Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 837426, 11 pages
Research Article

The Nonlocal -Laplacian Evolution for Image Interpolation

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Received 24 June 2011; Accepted 16 August 2011

Academic Editor: P. Liatsis

Copyright © 2011 Yi Zhan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper presents an image interpolation model with nonlocal -Laplacian regularization. The nonlocal -Laplacian regularization overcomes the drawback of the partial differential equation (PDE) proposed by Belahmidi and Guichard (2004) that image density diffuses in the directions pointed by local gradient. The grey values of images diffuse along image feature direction not gradient direction under the control of the proposed model, that is, minimal smoothing in the directions across the image features and maximal smoothing in the directions along the image features. The total regularizer combines the advantages of nonlocal -Laplacian regularization and total variation (TV) regularization (preserving discontinuities and 1D image structures). The derived model efficiently reconstructs the real image, leading to a natural interpolation, with reduced blurring and staircase artifacts. We present experimental results that prove the potential and efficacy of the method.