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Scaling theorems for zero crossings.

A L Yuille1, T A Poggio

  • 1Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139.

IEEE Transactions on Pattern Analysis and Machine Intelligence
|August 27, 2011
PubMed
Summary
This summary is machine-generated.

The Gaussian filter is the only linear filter that prevents new zero crossings in scaled signals. This finding applies to various differential operators, including image intensity ridges and ravines.

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Area of Science:

  • Computer Vision
  • Image Processing
  • Signal Analysis

Background:

  • Zero crossings in filtered signals are crucial for feature detection.
  • Previous work explored scale-space properties of signal derivatives.

Purpose of the Study:

  • To analyze zero crossings of the Laplacian of linear filter-applied signals.
  • To determine which filters avoid creating new zero crossings as scale increases.

Main Methods:

  • Mathematical analysis of signal filtering with linear filters.
  • Investigation of zero crossing properties as a function of filter scale.
  • Generalization to level crossings of linear differential operators.

Main Results:

  • The Gaussian filter is uniquely identified as the only filter that does not generate generic zero crossings with increasing scale.
  • This property extends to level crossings of linear differential operators, including image intensity ridges and ravines.
  • For the second derivative along the gradient, zero crossings are unavoidable unless filtering follows differentiation.

Conclusions:

  • The Gaussian filter's scale-space behavior is unique in its stability regarding zero crossings.
  • Understanding these properties is vital for robust feature detection in image analysis.
  • The findings offer insights into the mathematical foundations of scale-space theory.