The paper introduces a new region descriptor called region covariance, which is used for object detection and texture classification. The covariance of $d$-features, such as color vectors and derivatives of intensity, characterizes a region of interest. The authors propose a fast method for computing covariances using integral images, which reduces computational complexity and is independent of the region size. The covariance matrices, which do not lie on Euclidean space, are compared using a distance metric involving generalized eigenvalues. This metric is derived from the Lie group structure of positive definite matrices. The feature matching is performed using a nearest neighbor search, which is extremely rapid due to the integral image representation. In object detection, the method uses pixel locations, color values, and norms of first and second derivatives of intensities as features. Five covariance matrices are extracted from overlapping regions of an object's feature image, and these are used to locate the object in an arbitrary image. The dissimilarity between the object and target regions is measured using the proposed distance metric, and the best matching region is selected based on the smallest dissimilarity. For texture classification, the method extracts several features from each pixel and computes the covariance matrix of randomly selected regions. The texture is represented by multiple covariance matrices, and the class of the texture is determined by the majority voting among the k nearest neighbors in the training set. The method is tested on the Brodatz texture database and shows superior performance compared to other methods, including texton histograms and raw intensity values. The paper concludes by discussing the potential extensions of the method, such as object tracking and classification algorithms that utilize the Lie group structure of covariance matrices.The paper introduces a new region descriptor called region covariance, which is used for object detection and texture classification. The covariance of $d$-features, such as color vectors and derivatives of intensity, characterizes a region of interest. The authors propose a fast method for computing covariances using integral images, which reduces computational complexity and is independent of the region size. The covariance matrices, which do not lie on Euclidean space, are compared using a distance metric involving generalized eigenvalues. This metric is derived from the Lie group structure of positive definite matrices. The feature matching is performed using a nearest neighbor search, which is extremely rapid due to the integral image representation. In object detection, the method uses pixel locations, color values, and norms of first and second derivatives of intensities as features. Five covariance matrices are extracted from overlapping regions of an object's feature image, and these are used to locate the object in an arbitrary image. The dissimilarity between the object and target regions is measured using the proposed distance metric, and the best matching region is selected based on the smallest dissimilarity. For texture classification, the method extracts several features from each pixel and computes the covariance matrix of randomly selected regions. The texture is represented by multiple covariance matrices, and the class of the texture is determined by the majority voting among the k nearest neighbors in the training set. The method is tested on the Brodatz texture database and shows superior performance compared to other methods, including texton histograms and raw intensity values. The paper concludes by discussing the potential extensions of the method, such as object tracking and classification algorithms that utilize the Lie group structure of covariance matrices.