Tutorial: Dealing with rotation matrices and translation vectors in image-based applications: A tutorial
Author(s)
Date Issued
2019
Type
article
Volume
34
Issue
2
Start Page
38
End Page
53
Abstract
Rotation matrices are a convenient and intuitive way to de-
scribe algebraically the relative orientation of multiple cam-
eras or of the same camera shooting from different points of
view. However, the definition of a rotation matrix is prone to
intrinsic ambiguity, which often leads to a mismatch with the
physical rotation one wants to describe, even if the definition
is mathematically correct. This is a common source of errors
whenever it is required to compute a rotation matrix from
camera orientation data, or vice versa, to recover such data
from a given rotation matrix. This tutorial aims to describe
and solve the main factors that generate the ambiguity in us-
ing rotation matrices and to permit dealing with them properly
both in theory and in practice. Through a detailed analysis of
these factors, which ranges from basic mathematical aspects
to the notation used to refer to them, it is shown how to avoid
errors in the algebraic description of the relative orientation
of different cameras by means of rotation matrices. This work
is followed by another contribution, in which the interaction
between rotation matrices and translation vectors (used to de-
scribe the shifts between pairs of cameras) is also analyzed,
and a recommendation on how to define a common reference
system coherent with a camera (a crucial aspect to model the
camera acquisition geometry) is given. The two contributions
jointly embrace the entire description of the relative acquisi-
tion geometry of images taken from different points of view
and provide a complete and error-free methodology to recover
it or to extract useful data from it. This topic is particularly
important in a wide variety of aerospace applications, which
often rely on multiple imaging sensors whose information
should be merged, or on imaging devices carried by manned
or unmanned vehicles. Such applications range from flying ob-
ject detection to tridimensional reconstruction by using aerial
or satellite images to drone automatic navigation, to change
detection for area monitoring to georegistration by ground-to-
aerial image matching.