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ISPRS Commission III, Vol.34, Part 3A Photogrammetric Computer Vision“, Graz, 2002 
A MINIMAL SET OF CONSTRAINTS AND A MINIMAL 
PARAMETERIZATION FOR THE TRIFOCAL TENSOR 
C. Ressl 
Institute of Photogrammetry and Remote Sensing 
University of Technology, Vienna, Austria 
car@ipf.tuwien.ac.at 
KEY WORDS: Mathematics, Algorithms, Modelling, Orientation, Calibration, Theory 
ABSTRACT: 
The topic of this paper is the so-called trifocal tensor (TFT), which describes the relative orientation of three 
uncalibrated images. The TFT is made up of 27 homogenous elements but only has 18 DOF. Therefore, its 
elements have to fulfil 8 constraints - a new form for these constraints is presented in this paper. Furthermore, a 
new minimal parameterization for the TFT is presented having exactly 18 DOF and which is generally applicable 
for any arrangement of the three images - provided not all three projection centers coincide. Constraints and 
parameterization are found using the so-called correlation slices. 
1 INTRODUCTION 
The trifocal tensor (TFT) allows a linear formulation for 
the relative orientation of three uncalibrated images. So it 
basically plays the same role for three images as the funda- 
mental matrix [Loung, Faugeras 1996] plays for two. The 
TFT has been subject of much research in the past ten 
years. [Spetsakis, Aloimonos 1990] were the first to dis- 
cover redundancies within the contents of three calibrated 
images. For uncalibrated images [Shashua 1995| showed 
that 27 coefficients and one homologous triple of points in 
three views form together nine homogenous linear equa- 
tions (four of them being independent), which he called 
trilinearities, since they consist of products of three image 
coordinates and one of the 27 coefficients. Furthermore 
[Hartley 1994] showed that Shashua's 27 coefficients and 
a homologous triple of lines create two homogenous linear 
equations. Therefore, the TFT can be linearly computed 
using at least 7 points or 13 lines or a proper combination. 
He also proposed for this set of 3 x 3 x 3 coefficients the 
term trifocal tensor. 
Although the TFT is made up of 27 elements, it only has 
18 degrees of freedom (DOF): 3x11 DOF /image - 15 (abso- 
lute projective orientation). Therefore the TF'T's elements 
must fulfil 9 constraints (one of them is the fixation of the 
TFT's scale due to the scale ambiguity in the homogenous 
trilinear relations). If these constraints are neglected, er- 
rors in the image data used to compute the TFT might be 
absorbed by the redundant parameters, yielding a perhaps 
severely disturbed TFT - and thus a wrong image orien- 
tation. So, in the past few years attempts were made to 
derive algorithms that return a valid trifocal tensor. This 
can be done in two ways: a) by introducing the necessary 
number of constraints into the computation, or b) by using 
a minimal parameterization for the TFT having 18 DOF. 
For each of these methods some solutions were presented in 
the past: [Torr, Zisserman 1997], [Papadopoulo, Faugeras 
1998], [Canterakis 2000]. 
In this paper a new minimal set of constraints will be pre- 
sented. In that course we will also arrive at a new minimal 
parameterization for the TFT. The paper is organized in 
the following way: In section 2 the basic formulation of 
the TFT is presented. The properties of the tensor resp. 
of its slices will be summarized in section 3. After a sum- 
mary of the existing solutions in section 4, the new set 
of constraints is presented in section 5.2, followed by the 
minimal parameterization in section 6. 
1.1 Notation 
Geometric objects: 
object points: upper case Roman font, e.g. P 
image points: lower case Roman font, e.g. p 
special image lines: small Greek font, e.g. A 
general image line: 4 
Mathematical quantities: 
matrices: upper case Typewriter font, e.g. RT 
vectors: bold Roman font 
- euclidian/affine vectors, e.g. Oy, ei 
- projective vectors, e.g. Va1, V31, V31 
- X emphasizes, that x can be replaced 
by A- x anytime (A zZ 0); 
- € emphasizes, that the scale of X is 
determined by a specific relation; 
- X emphasizes, that the scale of X is 
fixed either by setting || x|| — 1 or 
max(X) = 1. 
scalars: lower case Roman or Greek font; e.g. u, u 
Special objects: 
Vzy epipole of image v, in image v; 
i.e. the mapping of O,, into image vx 
T21, Ta2, Ta3 the principal rays of image v; c.f. sec 2 
Tel, Ta2, Tez the principal planes of image v; 
The symbol ~ denotes equality up to scale. 
2 BASICS 
2.1 The central projection using homogenous co- 
ordinates 
The formulation of the image geometry and the underlying 
relations are based on the one used in [Ressl 2000]. The 
projective relation between a 3D object point P and its 
image point p can be represented in a very compact way 
using projective geometry. If the exterior orientation of 
an image v is given by the image’s projection center Oy 
and the rotation matrix Ry (from the image system to the 
object system), and if the interior orientation of the image 
is given by the principal point (xo yo), the principal dis- 
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