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This paper presents a novel algorithm for matching image interest points. Potential interest points are identified by searching for local peaks in Difference-of-Gaussian (DoG) images. We refine and assign rotation, scale and location for each keypoint by using the SIFT algorithm. Pseudo log-polar sampling grid is then applied to properly scaled image patches around each keypoint, and a weighted adaptive lifting scheme transform is designed for each ring of the log-polar grid. The designed adaptive transform for a ring in the reference keypoint and the general non-adaptive transform are applied to the corresponding ring in a test keypoint. Similarity measure is calculated by comparing the corresponding transform domain coefficients of the adaptive and non-adaptive transforms. We refer to the proposed versatile system of Rotation And Scale Invariant Matching as RASIM. Our experiments show that the accuracy of RASIM is more than SIFT, which is the most widely used interest point matching algorithm in the literature. RASIM is also more robust to image deformations while its computation time is comparable to SIFT.