Structural Similarity-Based Optimization Problems with \(L^1\)-Regularization: Smoothing Using Mollifiers

  • Daniel OteroEmail author
  • Davide La Torre
  • Edward R. Vrscay
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9164)


In this paper we propose a new method of solving optimization problems involving the structural similarity image quality measure with \(L^1\)-regularization. The regularization term \(\Vert x \Vert _1\) is approximated by a sequence of smooth functions \(\Vert x \Vert _1^\varepsilon \) by means of \(C^\infty _0\) functions known as mollifiers. Because the functions \(\Vert x \Vert _1^\varepsilon \) epi-converge to \(\Vert x \Vert _1\), the sequence of minimizers of the smooth objective functions converges to a minimizer of the non-smooth problem. This approach permits the use of gradient-based methods to solve the minimization problems as opposed to methods based on subdifferentials.


Discrete Cosine Transform Multivariate Gaussian Distribution Discrete Cosine Transform Coefficient Structural Similarity Index Measure Fidelity Term 
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.



We gratefully acknowledge that this research has been supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) in the form of a Discovery Grant (ERV).


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Daniel Otero
    • 1
    Email author
  • Davide La Torre
    • 2
    • 3
  • Edward R. Vrscay
    • 1
  1. 1.Department of Applied Mathematics, Faculty of MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of Economics, Management, and Quantitative MethodsUniversity of MilanMilanItaly
  3. 3.Department of Applied Mathematics and SciencesKhalifa UniversityAbu DhabiUAE

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