Analytical photogrammetry: a collinear theory
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Date
1971-07
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Ohio State University. Division of Geodetic Science
Abstract
A collinear theory of photogrammetry is developed through the employment of homogeneous coordinates and generalized matrix inverses. Chapter I outlines the basic concepts used in the succeeding chapters. In Chapter II Abee's theory of optical instruments is introduced as the foundation upon which the present theory is built. The distinguishing facit of Abbe's theory is its basic postulate which asserts (regardless of the mechanism by which it is accomplished) that there exists a one-to-one relationship between the object and its image. Through this postulate one arrives at the conclusion that a three-dimensional, non-singular projective transformation (collineation) relates object and image. Next, Chapter III investigates in same detail the photograph as a singular collineation. It is given two interpretations: 1. finite plane of discontinuity which includes the ESSA model, the nine-parameter perspective model, and the rectification model as examples: 2. infinite plane of discontinuity, which embraces the axonometry. The axonometry is the limiting case of narrow-angle photographs such as is used in astrometric work. Church's method of resection is shown to be an inner product formed from the above equations; and a variant of Church's formula is given. Von Gruber's rectification parameters are derived through the perspective equations. With the use of the generalized inverse, the inverse of the singular collineation is found and subsequently reduced to a more conventional form. Finally, two differential forms of the perspective equations are given. Chapter IV covers the theory of photographic stereo-pairs. First are given the general formulas which show that the object space can be obtained through a projective transform of the four photographic image coordinates. The standard parallax bar formulas are shown to be a speical case of the general formulas. Next is treated relative orientation for which a completely new form is given. In the final section, Chapter V, empirical instrumental relative orientation is shown to be analogous to the Gauss-Seidel methods.
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Prepared for Air Force Systems Command, Rome Air Development Center, Griffiss Air Force Base, New York: Contract No. F30602-70-C-0094, OSURF Project No. 2968