Skip to main content Accessibility help
×
Home
Hostname: page-component-56f9d74cfd-fv4mn Total loading time: 0.236 Render date: 2022-06-25T03:12:49.260Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

A NEW NONLOCAL NONLINEAR DIFFUSION EQUATION: THE ONE-DIMENSIONAL CASE

Published online by Cambridge University Press:  05 May 2022

G. ALETTI
Affiliation:
Environmental Science and Policy Department, Università degli Studi di Milano, 20133Milan, Italy e-mail: giacomo.aletti@unimi.it
A. BENFENATI
Affiliation:
Environmental Science and Policy Department, Università degli Studi di Milano, 20133Milan, Italy e-mail: alessandro.benfenati@unimi.it
G. NALDI*
Affiliation:
Advanced Applied Mathematical and Statistical Sciences Center, Università degli Studi di Milano, 20133Milan, Italy

Abstract

We prove a result on the existence and uniqueness of the solution of a new feature-preserving nonlinear nonlocal diffusion equation for signal denoising for the one-dimensional case. The 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.

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The three authors are members of the Italian Group GNCS of the Italian Institute ‘Istituto Nazionale di Alta Matematica’ and of the ADAMSS Center of the Università degli Studi di Milano (Italy).

References

Aletti, G., Lonardoni, D., Naldi, G. and Nieus, T., ‘From dynamics to links: a sparse reconstruction of the topology of a neural network’, Commun. Appl. Ind. Math. 10(2) (2019), 211.Google Scholar
Aletti, G., Moroni, M. and Naldi, G., ‘A new nonlocal nonlinear diffusion equation for data analysis’, Acta Appl. Math. 168(1) (2020), 109135.CrossRefGoogle Scholar
Alvarez, I., Guichard, F., Lions, P.-L. and Morel, J.-M., ‘Axioms and fundamental equations of image processing’, Arch. Ration. Mech. Anal. 123 (1993), 199257.CrossRefGoogle Scholar
Angenent, S., Pichon, E. and Tannenbaum, A., ‘Mathematical methods in medical image processing’, Bull. Amer. Math. Soc. (N.S.) 43 (2006), 365396.CrossRefGoogle ScholarPubMed
Benfenati, A. and Coscia, V., ‘Nonlinear microscale interactions in the kinetic theory of active particles’, Appl. Math. Lett. 26(10) (2013), 979983.CrossRefGoogle Scholar
Benfenati, A. and Coscia, V., ‘Modeling opinion formation in the kinetic theory of active particles I: spontaneous trend’, Ann. Univ. Ferrara 60 (2014), 3553.CrossRefGoogle Scholar
Bonsall, F. F., Lectures on Some Fixed Point Theorems of Functional Analysis (Tata Institute of Fundamental Research, Bombay, 1962).Google Scholar
Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext (Springer, New York, 2011).CrossRefGoogle Scholar
Evans, L. C., Partial Differential Equations, Graduate Studies in Mathematics, 19 (American Mathematical Society, Providence, RI, 1998).Google Scholar
Evans, L. C. and Gariepy, R. F., Measure Theory and Fine Properties of Functions, Textbooks in Mathematics (CRC Press, Boca Raton, 2015).CrossRefGoogle Scholar
Palazzolo, G., Moroni, M., Soloperto, A., Aletti, G., Naldi, G., Vassalli, M., Nieus, T. and Difato, F., ‘Fast wide-volume functional imaging of engineered in vitro brain tissues’, Sci. Rep. 7 (2017), Article no. 8499, 20 pages.CrossRefGoogle ScholarPubMed
Perona, P. and Malik, J., ‘Scale-space and edge detection using anisotropic diffusion’, IEEE Trans. Pattern Anal. Mach. Intell. 12 (1990), 629639.CrossRefGoogle Scholar
Piccinini, L. C., Stampacchia, G. and Vidossich, G., Ordinary Differential Equations in ${R}^n$: Problems and Methods, Applied Mathematical Sciences, 39 (Springer-Verlag, New York, 1984).Google Scholar
Sapiro, G., Geometric Partial Differential Equations and Image Analysis (Cambridge University Press, Cambridge, 2006).Google Scholar
Weickert, J., Anisotropic Diffusion in Image Processing, ECMI Series (B. G. Teubner, Stuttgart, 1998).Google Scholar

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A NEW NONLOCAL NONLINEAR DIFFUSION EQUATION: THE ONE-DIMENSIONAL CASE
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

A NEW NONLOCAL NONLINEAR DIFFUSION EQUATION: THE ONE-DIMENSIONAL CASE
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

A NEW NONLOCAL NONLINEAR DIFFUSION EQUATION: THE ONE-DIMENSIONAL CASE
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *