WIAS Preprint No. 2542, (2018)

Analysis and optimisation of a variational model for mixed Gaussian and Salt & Pepper noise removal


  • Calatroni, Luca
  • Papafitsoros, Kostas
    ORCID: 0000-0001-9691-4576

2010 Mathematics Subject Classification

  • 94A08 49K20 49J20 49J40 49Q20 65J20 34K29


  • Mixed noise removal, denoising, total variation, exact solutions, Gaussian noise, Salt and Pepper noise, bilevel optimisation, parameter learning




We analyse a variational regularisation problem for mixed noise removal that was recently proposed in [14]. The data discrepancy term of the model combines L1 and L2 terms in an infimal convolution fashion and it is appropriate for the joint removal of Gaussian and Salt & Pepper noise. In this work we perform a finer analysis of the model which emphasises on the balancing effect of the two parameters appearing in the discrepancy term. Namely, we study the asymptotic behaviour of the model for large and small values of these parameters and we compare it to the corresponding variational models with L1 and L2 data fidelity. Furthermore, we compute exact solutions for simple data functions taking the total variation as regulariser. Using these theoretical results, we then analytically study a bilevel optimisation strategy for automatically selecting the parameters of the model by means of a training set. Finally, we report some numerical results on the selection of the optimal noise model via such strategy which confirm the validity of our analysis and the use of popular data models in the case of "blind'' model selection.

Appeared in

  • Inverse Probl., 35 (2019), pp. 114001/1-114001/37, DOI 10.1088/1361-6420/ab291a with the new title Analysis and automatic parameter selection of a variational model for mixed Gaussian and salt-and-pepper noise removal

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