## Calculations on critical points under gaussian blurring (1999)

### Cached

### Download Links

- [give-lab.cs.uu.nl]
- [archive.cs.uu.nl]
- [www.cs.uu.nl]
- [www.cs.uu.nl]
- [www.doc.ic.ac.uk]
- [webdoc.sub.gwdg.de]
- DBLP

### Other Repositories/Bibliography

Venue: | In Nielsen et al |

Citations: | 10 - 8 self |

### BibTeX

@INPROCEEDINGS{Kuijper99calculationson,

author = {Arjan Kuijper and Luc Florack},

title = {Calculations on critical points under gaussian blurring},

booktitle = {In Nielsen et al},

year = {1999},

pages = {318--329},

publisher = {Springer -Verlag}

}

### OpenURL

### Abstract

Abstract. The behaviour of critical points of Gaussian scale-space images is mainly described by their creation and annihilation. In existing literature these events are determined in so-called canonical coordinates. A description in a user-defined Cartesian coordinate system is stated, as well as the results of a straightforward implementation. The location of a catastrophe can be predicted with subpixel accuracy. An example of an annihilation is given. Also an upper bound is derived for the area where critical points can be created. Experimental data of an MR, a CT, and an artificial noise image satisfy this result. 1

### Citations

660 |
The structure of images
- Koenderink
- 1984
(Show Context)
Citation Context ...he resulting structure has become known as linear Gaussian scale-space. In view of ample literature on the subject we will henceforth assume familiarity with the basics of Gaussian scale-space theory =-=[4,8,9,14,23,24,26,29,30]-=-. In their original accounts both Koenderink as well as Witkin proposed to investigate the “deep structure” of an image, i.e. structure at all levels of resolution simultaneously. Encouraged by the re... |

539 |
Scale-Space Filtering
- Witkin
- 1983
(Show Context)
Citation Context ...he resulting structure has become known as linear Gaussian scale-space. In view of ample literature on the subject we will henceforth assume familiarity with the basics of Gaussian scale-space theory =-=[4,8,9,14,23,24,26,29,30]-=-. In their original accounts both Koenderink as well as Witkin proposed to investigate the “deep structure” of an image, i.e. structure at all levels of resolution simultaneously. Encouraged by the re... |

140 |
Catastrophe Theory and its Applications
- Poston, Stewart
- 1996
(Show Context)
Citation Context ...ablish a generic underpinning of deep structure. Results from this could serve as a common basis for a diversity of multiresolution schemes. Such bottom-up approaches often rely on catastrophe theory =-=[1,6,25,27,28]-=-, which is in the context of the scale-space paradigm now fairly well-established. The application of catastrophe theory in Gaussian scale space has been studied e.g. by Damon [3]—probably the most co... |

133 |
Solid Shape
- Koenderink
- 1990
(Show Context)
Citation Context ...w fairly well-established. The application of catastrophe theory in Gaussian scale space has been studied e.g. by Damon [3]—probably the most comprehensive account on the subject—as well as by others =-=[7,10,11,12,13,15,16,17,18,19,20,21,22,23]-=- Closely related to the present article is the work by Florack and Kuijper [5], introducing new theoretical tools. We will summarise some results in section 2 and give an experimental verification of ... |

111 |
Singularity Theory I
- Arnold, Goryunov, et al.
- 1998
(Show Context)
Citation Context ...ablish a generic underpinning of deep structure. Results from this could serve as a common basis for a diversity of multiresolution schemes. Such bottom-up approaches often rely on catastrophe theory =-=[1,6,25,27,28]-=-, which is in the context of the scale-space paradigm now fairly well-established. The application of catastrophe theory in Gaussian scale space has been studied e.g. by Damon [3]—probably the most co... |

89 |
Catastrophe Theory for Scientists and Engineers. Dover books on advanced mathematics
- Gilmore
- 1993
(Show Context)
Citation Context ...ablish a generic underpinning of deep structure. Results from this could serve as a common basis for a diversity of multiresolution schemes. Such bottom-up approaches often rely on catastrophe theory =-=[1,6,25,27,28]-=-, which is in the context of the scale-space paradigm now fairly well-established. The application of catastrophe theory in Gaussian scale space has been studied e.g. by Damon [3]—probably the most co... |

81 |
Scale-space Theory in Computer Vision. The Kluwer
- Lindeberg
- 1994
(Show Context)
Citation Context ...he resulting structure has become known as linear Gaussian scale-space. In view of ample literature on the subject we will henceforth assume familiarity with the basics of Gaussian scale-space theory =-=[4,8,9,14,23,24,26,29,30]-=-. In their original accounts both Koenderink as well as Witkin proposed to investigate the “deep structure” of an image, i.e. structure at all levels of resolution simultaneously. Encouraged by the re... |

69 |
Stabilité Structurelle et Morphogenèse
- Thom
(Show Context)
Citation Context |

43 |
Local morse theory for solutions to the heat equation and gaussian blurring
- Damon
- 1995
(Show Context)
Citation Context ...he theory [1,6,25,27,28], which is in the context of the scale-space paradigm now fairly well-established. The application of catastrophe theory in Gaussian scale space has been studied e.g. by Damon =-=[3]-=-—probably the most comprehensive account on the subject—as well as by others [7,10,11,12,13,15,16,17,18,19,20,21,22,23] Closely related to the present article is the work by Florack and Kuijper [5], i... |

37 | The topological structure of scale-space images
- Florack, Kuijper
- 2000
(Show Context)
Citation Context ...on [3]—probably the most comprehensive account on the subject—as well as by others [7,10,11,12,13,15,16,17,18,19,20,21,22,23] Closely related to the present article is the work by Florack and Kuijper =-=[5]-=-, introducing new theoretical tools. We will summarise some results in section 2 and give an experimental verification of the theory on both real and artificial M. Nielsen et al. (Eds.): Scale-Space’9... |

36 |
Basic theory on normalization of a pattern (in case of typical one-dimensional pattern
- Iijima
- 1962
(Show Context)
Citation Context |

36 |
Scale-space behaviour of local extrema and blobs
- Lindeberg
- 1992
(Show Context)
Citation Context ...w fairly well-established. The application of catastrophe theory in Gaussian scale space has been studied e.g. by Damon [3]—probably the most comprehensive account on the subject—as well as by others =-=[7,10,11,12,13,15,16,17,18,19,20,21,22,23]-=- Closely related to the present article is the work by Florack and Kuijper [5], introducing new theoretical tools. We will summarise some results in section 2 and give an experimental verification of ... |

30 |
On the history of Gaussian scale-space axiomatics
- Weickert, Ishikawa, et al.
(Show Context)
Citation Context |

26 |
Image Structure, volume 10 of Computational Imaging and Vision
- Florack
- 1997
(Show Context)
Citation Context |

26 |
Superficial and deep structure in linear diffusion scale space: isophotes, critical points and separatrices
- Griffin, Colchester
- 1995
(Show Context)
Citation Context ...w fairly well-established. The application of catastrophe theory in Gaussian scale space has been studied e.g. by Damon [3]—probably the most comprehensive account on the subject—as well as by others =-=[7,10,11,12,13,15,16,17,18,19,20,21,22,23]-=- Closely related to the present article is the work by Florack and Kuijper [5], introducing new theoretical tools. We will summarise some results in section 2 and give an experimental verification of ... |

26 |
On the classification of toppoints in scale space
- Johansen
- 1994
(Show Context)
Citation Context |

22 |
Representing signals by their toppoints in scale-space
- Johansen, Skelboe, et al.
- 1986
(Show Context)
Citation Context |

22 |
ªTopological Numbers and Singularities in Scalar Images Scale-Space Evolution Properties,º Gaussian Scale-Space Theory
- Kalitzin
- 1997
(Show Context)
Citation Context |

21 |
A hitherto unnoticed singularity of scale-space
- Koenderink
- 1989
(Show Context)
Citation Context |

21 |
The Structure of TwoDimensional Scalar Fields with Applications to Vision
- Koenderink, Doorn
- 1979
(Show Context)
Citation Context |

14 |
Mathematical Studies on Feature Extraction in Pattern Recognition
- Otsu
- 1981
(Show Context)
Citation Context |

12 |
Spatial derivatives and the propagation of noise in Gaussian scale space
- Blom, Romeny, et al.
- 1993
(Show Context)
Citation Context ... σ) ∂xm ∂yn � def m+n =(−1) u(x ′ ,y ′ ) ∂m+nφ(x ′ − x, y ′ − y; σ) ∂x ′m dx ∂y ′n ′ dy ′ , where u(x, y) is the input image and φ(x, y; σ) a normalised Gaussian of scale σ. It has been shown by Blom =-=[2]-=- that we can take derivatives up to fourth order without problems with respect to the results, provided scale is somewhat larger than pixelscale. It is important to note that Eqs. (4–9) hold in any Ca... |

9 |
Local analysis of image scale space
- Johansen
- 1997
(Show Context)
Citation Context |

6 |
Structural Stability and Morphogenesis (translated by
- Thom
- 1975
(Show Context)
Citation Context |

5 |
The structure of the visual field
- Koenderink
(Show Context)
Citation Context |

5 |
On the behaviour in scale-space of local extrema and blobs
- Lindeberg
- 1992
(Show Context)
Citation Context |

1 |
A multiresolution hierarchical appraoch to image segmentation based on intensity extrema
- Lifshitz, Pizer
- 1990
(Show Context)
Citation Context |

1 | On the Behaviour of Critical Points under Gaussian Blurring Submitted to Scale-Space '99
- Florack, Kuijper
(Show Context)
Citation Context ...studied e.g. by Damon [3]---probably the most comprehensive account on the subject---as well as by others [7, 10--13, 15--22] Closely related to the present article is the work by Florack and Kuijper =-=[5]-=-, introducing new theoretical tools. We will summarise their results in section 2 and give an experimental verification of the theory on both real and artificial data sets in section 3. This verificat... |