|
The mathematical solution for the true positions of features in space from the position of their images on pairs of radiographs has been discussed in Ballistic Research Laboratory reports by Grabarek and Herr (BRL TN 1634, September 1966, AD 807619), and by Henry C. Dubin (BRL MR 2470, April 1975, AD B003797L, now available for unlimited distribution). A truly general solution should not require, or be limited to, orthogonality of the image pairs. Dubin provides such a solution. Because of errors in setup and measurement, the two lines presumed to connect the respective source and image points through the common object point do not necessarily intersect. Using vector notation and partial derivatives, he obtains the line of minimum length between these two vectors. The midpoint of this line is the best estimate for the object position, and the length is a measure of the error of the estimate. The redundant image coordinate in Dubin's method contributes to increasing the accuracy of the estimate of the position of features. The solution of Grabarek and Herr uses analytic geometry and the assumption that the two lines between the respective sources and images intersect, and requires an orthogonal radiographic setup. This approach forgoes generality and some available accuracy. Driven by the need to provide as simple an approach as possible, this paper presents two similar derivations. The author uses analytic geometry to rederive the equations of Grabarek and Herr from a simpler perspective. The form of the solution provides a conceptual bridge to a more direct derivation by the author, using trigonometry. Both derivations are for orthogonal radiographic setups. The trigonometric approach does not require complicated computation of magnification factors, is more easily understood in terms of the geometry of the setup, and is easily implemented in computer or calculator programs to reduce orthogonal radiographs.
|
