There are many imaging techniques that generate three-dimensional volume images today. With higher precision in the image acquisition equipment, storing and processing these images require increasing amount of data processing capacity. Traditionally, three-dimensional images are represented by cubic (or cuboid) picture elements on a cubic grid. The two-dimensional hexagonal grid has some advantages over the traditionally used square grid. For example, less samples are needed to get the same reconstruction quality, it is less rotational dependent, and each picture element has only one type of neighbor which simplifies many algorithms. The corresponding three-dimensional grids are the face-centered cubic (fcc) grid and the body-centered cubic (bcc) grids. In this thesis, image representations using non-standard grids is examined. The focus is on the fcc and bcc grids and tools for processing images on these grids, but distance functions and related algorithms (distance transforms and various representations of objects) are defined in a general framework allowing any point-lattice in any dimension. Formulas for point-to-point distance and conditions for metricity are given in the general case and parameter optimization is presented for the fcc and bcc grids. Some image acquisition and visualization techniques for the fcc and bcc grids are also presented. More theoretical results define distance functions for grids of arbitrary dimensions. Less samples are needed to represent images on non-standard grids. Thus, the huge amount of data generated by for example computerized tomography can be reduced by representating the images on non-standard grids such as the fcc or bcc grids. The thesis gives a tool-box that can be used to acquire, process, and visualize images on high-dimensional, non-standard grids.