Multiscale manipulations are central to image editing but also prone to halos. Achieving artifact-free results requires sophisticated edge-aware techniques and careful parameter tuning. These shortcomings were recently addressed by the local Laplacian filters, which can achieve a broad range of effects using standard Laplacian pyramids. However, these filters are slow to evaluate and their relationship to other approaches is unclear. In this article, we show that they are closely related to anisotropic diffusion and to bilateral filtering. Our study also leads to a variant of the bilateral filter that produces cleaner edges while retaining its speed. Building upon this result, we describe an acceleration scheme for local Laplacian filters on gray-scale images that yields speedups on the order of 50×. Finally, we demonstrate how to use local Laplacian filters to alter the distribution of gradients in an image. We illustrate this property with a robust algorithm for photographic style transfer.