Two semimetrics on probability distributions are proposed, based on a difference between features chosen from each, where these features can be in either the spatial or Fourier domains. The features are chosen so as to maximize the distinguishability of the distributions, by optimizing a lower bound of power for a statistical test using these features. The result is a parsimonious and interpretable indication of how and where two distributions differ, which can be used even in high dimensions, and when the difference is localized in the Fourier domain. A real-world benchmark image data demonstrates that the returned features provide a meaningful and informative indication as to how the distributions differ.