Fundamental difficulties with projective normalization of planar curves
Åström, Kalle LU
(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)
- author
- Åström, Kalle
LU
- organization
- publishing date
- 1994
- 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
- 978-3-540-58240-3
- 978-3-540-48583-4
- 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
- 2024-01-13 03:29:17
@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 = {{978-3-540-58240-3}},
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}},
}