A selected morphology operation on a binary image is called a thickening by Serra. It is simply a union of the image with a hit or miss transform. The definition can be extended to a gray level image by applying the operation to each binary level in a threshold decomposition. A difficulty is that the operation is not increasing, and the resulting threshold decomposition consists of stacks that contain holes. In this sense, the thickening operator leaves an image that is a multivalued function. A single valued function can be defined as a projection of the stacks onto the spatial dimension. This is called a projected thickening, and is different than the traditional umbra representation of functions.