Some “Weberized” \(L^2\)-Based Methods of Signal/Image Approximation

  • Ilona A. Kowalik-Urbaniak
  • Davide La Torre
  • Edward R. VrscayEmail author
  • Zhou Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8814)


We examine two approaches of modifying \(L^2\)-based approximations so that they conform to Weber’s model of perception, i.e., higher/lower tolerance of deviation for higher/lower intensity levels. The first approach involves the idea of intensity-weighted \(L^2\) distances. We arrive at a natural weighting function that is shown to conform to Weber’s model. The resulting “Weberized \(L^2\) distance” involves a ratio of functions. The importance of ratios in such distance functions leads to a consideration of the well-known logarithmic \(L^2\) distance which is also shown to conform to Weber’s model.

In fact, we show that the imposition of a condition of perceptual invariance in greyscale space \({\mathbb {R}}_g \subset {\mathbb {R}}\) according to Weber’s model leads to the unique (unnormalized) measure in \({\mathbb {R}}_g\) with density function \(\rho (t)=1/t\). This result implies that the logarithmic \(L^1\) distance is the most natural “Weberized” image metric. From this result, all other logarithmic \(L^p\) distances may be viewed as generalizations.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Ilona A. Kowalik-Urbaniak
    • 1
  • Davide La Torre
    • 2
    • 3
  • Edward R. Vrscay
    • 1
    Email author
  • Zhou Wang
    • 4
  1. 1.Department of Applied Mathematics, Faculty of MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of Applied Mathematics and SciencesKhalifa UniversityAbu DhabiUnited Arab Emirates
  3. 3.Department of Economics, Management and Quantitative MethodsUniversity of MilanMilanItaly
  4. 4.Department of Electrical and Computer Engineering, Faculty of EngineeringUniversity of WaterlooWaterlooCanada

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