基于黎曼流形的矩阵结构建模，提升极化SAR图像分类精度

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.