Gradient Space under Orthography and Perspective
CARNEGIE-MELLON UNIV PITTSBURGH PA DEPT OF COMPUTER SCIENCE
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Machworths gradient space has proven to be a useful tool for image understanding. However, descriptions of its important properties have been somewhat scattered in the literature. This paper develops and summarizes the fundamental properties of the gradient space under orthography and perspective, and for curved surfaces. 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 in the paper 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.
- Theoretical Mathematics