions and dense noise simultaneously. The discussion progresses to tensor-based extensions, addressing the challenges associated with higher-dimensional data structures. We present the Infrared Patch-Image model and its tensor-based counterpart, showing how robust PCA techniques can effectively separate small targets from complex backgrounds in IR imagery. Advanced regularisation techniques, including total variation constraints, are discussed to further enhance detection performance by incorporating prior knowledge about background structure and target characteristics. This chapter establishes the theoretical foundations and practical implementations of robust PCA, demonstrating its evolution from classical PCA to modern tensor-based methods, with particular emphasis on addressing real-world challenges in small target detection.