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First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. This paper proposes a novel distance and inner product on the space of positive, regularized self-adjoint operators. By considering the correspondence between a Hilbert space and the above space through log/exp map, the authors successfully introduce a geometrically natural inner product on the space. The distance can be regarded as an infinite dimensional generalization of the log-Euclidean distance defined for the finite dimensional positive definite matrices. The proposed distance has been applied to define positive definite kernels on the kernel covariance expressions of images, and has shown better performance for some image classification examples.