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Fundamental difficulties with projective normalization of planar curves

På svenska

Author

Editor

  • Joseph L. Mundy
  • Andrew Zisserman
  • David Forsyth

Summary, in English

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.

Publishing year

1994

Language

English

Pages

199-214

Publication/Series

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volume

825 LNCS

Document type

Conference paper

Publisher

Springer

Topic

  • Mathematics

Keywords

  • computational geometry
  • computer vision
  • projective normalization
  • planar curves
  • projective invariants
  • continuous affine invariants
  • Hausdorff metric
  • compactness
  • projectively equivalent curves
  • uniqueness

Conference name

Second Joint European - US Workshop Applications of Invariance in Computer Vision

Conference date

1993-10-09 - 1993-10-14

Conference place

Ponta Delgada, Azores, Portugal

Status

Published

ISBN/ISSN/Other

  • ISSN: 1611-3349
  • ISSN: 0302-9743
  • ISBN: 978-3-540-48583-4
  • ISBN: 978-3-540-58240-3