International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

# Invited Talk

### Image processing and non-local continuous models

E. Cuesta and A. Durán

Abstract
In this talk non-local PDE's models of Volterra type [1] $(1) {u}(t) = {u}_0 + \int_0^t {K}(t-s)A{u}(t) ds, t\ge 0,$ are considered in the context of image processing. In (1) ${u}_0$ stands for the original image vector-arranged, ${u}(t)$ stands for the evolved ${u}_0$ at time stage $t$, $A$ is a linear operator (typically a discrete 2D-Laplacian), and ${K}(t)$ is a convenient convolution kernel. The models (1) are linear, and well-posedness and stability are guaranteed under very general conditions on ${K}(t)$. The talk will discuss the adaptation of (1) to image processing problems. It will focus on scale-space properties, the preservation of relevant quantities, and the comparison with local nonlinear PDE-based models recently proposed in the literature [2]. Acknowledgement: The authors have been supported by MICINN project MTM2010-19510/MTM.

Bibliography
[1] E. Cuesta, M. Kirane and S. A. Malik, Image structure preserving denoising using generalized fractional time integrals, Signal Processing 92 (2012) 553-563.
[2] P. Guidotti and J. V. Lambers, Two new nonlinear nonlocal diffusions for noise reduction, J. Math. Imaging Vis. 33 (2009) 25-37.

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