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On the Behavior of Spatial Critical Points under Gaussian Blurring A Folklore Theorem and Scale-Space Constraints

  • Marco Loog
  • Johannes JisseDuistermaat
  • Luc M. J. Florack
Conference paper
Part of the Lecture Notes in Computer Science 2106 book series (LNCS, volume 2106)

Abstract

The main theorem we present is a version of a “Folklore Theorem” from scale-space theory for nonnegative compactly supported functions from ℝn to ℝ. The theorem states that, if we take the scale in scale-space sufficiently large, the Gaussian-blurred function has only one spatial critical extremum, a maximum, and no other critical points.

Two other interesting results concerning nonnegative compactly supported functions, we obtain are
  1. 1.

    a sharp estimate, in terms of the radius of the support, of the scale after which the set of critical points consists of a single maximum;

     
  2. 2.

    all critical points reside in the convex closure of the support of the function

     

These results show, for example, that all catastrophes take place within a certain compact domain determined by the support of the initial function and the estimate mentioned in 1.

To illustrate that the restriction of nonnegativity and compact support cannot be dropped, we give some examples of functions that fail to satisfy the theorem, when at least one assumption is dropped.

Keywords and Phrases

Large-scale behavior loss of detail nonnegative function compact support spatial critical point deep structure. 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Marco Loog
    • 1
  • Johannes JisseDuistermaat
    • 2
  • Luc M. J. Florack
    • 2
  1. 1.Image Sciences InstituteUniversity Medical CenterUtrechtThe Netherlands
  2. 2.Department of MathematicsUtrecht UniversityUtrechtThe Netherlands

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