APPROXIMATION OF NON PROJECTIVE MAPPING AND THEIR EFFICIENT
APPLICATION FOR A GEOMETRICALLY BASED 3D POINT DETERMINATION USING
MULTIPLE VIEWS
Kirsten Wolff
Institute of Geodesy and Photogrammetry
ETH Hoenggerberg
CH-8049 Zurich, Switzerland
wolff@geod.baug.ethz.ch
Commission III, WG III/8
KEY WORDS: Orientation, Geometric, Distortion, Matching, Underwater
ABSTRACT:
In this paper an unusual taxonomy for optical mappings is introduced based on their geometric characteristics: 1. type of projection
center (single viewpoint, non single viewpoint) and 2. type of transformation (projective, non projective). Under this background we
survey the multi media geometry (refraction resulting from different optical media). Strict physical models of non projective mappings
can be very complex in dependency on their geometric nature. Therefor different methods for reducing the complexity exist. This paper
describes a method of ascertaining a virtual camera to approximate non projective mappings by a projective model and their application
for a 3D point determination using multiple views with non projective multi media geometry. As will be seen, the approximation can be
used without loosing the quality of the strict model significantly. For the matching process we introduce a new algorithm for multiple
views based on geometric constrains alone which uses all images simultaneously.
1 INTRODUCTION Table 1: Classification of imaging systems
1.1 Motivation Class| Mapping Viewpoint Imaging System Modeling
distortion
The nature of an optical mapping process between a 3D object 1 Projective | Single Pinhole -
space and a 2D image space depends on the geometry of the Viewpoint
imaging system, its physical laws and the scene structure. 2 Non Single Wide-angle cam- | based on
Projective Viewpoint | eras, fish-eye | position
In this context we use the term imaging system instead of camera cameras, _ Cen- | in image
system, because it should contains all parameters, which have an ral atadiopiric space
5 . ; E ; E cameras, Approxi-
effect on the nature of the optical mapping. In particular the effect mation of objective
on the way of mapping light-rays, influenced by light refraction distortion...
or reflection. 3 Non Non- Wide-angle cam- | based on
; : ; Projective Single cras, fish-eye | position
Based on the different nature of optical mappings and their char- View- cameras , catadiop- | in object
acteristic, several kinds of classifications exist (e.g in (Hartley points tric cameras, camera space.
and Zisserman, 2003) whether they have a finite centre or a cen- clusters, moving |
tre “at infinity”, or whether they preserve straight lines or not). cameras, multi-
media geometry,
objective distortion
An unusual feature for a classification of mappings is the kind of
image distortion, resulting from light refraction or reflexion of the
mapping rays. Therefore we introduce a classification of imag-
ing systems, based on the following geometric features, which
influence the characteristics of image distortions:
mappings in (Wolff and Fórstner, 2001) and on the taxonomy of
distortions published in (Swaminathan et al., 2003).
: T. ERE : 1. Class | is the perspective mapping, also named pinhole cam-
Geometrical characteristics of optical mappings: ; s Jans : :
era. It is the most specialized and simplest model, where
the straight projecting rays intersect in a single viewpoint
(the pinhole) and preserves straight lines. This results in
no image distortions (not taking distortions into account,
which result from the perspective mapping). All cameras
modelling central projection are specialisations of the gen-
eral projective camera, therefore we use the term projective
mapping which could be presented by a projective model.
|. Single Viewpoint (SVP) or Non Single Viewpoint
(NSVP)
mapping rays intersect in one single point or not
2. Invariance or variance of straight lines
straight lines in the scene appear as straight lines in
the 2D image space (projective mapping) or appear as
curves s : ;
Every deviation from this model causes image distortion.
From the combination of these geometric features, we get a c/as- 2. Class 2 is created by mappings with single viewpoints, not
sification of imaging systems with three different classes, sum- preserving straight lines. However, this leads to image dis-
marised in table 1. lt is based on the classification of optical tortions, which depends on the image position. Its influence
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