Anomalous diffusion, dilation, and erosion in image processing

ABSTRACT

In this paper, anomalous sub- and super-diffusion arising in image processing is considered and is modelled by a diffusion equation with fractional time derivative. It might serve as a building block for the construction of various filters. The resulting partial differential equation is discretized in space with centred differences and in time with the explicit or implicit Euler method, respectively. A numerical investigation is performed to illustrate new and interesting results. Additionally, the time derivative of the partial differential equation describing dilation and erosion is replaced by a fractional time derivative and then solved numerically. Interesting new questions arise from the presented numerical results. A short summary and outlook conclude this article.

KEYWORDS:

• Anomalous diffusion
• fractional derivative
• Caputo
• image processing
• dilation
• erosion
• co-histogram

2010 AMS CLASSIFICATIONS:

• 34A08
• 35R11
• 68U10

Acknowledgments

The authors would like to thank the guest editors and the two anonymous referees for their valuable and detailed comments that led to a major improvement of the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Andreas Kleefeld  http://orcid.org/0000-0001-8324-821X