Tutorial: Dealing with rotation matrices and translation vectors in image-based applications: A common reference system for cameras
Author(s)
Date Issued
2019
Type
article
Volume
34
Issue
2
Start Page
54
End Page
68
Abstract
This tutorial follows and completes a previous work concern-
ing how to solve the factors of ambiguity intrinsic in the defini-
tion of rotation matrices, to describe the relative orientation of
one or more cameras acquiring images from different points of
view. The two works, jointly, aim to provide an error-avoiding
methodology to describe the whole relative acquisition geome-
try (pose). This is a crucial task for those aerospace applications
based on the use of imaging devices, like flying objects detection
and tracking, automatic guidance, tridimensional reconstruc-
tion, images georegistration, and change detection. To describe
the pose, not only the orientation but also the relative shift
should be taken into account. This can be conveniently modeled
through a translation vector. Using it together with a rotation
matrix it is possible to achieve the goal, but it is also prone to
ambiguity. In this contribution, the main factors that result in
such ambiguity are addressed. Through a detailed analysis, it
is shown how to solve them, in order to manage the interac-
tion between translation vectors and rotation matrices prop-
erly. This avoids the errors that frequently occur in practical
applications, whenever it is required to find the transformation
that makes the reference system of a camera coincident with the
reference system of another, or to switch from the expression of
the coordinates of a point of the scene in the reference system
of a camera to its expression in the reference system of another.
One of the reasons why errors are likely to be made in describ-
ing the pose is the lack of a commonly adopted choice for the
camera reference system. To obviate this lack, the manuscript
also presents a recommended definition for it. For the sake of
completeness, the definition includes a second reference system
for identifying points on the camera sensor plane, in order to permit describing also the projective transformation operated by the camera during the image formation process.