Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Fundamental difficulties with projective normalization of planar curves

Åström, Kalle LU orcid (1994) Second Joint European - US Workshop Applications of Invariance in Computer Vision In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 825 LNCS. p.199-214
Abstract

In this paper projective normalization and projective invariants of planar curves are discussed. It is shown that there exists continuous affine invariants. It is shown that many curves can be projected arbitrarily close to a circle in a strengthened Hausdorff metric. This does not infer any limitations on projective invariants, but it is clear that projective normalization by maximizing compactness is unsuitable. It is also shown that arbitrarily close to each of a finite number of closed planar curves there is one member of a set of projectively equivalent curves. Thus there can not exist continuous projective invariants, and a projective normalisation scheme can not have both the properties of continuity and uniqueness. Although... (More)

In this paper projective normalization and projective invariants of planar curves are discussed. It is shown that there exists continuous affine invariants. It is shown that many curves can be projected arbitrarily close to a circle in a strengthened Hausdorff metric. This does not infer any limitations on projective invariants, but it is clear that projective normalization by maximizing compactness is unsuitable. It is also shown that arbitrarily close to each of a finite number of closed planar curves there is one member of a set of projectively equivalent curves. Thus there can not exist continuous projective invariants, and a projective normalisation scheme can not have both the properties of continuity and uniqueness. Although uniqueness might be preferred it is not essential for recognition. This is illustrated with an example of a projective normalization scheme for non-algebraic, both convex and non-convex, curves.

(Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
keywords
computational geometry, computer vision, projective normalization, planar curves, projective invariants, continuous affine invariants, Hausdorff metric, compactness, projectively equivalent curves, uniqueness
host publication
Applications of Invariance in Computer Vision - 2nd Joint European - US Workshop, Proceedings
series title
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
editor
Mundy, Joseph L. ; Zisserman, Andrew and Forsyth, David
volume
825 LNCS
pages
16 pages
publisher
Springer
conference name
Second Joint European - US Workshop Applications of Invariance in Computer Vision
conference location
Ponta Delgada, Azores, Portugal
conference dates
1993-10-09 - 1993-10-14
external identifiers
  • scopus:84984826281
ISSN
0302-9743
1611-3349
ISBN
978-3-540-48583-4
978-3-540-58240-3
DOI
10.1007/3-540-58240-1_11
language
English
LU publication?
yes
id
a64fcc4b-30bc-49f4-9774-b4d2c7250b55 (old id 787631)
date added to LUP
2016-04-04 12:05:20
date last changed
2021-01-03 07:46:58
@inproceedings{a64fcc4b-30bc-49f4-9774-b4d2c7250b55,
  abstract     = {{<p>In this paper projective normalization and projective invariants of planar curves are discussed. It is shown that there exists continuous affine invariants. It is shown that many curves can be projected arbitrarily close to a circle in a strengthened Hausdorff metric. This does not infer any limitations on projective invariants, but it is clear that projective normalization by maximizing compactness is unsuitable. It is also shown that arbitrarily close to each of a finite number of closed planar curves there is one member of a set of projectively equivalent curves. Thus there can not exist continuous projective invariants, and a projective normalisation scheme can not have both the properties of continuity and uniqueness. Although uniqueness might be preferred it is not essential for recognition. This is illustrated with an example of a projective normalization scheme for non-algebraic, both convex and non-convex, curves.</p>}},
  author       = {{Åström, Kalle}},
  booktitle    = {{Applications of Invariance in Computer Vision - 2nd Joint European - US Workshop, Proceedings}},
  editor       = {{Mundy, Joseph L. and Zisserman, Andrew and Forsyth, David}},
  isbn         = {{978-3-540-48583-4}},
  issn         = {{0302-9743}},
  keywords     = {{computational geometry; computer vision; projective normalization; planar curves; projective invariants; continuous affine invariants; Hausdorff metric; compactness; projectively equivalent curves; uniqueness}},
  language     = {{eng}},
  pages        = {{199--214}},
  publisher    = {{Springer}},
  series       = {{Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)}},
  title        = {{Fundamental difficulties with projective normalization of planar curves}},
  url          = {{http://dx.doi.org/10.1007/3-540-58240-1_11}},
  doi          = {{10.1007/3-540-58240-1_11}},
  volume       = {{825 LNCS}},
  year         = {{1994}},
}