Advertisement

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)

Abstract

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Forte, B., Vrscay, E.R.: Solving the inverse problem for function and image approximation using iterated function systems. Dynamics of Continuous, Discrete and Impulsive Systems 1, 177–231 (1995)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Girod, B.: What’s wrong with mean squared error? In: Watson, A.B. (ed.) Digital Images and Human Vision. MIT Press, Cambridge (1993)Google Scholar
  3. 3.
    Lee, S., Pattichis, M.S., Bovik, A.C.: Foveated video quality assessment. IEEE Trans. Multimedia 4(1), 129–132 (2002)CrossRefGoogle Scholar
  4. 4.
    Oppenheim, A.V., Schafer, R.W., Stockham Jr, T.G.: Nonlinear filtering of multiplied and convolved signals. Proc. IEEE 56(8), 1264–1291 (1968)CrossRefGoogle Scholar
  5. 5.
    Shen, J.: On the foundations of vision modeling I. Weber’s law and Weberized TV restoration. Physica D 175, 241–251 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Shen, J., Jung, Y.-M.: Weberized Mumford-Shah model with Bose-Einstein photon noise. Appl. Math. Optim. 53, 331–358 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Wandell, B.A.: Foundations of Vision. Sinauer Publishers, Sunderland (1995)Google Scholar
  8. 8.
    Wang, Z., Bovik, A.C.: Mean squared error: Love it or leave it? A new look at signal fidelity measures. IEEE Sig. Proc. Mag. 26, 98–117 (2009)CrossRefGoogle Scholar
  9. 9.
    Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: From error visibility to structural similarity. IEEE Trans. Image Proc. 13(4), 600–612 (2004)CrossRefGoogle Scholar
  10. 10.
    Wang, Z., Li, Q.: Information content weighting for perceptual image quality assessment. IEEE Trans. Image Proc. 20(5), 1185–1198 (2011)CrossRefGoogle Scholar

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

Personalised recommendations