Irregular Tensor Low-Rank Representation for Hyperspectral Image Representation

TL;DR

Hyperspectral images suffer spectral variation and irregular spatial distributions that limit regular tensor low-rank models. This work introduces ITLRR, which partitions HSIs into irregular 3D cubes via ERS, fills with a complementary tensor to form regular patches, and enforces a local non-convex tensor rank constraint $||\mathcal{A}||_{S_p}^p$ together with a global discriminative term $-\|\mathcal{L}^o\|_{*}$, solved by an alternating augmented Lagrangian method. It achieves superior performance over state-of-the-art tensor-based and deep-learning approaches on four public datasets, especially under limited training data, demonstrating effective modeling of irregular spatial distributions while preserving discriminability. The method provides a principled framework for irregular-tensor representation and offers a practical pathway to integrate with broader learning architectures. Code is publicly available at the provided repository.

Abstract

Spectral variations pose a common challenge in analyzing hyperspectral images (HSI). To address this, low-rank tensor representation has emerged as a robust strategy, leveraging inherent correlations within HSI data. However, the spatial distribution of ground objects in HSIs is inherently irregular, existing naturally in tensor format, with numerous class-specific regions manifesting as irregular tensors. Current low-rank representation techniques are designed for regular tensor structures and overlook this fundamental irregularity in real-world HSIs, leading to performance limitations. To tackle this issue, we propose a novel model for irregular tensor low-rank representation tailored to efficiently model irregular 3D cubes. By incorporating a non-convex nuclear norm to promote low-rankness and integrating a global negative low-rank term to enhance the discriminative ability, our proposed model is formulated as a constrained optimization problem and solved using an alternating augmented Lagrangian method. Experimental validation conducted on four public datasets demonstrates the superior performance of our method compared to existing state-of-the-art approaches. The code is publicly available at https://github.com/hb-studying/ITLRR.

Figures (9)

• Figure 1: (a) presents a 3D representation of hyperspectral data, showcasing the distribution of ground truth. (b) displays an image patched into blocks. (c) illustrates an image segmented into superpixels, allowing for a more effective representation of spatial structures.
• Figure 2: Framework of the proposed method. (I) stands for the origin HSI with spectral variation. The same color indicates that those pixels belong to the same class, and the black dot represents the data noise. We first divide the input HSI into several irregular 3D cubes by a typical superpixel segmentation method ERS; Then we complete the irregular 3D data cubes with black shape-complementary data cubes to constitute regular tensor patches (III), and design a novel ITLRR model (IV) that can pursue the tensor low-rank representation, which only relies on the original irregular data cubes and ignores the black complementary 3D patches. Besides, the ITLRR is based on a non-convex tensor nuclear norm that can better approximate the tensor low-rankness (IV). The above irregular tensor low-rank representation processes the 3D patches individually, which may lead to the over-smoothing problem, i.e., two disconnected areas with the same material do not share a similar representation (e.g., E1 and E2 in (IV)). To address the problem, we propose to add a negative low-rank term on the whole HSI to enhance the overall discriminative ability (V). Finally, we pack the low-ranked irregular 3D cubes to get the final low-rank HSI representation (VI).
• Figure 3: The illustration of the operation $\oplus$ to combine three irregular 3D patches.
• Figure 4: Flowchart for the low-rank approximation of irregular tensors.
• Figure 5: Comparison of the classification accuracy of different methods under various percentages of training samples on four datasets.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1: Tensor nuclear norm
• Definition 2: The $\oplus$ operator