Article

Image Processing Done Right

Authors:

Jan J. Koenderink,

Andrea J. van DoornAuthors Info & Claims

ECCV '02: Proceedings of the 7th European Conference on Computer Vision-Part I

Pages 158 - 172

Published: 28 May 2002 Publication History

Abstract

A large part of "image processing" involves the computation of significant points, curves and areas ("features"). These can be defined as loci where absolute differential invariants of the image assume fiducial values, taking spatial scale and intensity (in a generic sense) scale into account. "Differential invariance" implies a group of "similarities" or "congruences". These "motions" define the geometrical structure of image space. Classical Euclidian invariants don't apply to images because image space is non-Euclidian. We analyze image structure from first principles and construct the fundamental group of image space motions. Image space is a Cayley-Klein geometry with one isotropic dimension. The analysis leads to a principled definition of "features" and the operators that define them.

References

[1]

Adams, A.: The Print: Contact Printing and Enlarging . Basic Photo 3 . Morgan and Lester, New York, 1950.

[2]

Cayley, A.: Sixth Memoir upon Quantics. Phil. Trans. Roy. Soc. Lond. 149 (1859) 61-70.

[3]

Clifford, W. K.: Preliminary sketch of biquaternions. Proc. Lond. Math. Soc. (1873) 381-395.

[4]

Florack, L.: Image Structure . Kluwer, Dordrecht, 1997.

[5]

Gauss, C. F.: Algemeine Flächentheorie . (German translation of Disquisitiones generales circa Superficies Curvas ), Hrsg. A. Wangerin, Ostwalds Klassiker der exakten Wissenschaften 5 . Engelmann, Leipzig, 1889 (orig. 1827).

[6]

Ginneken, B. van, Haar Romeny, B. M. ter: Applications of locally orderless Images. In: Scale-Space Theories in Computer Vision . Eds. M. Nielsen, P. Johansen, O. F. Olsen and J. Weickert. Second Int. Conf. Scale-Space'99, Lect. Notes in Computer Science 1682 . Springer, Berlin (1999) 10-21.

[7]

Jaglom, I. M.: A simple non-Euclidian geometry and its physical basis: an elementary account of Galilean geometry and the Galilean principle of relativity . Transl. A. Shenitzer, ed. ass. B. Gordon. Springer, New York, 1979.

[8]

Jaglom, I. M.: Complex numbers in geometry . Transl. E. J. F. Primrose. Academic Paperbacks, New York, 1968.

[9]

Jaynes, E. T.: Prior Probabilities. IEEE Trans. on Systems Science and Cybernetics 4 (3) (1968) 227-241.

[10]

Klein, F.: Über die sogenannte nichteuklidische Geometrie. Mathematische Annalen Bd. 6 (1871) 112-145.

[11]

Klein, F.: Vergleichende Betrachtungen über neue geometrische Forschungen (the "Erlangen Program"). Programm zu Entritt in die philosophische Fakultät und den Senat der Universität zu Erlangen. Deichert, Erlangen, 1872.

[12]

Lindeberg, T.: Scale-Space Theory in Computer Vision . Kluwer, Dordrecht, 1994.

[13]

Pottmann, H., Opitz, K.: Curvature analysis and visualization for functions defined on Euclidian spaces or surfaces. Computer Aided Geometric Design 11 (1994) 655- 674.

[14]

Sachs, H.: Ebene Isotrope Geometrie . Vieweg, Braunschweig/Wiesbaden, 1987.

[15]

Sachs, H.: Isotrope Geometrie des Raumes . Vieweg, Braunschweig/Wiesbaden, 1990.

[16]

Scheffer, G.: Verallgemeinerung der Grundlagen der gewöhnlichen komplexen Funktionen. Sitz. ber. Sächs. Ges. Wiss., Math.-phys. Klasse, Bnd. 45 (1893) 828-842.

[17]

Strubecker, K.: Differentialgeometrie des isotropen Raumes I. Sitzungsberichte der Akademie der Wissenschaften Wien 150 (1941) 1-43.

[18]

Strubecker, K.: Differentialgeometrie des isotropen Raumes II. Math. Z. 47 (1942) 743-777.

[19]

Strubecker, K.: Differentialgeometrie des isotropen Raumes III. Math. Z. 48 (1943) 369-427.

[20]

Strubecker, K.: Differentialgeometrie des isotropen Raumes IV. Math. Z. 50 (1945) 1-92.

Cited By

Dalstein BRonfard Rvan de Panne M(2015)Vector graphics animation with time-varying topologyACM Transactions on Graphics10.1145/276691334:4(1-12)Online publication date: 27-Jul-2015
https://dl.acm.org/doi/10.1145/2766913
Oxholm GNishino K(2010)Membrane nonrigid image registrationProceedings of the 11th European conference on Computer vision: Part II10.5555/1888028.1888086(763-776)Online publication date: 5-Sep-2010
https://dl.acm.org/doi/10.5555/1888028.1888086
Griffin L(2009)Symmetries of 2-D ImagesJournal of Mathematical Imaging and Vision10.1007/s10851-009-0148-z34:3(259-269)Online publication date: 1-Jul-2009
https://dl.acm.org/doi/10.1007/s10851-009-0148-z
Show More Cited By

Image Processing Done Right
1. Computing methodologies
  1. Artificial intelligence
  2. Computer graphics
    1. Shape modeling
2. Theory of computation
  1. Randomness, geometry and discrete structures

Recommendations

Enhancing large image database indexing and retrieval performance through integrating database structure with image features

In this paper, a two-dimensional hash table structure based on image discriminator measure is proposed to design an image database indexing and retrieval system. Unlike many other techniques such as R-tree, FastMap, MetricMap and SparseMap where a false ...
P---M equation based multiscale decomposition and its application to image fusion

P---M equation proposed by Perona and Malik can not only perform scale-space, but also preserve edges while smoothing an image. In this paper, we employ this property to construct a new multiscale decomposition method, by which an image can be decomposed ...
Image processing in the context of imaging science
ICIP '95: Proceedings of the 1995 International Conference on Image Processing (Vol.2)-Volume 2 - Volume 2

The rate of discovery in the natural sciences has been accelerated by powerful computer-based imaging methods, which enable scientists to map certain object properties into an 'image space'. These methods also support the goals of medicine, education, ...

Comments

Information & Contributors

Information

Published In

cover image Guide Proceedings

ECCV '02: Proceedings of the 7th European Conference on Computer Vision-Part I

May 2002

810 pages

ISBN:3540437452

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 28 May 2002

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

11
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 11 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Dalstein BRonfard Rvan de Panne M(2015)Vector graphics animation with time-varying topologyACM Transactions on Graphics10.1145/276691334:4(1-12)Online publication date: 27-Jul-2015
https://dl.acm.org/doi/10.1145/2766913
Oxholm GNishino K(2010)Membrane nonrigid image registrationProceedings of the 11th European conference on Computer vision: Part II10.5555/1888028.1888086(763-776)Online publication date: 5-Sep-2010
https://dl.acm.org/doi/10.5555/1888028.1888086
Griffin L(2009)Symmetries of 2-D ImagesJournal of Mathematical Imaging and Vision10.1007/s10851-009-0148-z34:3(259-269)Online publication date: 1-Jul-2009
https://dl.acm.org/doi/10.1007/s10851-009-0148-z
Kuijper A(2008)Exploring and exploiting the structure of saddle points in Gaussian scale spaceComputer Vision and Image Understanding10.1016/j.cviu.2008.06.001112:3(337-349)Online publication date: 1-Dec-2008
https://dl.acm.org/doi/10.1016/j.cviu.2008.06.001
Pottmann HLiu Y(2007)Discrete surfaces in isotropic geometryProceedings of the 12th IMA international conference on Mathematics of surfaces XII10.5555/1770873.1770894(341-363)Online publication date: 4-Sep-2007
https://dl.acm.org/doi/10.5555/1770873.1770894
Griffin L(2007)The Second Order Local-Image-Structure SolidIEEE Transactions on Pattern Analysis and Machine Intelligence10.1109/TPAMI.2007.106629:8(1355-1366)Online publication date: 1-Aug-2007
https://dl.acm.org/doi/10.1109/TPAMI.2007.1066
Phillips CSiddiqi K(2006)Greyscale photograph geometry informed by dodging and burningProceedings of the 5th Indian conference on Computer Vision, Graphics and Image Processing10.1007/11949619_2(13-24)Online publication date: 13-Dec-2006
https://dl.acm.org/doi/10.1007/11949619_2
Georgiev T(2006)Covariant derivatives and visionProceedings of the 9th European conference on Computer Vision - Volume Part IV10.1007/11744085_5(56-69)Online publication date: 7-May-2006
https://dl.acm.org/doi/10.1007/11744085_5
Bauckhage CTsotsos J(2005)Image space I and eigen curvature for illumination insensitive face detectionProceedings of the Second international conference on Image Analysis and Recognition10.1007/11559573_57(456-463)Online publication date: 28-Sep-2005
https://dl.acm.org/doi/10.1007/11559573_57
Bauckhage CTsotsos J(2005)Separable linear classifiers for online learning in appearance based object detectionProceedings of the 11th international conference on Computer Analysis of Images and Patterns10.1007/11556121_43(347-354)Online publication date: 5-Sep-2005
https://dl.acm.org/doi/10.1007/11556121_43
Show More Cited By

View Options

View options

Figures

Tables

Media

View Table of Conten