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Camillo Ressl 
AN INTRODUCTION TO THE RELATIVE ORIENTATION 
USING THE TRIFOCAL TENSOR 
Camillo RESSL 
University of Technology, Vienna, Austria 
Institute of Photogrammetry and Remote Sensing 
car Q ipf.tuwien.ac.at 
Working Group III/1 
KEY WORDS: calibration, computer vision, mathematical models, orientation 
ABSTRACT 
About five years ago, in Computer Vision, a linear representation for the relative orientation of three images by means 
of the so-called Trifocal Tensor (TFT) was discovered using more parameters than necessary having no easily 
comprehensible geometric meaning compared to the notion of inner orientation (IOR) and exterior orientation (XOR). 
The relative orientation’s fundamental condition of intersecting projection-rays for each homologous point-triple is 
described by four homogenous linear equations. The TFT also allows the usage of homologous image-lines for the 
relative orientation which is not possible for the relative orientation of two images. Each triple of homologous lines 
gives two linear equations. The TFT is made of 27 elements and so it can be computed linearly up to scale (since only 
homogenous equations are used) using > 7 point-triples or > 13 line-triples or combinations by means of a linear least- 
squares-adjustment minimising algebraic-error. Certain arrangements of the TFT's elements form matrices with 
interesting geometric properties, which can be used to compute the images IOR and XOR out of the tensor. 
1 INTRODUCTION 
The basic requirement for doing object reconstruction with a set of photographs is image orientation; i.e. the estimation 
of the XOR and maybe the IOR of all the photographs. For some of these tasks the reference to a global system of 
coordinates ('absolute orientation') is either not necessary or done later: in other words, no or insufficient control 
features may be available. In such cases one works with the so-called ‘relative orientation’. This is the alignment of at 
least two images in such a way that homologous projection rays intersect each other in a point in space. 
The relative orientation of images is determined using only the observed image coordinates which are subject to 
accidental errors. To decrease the error's disturbing influence on the estimated unknowns an adjustment is done during 
which the sum of squares of the errors is minimized. For this, two statistically well-founded models exist: the Gauss- 
Markoff-model, in which each observation can be expressed in terms of the unknowns (aka ‘adjustment by indirect 
observations’), and the Gauss-Helmert-model, in which only combinations of the observations can be expressed in 
terms of the unknowns (aka ‘general case of least squares adjustment’). 
The unknowns which are estimated in these two models are distinguished by the following properties: They are 
unbiased and have least variance (‘best unbiased estimation’), the sum of the squared discrepancies is minimized 
(‘least-squares-estimation’) and in the case of observations having a normal distribution they are a ‘maximum- 
likelihood-estimation’;[Koch 1987]. In general, one works with a least-squares-estimation. However, it is important to 
point out that such a least-squares-estimation is a best unbiased estimation only if the sum of the squares of the original 
observations’ errors (so-called ‘measurement-error’ or ‘reprojection-error’) is minimized and not the sum of squares of 
some other quantities (so-called ‘algebraic-error’). 
For such an adjustment, however, linear equations are required. Since the equations of the central projection are non- 
linear, they have to be linearized, and for this approximate values of the XOR (and under certain circumstances also for 
the IOR) of the photographs are necessary, but in many cases the determination of these approximate values is quite 
tedious. 
In the relatively young discipline of computer vision central perspective images are also worked with. Due to the highly 
non-linear character of the central-projection formulated in terms of XOR and IOR in computer vision a linear 
fepresentation for the central perspective relation between object and image is aimed for. This linear representation is 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 769