Images of the corona contain information over a wide range of spatial scales and various structures such as coronal holes, coronal loops, coronal mass ejections (CMEs), and helmet streamers with differences in brightness. Therefore, processing these images is important for determining various types of information. However, it is very difficult to elucidate the intricate structure of the corona because of the steep radial gradient in terms of brightness and the large differences in brightness within local areas of coronal images. In this study, we present a filter inspired by the normalizing-radial-graded filter, which we call the radial local multi-scale filter (RLMF). The RLMF method extracts radial vectors and performs local multi-scale filtering based on each vector, before normalizing the vector in order to enhance coronal images. This method facilitates the enhancement of large-scale structural characteristics and fine details in low contrast regions. To demonstrate the power of the RLMF, we applied the proposed method to Large Angle and Spectrometric Coronagraph C2 and K-Coronagraph observations. The RLMF method more clearly revealed the fine details of the corona and CMEs compared with the original observations.