OPTIMISATION OF BUNDLE ADJUSTMENTS FOR STEREO PHOTOGRAPHY 
Bruce King 
Department of Civil Engineering and Surveying 
University of Newcastle, Australia 
Commission 5 
ABSTRACT 
Conventional bundle adjustments ignore the invariant geometric relationships that exist between camera pairs in a bundle of 
stereo photography. Two models for optimising conventional bundle adjustments to take advantage of these relationships 
are developed. These models are compared with a conventional bundle adjustment. Initial results indicate that both 
presented models yield improved accuracies when compared to a conventional bundle adjustment. 
KEY WORDS: Optimisation, Bundle adjustment, Stereo photography. 
INTRODUCTION 
In the field of non-topographic photogrammetry bundle 
adjustments are used in a wide range of applications. 
Whether implemented as a DLT or self-calibrating model 
the purpose of the bundle adjustment is to minimise the 
residuals of all the observations. Each camera thus finds a 
position and orientation that reflects this minimisation. This 
approach is well established for single camera imaging 
geometry as typified by Fraser (1991). 
When the imaging is performed by a stereo camera 
system, inherent in each pair of photographs is the 
invariant geometrical relationship of the two cameras. f 
such a bundle of stereopairs were reduced by a 
conventional bundle adjustment the invariant camera 
relationships between the two images of each stereopair 
would be ignored in favour of optimising each camera's 
position and orientation based on the observations and 
their random errors. 
This paper reports on two methods that have been 
developed to optimise a conventional bundle adjustment 
for use with stereo photography so that the invariant 
relationships are retained for all stereopairs. The 
mathematical models are developed and the results of a 
trial comparing the two models with a conventional bundle 
adjustment are presented. 
THE BUNDLE ADJUSTMENT 
Slama (1980, Ch2) gives the generally accepted model of 
the observation equations for a conventional bundle 
adjustment. These are of the following form: 
V.BA «C (1) 
= 458 Al - e 
V=iV B=|-1 0 A=} 1 C- Cl. 
0 1 A G 
  
  
V = vector of plate observation residuals, 
V = vector of exterior orientation parameter 
observation residuals; 
V - vector of object coordinate observation 
residuals, 
B = matrix of partial derivatives wrt exterior 
orientation parameters; 
B = matrix of partial derivatives wrt object 
coordinates; 
A = vector of exterior orientation parameter 
corrections; 
A = vector of object coordinate corrections; 
e = vector of plate observation descrepancies, 
C = vector of exterior orientation parameter 
descrepancies, 
C - vector of object coordinate discrepancies; 
The structure of the camera parameter portion of the 
normal equation matrix produced by the least squares 
solution to this model is shown in figure 1. 
  
  
  
  
  
Figure 1 Normal equation structure of the camera 
parameter portion of a conventional bundle adjustment of 
stereopairs. 
Each camera is represented by a 6x6 symmetric sub- 
matrix. The total number of camera parameters to be 
solved for a bundle of s stereopairs is 12s. An overview of 
photogrammetric bundle adjustment programs can be 
found in Karara (1989, Ch6). Bundle adjustments 
developed in this study were based upon this model. 
MODELLING OF CAMERA INVARIANCE 
The invariance that exists between the cameras of a 
stereopair may be divided into two relationships: