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Fundamental Difficulties With Projective Normalization of Planar Curves

Åström, Karl LU (1993) Second Joint European - US Workshop Applications of Invariance in Computer Vision In Applications of Invariance in Computer Vision. Second Joint European - US Workshop Proceedings 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)
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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
in
Applications of Invariance in Computer Vision. Second Joint European - US Workshop Proceedings
pages
199 - 214
publisher
Springer
conference name
Second Joint European - US Workshop Applications of Invariance in Computer Vision
ISBN
3 540 58240 1
language
English
LU publication?
yes
id
a64fcc4b-30bc-49f4-9774-b4d2c7250b55 (old id 787631)
date added to LUP
2008-03-31 15:31:15
date last changed
2016-04-16 10:06:27
@misc{a64fcc4b-30bc-49f4-9774-b4d2c7250b55,
  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 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},
  author       = {Åström, Karl},
  isbn         = {3 540 58240 1},
  keyword      = {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    = {ARRAY(0x88c5818)},
  series       = {Applications of Invariance in Computer Vision. Second Joint European - US Workshop Proceedings},
  title        = {Fundamental Difficulties With Projective Normalization of Planar Curves},
  year         = {1993},
}