Local Log-Euclidean Covariance Matrix (L2ECM) for Image Representation and Its Applications

This paper presents Local Log-Euclidean Covariance Matrix (LECM) to represent neighboring image properties by capturing correlation of various image cues. Our work is inspired by the structure tensor which computes the second-order moment of image gradients for representing local image properties, and the Diffusion Tensor Imaging which produces tensor-valued image characterizing the local tissue structure. Our approach begins with extraction of raw features consisting of multiple image cues. For each pixel we compute a covariance matrix in its neighboring region, producing a tensor-valued image. The covariance matrices are symmetric and positive-definite (SPD) which forms a Riemannian manifold. In the Log-Euclidean framework, the SPD matrices form a Lie group equipped with Euclidean space structure, which enables common Euclidean operations in the logarithm domain. Hence, we compute the covariance matrix logarithm, obtaining the pixel-wise symmetric matrix. After half-vectorization we obtain the vector-valued LECM image, which can be flexibly handled with Euclidean operations while preserving the geometric structure of SPD matrices. The LECM features can be used in diverse image or vision tasks. We demonstrate some applications of its statistical modeling by simple second-order central moment and achieve promising performance.