Riemannian Complex Hermit Positive Definite Convolution Network for Polarimetric SAR Image Classification

TL;DR

This paper introduces HPDNet, a novel deep learning framework that directly processes polarimetric SAR covariance matrices on the Riemannian manifold, preserving their geometric structure for improved classification accuracy.

Contribution

The paper proposes HPDNet, which processes complex Hermitian positive definite matrices directly on the Riemannian manifold, incorporating a new LogEig layer and a fast eigenvalue decomposition method.

Findings

Outperforms state-of-the-art methods on three PolSAR datasets.

Effectively preserves the geometric structure of covariance matrices.

Shows improved classification in heterogeneous regions.

Abstract

Deep learning has been extensively utilized for PolSAR image classification. However, most existing methods transform the polarimetric covariance matrix into a real- or complex-valued vector to comply with standard deep learning frameworks in Euclidean space. This approach overlooks the inherent structure of the covariance matrix, which is a complex Hermitian positive definite (HPD) matrix residing in the Riemannian manifold. Vectorization disrupts the matrix structure and misrepresents its geometric properties. To mitigate this drawback, we propose HPDNet, a novel framework that directly processes HPD matrices on the Riemannian manifold. The HPDnet fully considers the complex phase information by decomposing a complex HPD matrix into the real- and imaginarymatrices. The proposed HPDnet consists of several HPD mapping layers and rectifying layers, which can preserve the geometric structure of the data and transform them into a more separable manifold representation. Subsequently, a complex LogEig layer is developed to project the manifold data into a tangent space, ensuring that conventional Euclidean-based deep learning networks can be applied to further extract contextual features for classification. Furthermore, to optimize computational efficiency, we design a fast eigenvalue decomposition method for parallelized matrix processing. Experiments conducted on three real-world PolSAR datasets demonstrate that the proposed method outperforms state-of-the-art approaches, especially in heterogeneous regions.