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Keypoint detection and matching is of fundamental importance for many applications in computer and robot vision. The association of points across different views is problematic because image features can undergo significant changes in appearance. Unfortunately, state-of-the-art methods, like the scale-invariant feature transform (SIFT), are not resilient to the radial distortion that often arises in images acquired by cameras with microlenses and/or wide field-of-view. This paper proposes modifications to the SIFT algorithm that substantially improve the repeatability of detection and effectiveness of matching under radial distortion, while preserving the original invariance to scale and rotation. The scale-space representation of the image is obtained using adaptive filtering that compensates the local distortion, and the keypoint description is carried after implicit image gradient correction. Unlike competing methods, our approach avoids image resampling (the processing is carried out in the original image plane), it does not require accurate camera calibration (an approximate modeling of the distortion is sufficient), and it adds minimal computational overhead. Extensive experiments show the advantages of our method in establishing point correspondence across images with radial distortion.