Edward M. Mikhail 
PHOTOGRAMMETRIC INVARIANCE 
H. J. THEISS, E. M. MIKHAIL", I. M. ALY, J. S. BETHEL’, C. LEE" 
"Purdue University 
Geomatics Engineering Department 
theiss, mikhail, aly, bethel, changno@ecn.purdue.edu 
Working Group III/4 
KEY WORDS: Image transfer, Mathematical models, Orientation, Photogrammetry, Reconstruction 
ABSTRACT 
Two tasks, image transfer and object reconstruction, are investigated using three different approaches. The first 
approach, based on the fundamental (F) matrix, compares the use of all three F matrices with epipolar constraints to the 
use of only two F matrices. The second approach uses the four trilinearity equations and employs strategies to deal with 
the dependency among the equations and the parameters. The third approach is a new one based on collinearity that 
uses independent equations and therefore yields a rigorous solution. 
1 INTRODUCTION 
The goal of this research is to investigate and improve invariance techniques to assist in performing photogrammetric 
tasks. The topics of image transfer and object reconstruction constitute the main sections of this paper. The invariance 
relationship among image coordinate observations that must be solved to perform image transfer is also a necessary step 
before doing object reconstruction. Three main approaches — based on the fundamental matrix, the trilinearity 
equations, and a new collinearity approach — are presented with relevant equations; and results are tabulated for 
experiments with both simulated and real image data. 
2 IMAGE TRANSFER 
Image transfer is an application performed on a triplet of images. Given two pairs of measured image coordinates, the 
third pair can be calculated using a previously established relationship between pairs of image coordinates on all three 
images. Three basic approaches for establishing the image-to-image relationship are discussed in this paper: the 
fundamental matrix approach, the trilinearity approach, and the collinearity approach. 
2.1 Fundamental Matrix (F) Model 
The fundamental matrix relates the image coordinates of 3D objects that appear on two images, and j, as follows: 
Esuürksif-o (1) 
where (x, y) are measured image coordinates. 
The 3 by 3 F matrix has eight unknown parameters since it is determinable up to a scale factor. In fact there are seven 
independent parameters, since F is of rank two and its determinant must be constrained to equal zero. 
Previous techniques based on the F matrix have enforced the relationship between only two of the three existing image 
pairs. In other words, if image coordinates are to be computed on image 3, then we would solve for the elements of 
only Fız and F»;. With eight common points between images 1 and 3. we can linearly solve for the eight elements of 
F,; using equation (1). Eight common points between images 2 and 3, which can contain any or all of those used 
between 1 and 3, can be used in the same way to solve for the elements of F»;. The solution can be refined by imposing 
the determinant equals zero constraint on each of the two F matrices, thus reducing the number of independent 
  
584 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 
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