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

Abstract

This paper presents Local Log-Euclidean Covariance Matrix (L^2ECM) 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 L^2ECM image, which can be flexibly handled with Euclidean operations while preserving the geometric structure of SPD matrices. The L^2ECM 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.