Book excerpt: Over the last decade, digital imaging has become ubiquitous. The advent of cheap digital cameras, and the inclusion of cameras in almost all mobile devices, has made photography one of the basic ways in which people record and communicate experiences. The ubiquity of cameras has imposed new constraints on their physical form. Camera modules are expected to be thin, light, and cheap. These restrictions make the production of high-quality images challenging. We turn to increasingly sophisticated algorithmic tools to transform the raw data captured by a camera into a photograph. This dissertation focuses on one such family of algorithmic tools: those expressible as a Gauss transform. One popular technique in this family is the bilateral filter, which smooths the fine detail in an image without crossing strong edges. It can be used to isolate and control the sharpness, tone, and contrast of a photograph at various scales. Its relatives, the joint-bilateral filter and the joint-bilateral upsample, allow for the fusion of data from multiple images. Another popular technique in the same family is non-local means, which denoises an image by replacing each pixel with the average color of all other pixels in the image with a similar local neighborhood. A naive implementation of these algorithms is prohibitively slow. This dissertation unifies these algorithms under a common framework, describes a variety of applications of the transform in photographic image processing, and presents two new data structures to accelerate the computation of such transforms: the permutohedral lattice, and the Gaussian kd-tree.