Images of the corona have a high dynamic range which is excellent for quantitative photometric analysis. To understand the processes governing the solar corona, it is essential to have information about the absolute brightness as well as the underlying structure. However, due to the steep radial gradient of brightness in the images, and to the fact that structures closer to the solar disk have higher contrast than structures further from the disk, human vision cannot view the intricate structure of the corona in such images. The recently developed normalizing-radial-graded filter (NRGF) is an effective way for revealing the coronal structure. In this work, we present a more adaptive filter inspired by the NRGF, which we call the Fourier normalizing-radial-graded filter (FNRGF). It approximates the local average and the local standard deviation by a finite Fourier series. This method enables the enhancement of finer details, especially in regions of lower contrast. We also show how the influence of additive noise is reduced by a modification to the FNRGF. To illustrate the power of the method, the FNRGF is applied to images of emission from coronal forbidden lines observed during the 2010 July 11 total solar eclipse. It is also successfully applied to space-based observations of the low corona in the extreme ultraviolet and to white light coronagraph observations, thus demonstrating the validity of the FNRGF as a new tool that will help the interpretation of the information embedded in most types of coronal images.