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This paper studies phase singularities (PSs) for image representation. We show that PSs calculated with Laguerre-Gauss filters contain important information and provide a useful tool for image analysis. PSs are invariant to image translation and rotation. We introduce several invariant features to characterize the core structures around PSs and analyze the stability of PSs to noise addition and scale change. We also study the characteristics of PSs in a scale space, which lead to a method to select key scales along phase singularity curves. We demonstrate two applications of PSs: object tracking and image matching. In object tracking, we use the iterative closest point algorithm to determine the correspondences of PSs between two adjacent frames. The use of PSs allows us to precisely determine the motions of tracked objects. In image matching, we combine PSs and scale-invariant feature transform (SIFT) descriptor to deal with the variations between two images and examine the proposed method on a benchmark database. The results indicate that our method can find more correct matching pairs with higher repeatability rates than some well-known methods.