Abstract

Given an optical photograph with images of six control points (whose three-space coordinates are known), and the two image coordinates of an unknown ground point, it is possible to determine the equation of the line of sight from the camera station to the unknown point via a direct linear transformation (DLT) approach.¹ To do this requires writing the projective equations for the control points in a form that is linear in the unknowns containing camera position, orientation, and (assumed affine) film distortions. These equations are written so each image coordinate of each control point appears in only one equation. The result is a linear system solvable for eleven camera variables, which can then be used with the unknown-point image coordinates to determine the line of sight. In the spotlight mode of synthetic-aperture radar (SAR), similar DLT equations are developed to determine the projection circle which is the intersection of the isorange sphere and iso-Doppler cone for a reflecting point. To avoid singularity, the Doppler equations are used to eliminate one image coordinate of each control point in the range equations, where that coordinate appears to the first power. The result is a linear system solvable for thirteen camera variables, which can then be used to determine the projection circle for an unknown ground point. Eight control points are needed for solution. Given optical and SAR images of a ground point, it is now possible to triangulate by intersecting two optical lines of sight, two SAR projection circles, or a line of sight and a projection circle, depending on the available imagery.

© 1986 Optical Society of America