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A New Nonlocal Nonlinear Diffusion Equation for Data Analysis

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Abstract

In this paper we introduce and study a new feature-preserving nonlinear nonlocal diffusion equation for denoising signals. The proposed partial differential equation is based on a novel diffusivity coefficient that uses a nonlocal automatically detected parameter related to the local bounded variation and the local oscillating pattern of the noisy input signal. We provide a mathematical analysis of the existence of the solution in the two dimensional case, but easily extensible to the one-dimensional model. Finally, we show some numerical experiments, which demonstrate the effectiveness of the new approach.

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Acknowledgements

The authors are also extremely grateful to T. Nieus (UniMI) for providing them with data of simulated membrane potentials, and to F. Difato (IIT-Ge) for providing the recorded calcium signals.

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Authors and Affiliations

  1. Department of Environmental Science and Policy & ADAMSS Center, Università di Milano, Via Celoria 2, Milano, 20133, Italy

    Giacomo Aletti & Giovanni Naldi

  2. IIT Genova, Via Morego, 30, Genova, 16163, Italy

    Monica Moroni

Authors
  1. Giacomo Aletti
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  2. Monica Moroni
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  3. Giovanni Naldi
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Correspondence to Giovanni Naldi.

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This work was funded by ADAMSS Center, G. Naldi acknowledges the support of the Hausdorff Research Institute for Mathematics during the Special Trimester “Mathematics of Signal Processing”. G. Aletti and G. Naldi are members of “Gruppo Nazionale per il Calcolo Scientifico (GNCS)” of the INDAM.

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Aletti, G., Moroni, M. & Naldi, G. A New Nonlocal Nonlinear Diffusion Equation for Data Analysis. Acta Appl Math 168, 109–135 (2020). https://doi.org/10.1007/s10440-019-00281-1

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