I present a family of algorithms to reduce noise in astrophysical images and image sequences, preserving more information from the original data than is retained by conventional techniques. The family uses locally adaptive filters ("noise gates") in the Fourier domain to separate coherent image structure from background noise based on the statistics of local neighborhoods in the image. Processing of solar data limited by simple shot noise or by additive noise reveals image structure not easily visible in the originals, preserves photometry of observable features, and reduces shot noise by a factor of 10 or more with little to no apparent loss of resolution. This reveals faint features that were either not directly discernible or not sufficiently strongly detected for quantitative analysis. The method works best on image sequences containing related subjects, for example movies of solar evolution, but is also applicable to single images provided that there are enough pixels. The adaptive filter uses the statistical properties of noise and of local neighborhoods in the data to discriminate between coherent features and incoherent noise without reference to the specific shape or evolution of those features. The technique can potentially be modified in a straightforward way to exploit additional a priori knowledge about the functional form of the noise.