I think that if you talk about the amount of detail in an image, the discrete wavelet transform (DWT) fits your description perfectly. It isn't completely dissimilar from the discrete Fourier transform (DFT) in that it, too, operates in terms of fine and coarse scale components of a signal, but it's also very localized unlike the DFT. A fantastic introduction for one-dimensional signals by I. Selesnick is here.
A wavelet transform essentially is a series of nested orthogonal band-pass filters that in the end create signals of different spectral components, so in this sense you can use either wavelet of Fourier transform. However, if you would like to actually plot the components separately from each other, you have to use WFT because is also gives you the right window and localization in space.
If you want to simply compute the amount of detail on each scale level, calculating total energy of each band in interest in the Fourier transform would suffice:
$$D_{\beta} = \sum_{\omega_{\beta} \in \beta}\left\vert S^f(\omega_{\beta})\right\vert^2$$
where $S^f(\omega)$ is the Fourier transform of some signal $s(t)$, and $\beta$ is some interval of frequencies in the Fourier domain.