Simultaneous Invariance with Respect to Translation, Rotation and Scaling

  • Hermann Haken
Part of the Springer Series in Synergetics book series (SSSYN, volume 50)


In this chapter we wish to show how pattern recognition can be made invariant with respect to the above-mentioned transformations. To this end we write the test pattern vectors q and prototype pattern vectors υ k by means of their components and replace the discrete index j by the Cartesian coordinates x and y
$${q_j} \to q(x,y),{\upsilon _{kj}} \to {\upsilon _k}(x,y),$$
where q and υ represent for instance the value of grey tones at the position x,y.


Test Pattern Synaptic Strength Variational Derivative Pattern Vector Adjoint Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. G. Baumgartner, E. Peterhaus, R. von der Heydt: In Computational Systems — Natural and Artificial, ed. by H. Haken, Springer Ser. Syn. Vol. 38 ( Springer, Berlin, Heidelberg 1987 )Google Scholar
  2. H. Haken, J.A.S. Kelso, H. Bunz: Biol. Cybern. 51, 347–356 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  3. D.O. Hebb: The Organization of Behavior. A Neuropsychological Theory (Wiley, New York 1948) B. Julesz: Foundations of Cyclopean Perception (University of Chicago Press, Chicago 1971) S.M. Kosslyn: In Cognition and Categorization, ed. by E. Rosch, B.B. Lloyd (Hillsdale, NJ 1978 ) S.M. Kosslyn: Image and Mind ( MIT Press, Cambridge, MA 1980 )Google Scholar
  4. S.M. Kosslyn: Psychological Review 88, 46–66 (1981)CrossRefGoogle Scholar
  5. S.M. Kosslyn: Ghosts in the Mind’s Machine: Creating and Using Images in the Brain ( Norton, New York 1983 )Google Scholar
  6. S.M. Kosslyn, T.M. Ball, B.J. Reiser: J. Exp. Psychol.: Human Perception and Performance 4, 47–60 (1978)CrossRefGoogle Scholar
  7. H. Land: Experiments in color vision, Sci. Am. 200 (1959)Google Scholar
  8. J. Rubner, K. Schulten: Biol. Cybern. 62, 193–199 (1990)CrossRefGoogle Scholar
  9. R.N. Shepard, T. Metzler: Science 171, 701–703 (1971)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Hermann Haken
    • 1
  1. 1.Institut für Theoretische Physik und SynergetikUniversität StuttgartStuttgartGermany

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