Fundamental difficulties with projective normalization of planar curves
(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.199214 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 nonalgebraic, both convex and nonconvex, 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
 19931009  19931014
 external identifiers

 scopus:84984826281
 ISSN
 03029743
 16113349
 ISBN
 9783540582403
 9783540485834
 DOI
 10.1007/3540582401_11
 language
 English
 LU publication?
 yes
 id
 a64fcc4b30bc49f49774b4d2c7250b55 (old id 787631)
 date added to LUP
 20160404 12:05:20
 date last changed
 20240113 03:29:17
@inproceedings{a64fcc4b30bc49f49774b4d2c7250b55, 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 nonalgebraic, both convex and nonconvex, 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 = {{9783540582403}}, issn = {{03029743}}, keywords = {{computational geometry; computer vision; projective normalization; planar curves; projective invariants; continuous affine invariants; Hausdorff metric; compactness; projectively equivalent curves; uniqueness}}, language = {{eng}}, pages = {{199214}}, 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/10.1007/3540582401_11}}, doi = {{10.1007/3540582401_11}}, volume = {{825 LNCS}}, year = {{1994}}, }