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Recognition of Planar Point Configurations using the Density of Affine Shape

Berthilsson, Rikard LU and Heyden, Anders LU (1998) Computer Vision - ECCV'98 5th European Conference on Computer Vision In [Host publication title missing] 1. p.72-88
Abstract
We study the statistical theory of shape for ordered finite point configurations, or otherwise stated, the uncertainty of geometric invariants. Such studies have been made for affine invariants, where a bound on errors is used instead of errors described by density functions, and a first-order approximation gives an ellipsis as uncertainty region. Here, a general approach for defining shape and finding its density, expressed in the densities for the individual points, is developed. No approximations are made, resulting in an exact expression of the uncertainty region. Similar results have been obtained for the special case of the density of the cross ratio. In particular, we concentrate on the affine shape, where often analytical... (More)
We study the statistical theory of shape for ordered finite point configurations, or otherwise stated, the uncertainty of geometric invariants. Such studies have been made for affine invariants, where a bound on errors is used instead of errors described by density functions, and a first-order approximation gives an ellipsis as uncertainty region. Here, a general approach for defining shape and finding its density, expressed in the densities for the individual points, is developed. No approximations are made, resulting in an exact expression of the uncertainty region. Similar results have been obtained for the special case of the density of the cross ratio. In particular, we concentrate on the affine shape, where often analytical computations are possible. In this case confidence intervals for invariants can be obtained from a priori assumptions on the densities of the detected points in the images. However, the theory is completely general and can be used to compute the density of any invariant (Euclidean, similarity, projective etc.) from arbitrary densities of the individual points. These confidence intervals can be used in such applications as geometrical hashing, recognition of ordered point configurations and error analysis of reconstruction algorithms. Finally, an example is given, illustrating an application of the theory for the problem of recognising planar point configurations from images taken by an affine camera. This case is of particular importance in applications where details on a conveyor belt are captured by a camera, with image plane parallel to the conveyor belt and extracted feature points from the images are used to sort the objects (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, error analysis, feature extraction, image recognition, image reconstruction, statistical analysis
in
[Host publication title missing]
volume
1
pages
72 - 88
publisher
Springer
conference name
Computer Vision - ECCV'98 5th European Conference on Computer Vision
external identifiers
  • Scopus:84957632590
ISBN
3 540 64569 1
language
English
LU publication?
yes
id
12347625-7e2a-4d26-a829-0be14e99f824 (old id 787214)
date added to LUP
2008-03-31 16:01:01
date last changed
2016-10-13 04:43:06
@misc{12347625-7e2a-4d26-a829-0be14e99f824,
  abstract     = {We study the statistical theory of shape for ordered finite point configurations, or otherwise stated, the uncertainty of geometric invariants. Such studies have been made for affine invariants, where a bound on errors is used instead of errors described by density functions, and a first-order approximation gives an ellipsis as uncertainty region. Here, a general approach for defining shape and finding its density, expressed in the densities for the individual points, is developed. No approximations are made, resulting in an exact expression of the uncertainty region. Similar results have been obtained for the special case of the density of the cross ratio. In particular, we concentrate on the affine shape, where often analytical computations are possible. In this case confidence intervals for invariants can be obtained from a priori assumptions on the densities of the detected points in the images. However, the theory is completely general and can be used to compute the density of any invariant (Euclidean, similarity, projective etc.) from arbitrary densities of the individual points. These confidence intervals can be used in such applications as geometrical hashing, recognition of ordered point configurations and error analysis of reconstruction algorithms. Finally, an example is given, illustrating an application of the theory for the problem of recognising planar point configurations from images taken by an affine camera. This case is of particular importance in applications where details on a conveyor belt are captured by a camera, with image plane parallel to the conveyor belt and extracted feature points from the images are used to sort the objects},
  author       = {Berthilsson, Rikard and Heyden, Anders},
  isbn         = {3 540 64569 1},
  keyword      = {computational geometry,computer vision,error analysis,feature extraction,image recognition,image reconstruction,statistical analysis},
  language     = {eng},
  pages        = {72--88},
  publisher    = {ARRAY(0xb537cf0)},
  series       = {[Host publication title missing]},
  title        = {Recognition of Planar Point Configurations using the Density of Affine Shape},
  volume       = {1},
  year         = {1998},
}