Gradient space under orthography and perspective

Abstract

Mackworth's gradient space has proved to be a useful tool for image understanding. However, descriptions of its important properties have been somewhat scattered in the literature. The fundamental properties of the gradient space under orthography and perspective, and for curved surfaces, are developed and summarized. While largely a recounting of previously published results, there are a number of new observations, particularly concerning the gradient space and perspective projection. In addition, the definition and use of vector gradients as well as surface gradients provides concise notation for several results. The properties explored include the orthographic and perspective projections themselves; the definition of gradients; the gradient space consequences of vectors (edges) belonging to one or more surfaces, and of several vectors being contained on a single surface; and the relationships between vanishing points, vanishing lines, and the gradient space. The paper is intended as a study guide for learning about the gradient space, as well as a reference for researchers working with gradient space.