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Projective Area-invariants as an extension of the Cross-Ratio

Nielsen, Lars and Sparr, Gunnar LU (1991) In Graphical Models 54(1). p.145-159
Abstract
Projective invariants provide a framework for computer vision where the image of an object is described by its intrinsic properties, independently of the particular view. It is advantageous if these intrinsic properties are defined in terms of computationally simple features. An area-measurement provides a good candidate that is easy to reliably compute from a particular image of the object. The main contributions of this paper are the definition and justification of area-invariants in projective geometry and the indication of its relevance in image analysis. A framework that covers one-dimensional intervals and two-dimensional figures has been developed. In the linear case, the invariants are linear only in two cases. The first case is... (More)
Projective invariants provide a framework for computer vision where the image of an object is described by its intrinsic properties, independently of the particular view. It is advantageous if these intrinsic properties are defined in terms of computationally simple features. An area-measurement provides a good candidate that is easy to reliably compute from a particular image of the object. The main contributions of this paper are the definition and justification of area-invariants in projective geometry and the indication of its relevance in image analysis. A framework that covers one-dimensional intervals and two-dimensional figures has been developed. In the linear case, the invariants are linear only in two cases. The first case is the well known cross-ratio, and the second case is called the polar case. The generalization to the plane can be done in different directions. One can use either points (on the line or in the plane) or the geometric figures (intervals, triangles, circles) as the basic entities involved. The first view was adopted already by Möbius, who generalized the cross-ratio in various directions. The second view used here leads to another generalization of the cross-ratio, where the invariants are relations between the areas of a class of geometric figures, related to each other in a certain manner. Remarkably enough, these invariants turn out to be linear if the figures involved are related in a pole/ polar configuration (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Graphical Models
volume
54
issue
1
pages
145 - 159
publisher
Elsevier
external identifiers
  • scopus:0026358093
ISSN
1524-0703
DOI
10.1016/1049-9660(91)90079-5
language
English
LU publication?
yes
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f64fa2ea-1295-407b-81a6-2823c89b73d7 (old id 788248)
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http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WDD-4DX42XN-P-1&_cdi=6764&_user=745831&_orig=search&_coverDate=07%2F31%2F1991&_sk=999459998&view=c&wchp=dGLbVtb-zSkWA&md5=f0316ea4abbb5288c2db5b125aefd38c&ie=/sdarticle.pdf
date added to LUP
2008-04-28 11:54:42
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2017-08-27 03:50:55
@article{f64fa2ea-1295-407b-81a6-2823c89b73d7,
  abstract     = {Projective invariants provide a framework for computer vision where the image of an object is described by its intrinsic properties, independently of the particular view. It is advantageous if these intrinsic properties are defined in terms of computationally simple features. An area-measurement provides a good candidate that is easy to reliably compute from a particular image of the object. The main contributions of this paper are the definition and justification of area-invariants in projective geometry and the indication of its relevance in image analysis. A framework that covers one-dimensional intervals and two-dimensional figures has been developed. In the linear case, the invariants are linear only in two cases. The first case is the well known cross-ratio, and the second case is called the polar case. The generalization to the plane can be done in different directions. One can use either points (on the line or in the plane) or the geometric figures (intervals, triangles, circles) as the basic entities involved. The first view was adopted already by Möbius, who generalized the cross-ratio in various directions. The second view used here leads to another generalization of the cross-ratio, where the invariants are relations between the areas of a class of geometric figures, related to each other in a certain manner. Remarkably enough, these invariants turn out to be linear if the figures involved are related in a pole/ polar configuration},
  author       = {Nielsen, Lars and Sparr, Gunnar},
  issn         = {1524-0703},
  language     = {eng},
  number       = {1},
  pages        = {145--159},
  publisher    = {Elsevier},
  series       = {Graphical Models},
  title        = {Projective Area-invariants as an extension of the Cross-Ratio},
  url          = {http://dx.doi.org/10.1016/1049-9660(91)90079-5},
  volume       = {54},
  year         = {1991},
}