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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 798080, 17 pages
Research Article

Integral Equation-Wavelet Collocation Method for Geometric Transformation and Application to Image Processing

1Department of Computer and Information Science, Faculty of Science and Technology, University of Macau, Macau
2Department of Mathematics and Computer Science, Guangxi Normal University of Nationalities, Chongzuo 532200, China
3College of Computer Science, Chongqing University, Chongging 40030, China
4Department of Mathematics, Xidian University, Xi’an 710126, China
5Sichuan Sunray Machinery Co., Ltd., Deyang 618000, China

Received 28 August 2013; Accepted 13 February 2014; Published 1 April 2014

Academic Editor: M. Mursaleen

Copyright © 2014 Lina Yang et al. 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.


Geometric (or shape) distortion may occur in the data acquisition phase in information systems, and it can be characterized by geometric transformation model. Once the distorted image is approximated by a certain geometric transformation model, we can apply its inverse transformation to remove the distortion for the geometric restoration. Consequently, finding a mathematical form to approximate the distorted image plays a key role in the restoration. A harmonic transformation cannot be described by any fixed functions in mathematics. In fact, it is represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the geometric restoration. In this paper, a novel wavelet-based method is presented, which consists of three phases. In phase 1, the partial differential equation is converted into boundary integral equation and representation by an indirect method. In phase 2, the boundary integral equation and representation are changed to plane integral equation and representation by boundary measure formula. In phase 3, the plane integral equation and representation are then solved by a method we call wavelet collocation. The performance of our method is evaluated by numerical experiments.