Modified Gabor Wavelets for Image Decomposition and Perfect Reconstruction

This article presents a scheme for image decomposition and perfect reconstruction based on Gabor wavelets. Gabor functions have been used extensively in areas related to the human visual system due to their localization in space and bandlimited properties. However, since the standard two-sided Gabor functions are not orthogonal and lead to nearly singular Gabor matrices, they have been used in the decomposition, feature extraction, and tracking of images rather than in image reconstruction. In an attempt to reduce the singularity of the Gabor matrix and produce reliable image reconstruction, in this article, the authors used single-sided Gabor functions. Their experiments revealed that the modified Gabor functions can accomplish perfect reconstruction. DOI: 10.4018/jcini.2009062302 IGI PUBLISHING This paper appears in the publication, International Journal of Cognitive Informatics and Natural Intelligence, Volume 3, Issue 4 edited by Yingxu Wang © 2009, IGI Global 701 E. Chocolate Avenue, Hershey PA 17033-1240, USA Tel: 717/533-8845; Fax 717/533-8661; URL-http://www.igi-global.com ITJ 5459 20 International Journal of Cognitive Informatics and Natural Intelligence, 3(4), 19-33, October-December 2009 Copyright © 2009, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. reducing the bandwidth required to transmit signals (Gabor, 1946; Gabor, 1947). Since then, the Gabor function has been used in different areas of research such as image texture analysis (Porat, 1989; du Buf, 1991), image segmentation (Billings, 1976; Bochum, 1999), motion estimation (Magarey, 1998), image analysis (Daugman, 1988), signal processing (Qiu, 1997; Bastiaans, 1981), and face authentication (Duc, 1999). It should be noted that most of those areas rely on analysis and feature extraction, and not reconstruction. In 1977, Cowan proposed that since visual mechanisms are indeed effectively bandlimited and localized in space, Gabor functions are suitable for their representation (Cowan, 1973). Other studies (Marcelja, 1980; Kulikowski, 1982; Pollen, 1985; Jones, 1987) assert that Gabor functions also well represent the characteristics of simple cortical cells, and present a viable model for such cells. Investigating “what does eye see best”, Watson et al. have demonstrated that the pattern of two-dimensional Gabor functions is optimal (Watson, 1983). The main difficulty with the Gabor functions is that they are not orthogonal. Therefore, they do not have a perfect reconstruction condition and no straightforward technique is available to extract the coefficients. However, there has been some attempt for reconstruct images with an acceptable accuracy. For example, Wundrich (Wundrich et. al, 2002) developed an iterative method to reconstruct images from the magnitude of the Gabor wavelet. It is shown that the image can be detected after 1300 iteration. In this article, we present an analytical approach to overcome the difficulty with Gabor functions, and demonstrate their usefulness in the decomposition and reconstruction of still images. We show that an image decomposed using modified Gabor wavelets can be reconstructed perfectly. The Gabor Decomposition Notation A one-dimensional (1D) signal u = [u1, u2,...,uN] T is considered as an N-element complex column vector. Such signals are also considered as N-periodic sequences over integers Z. If the kth coordinate of u is expressed as either uk or u(k), we have u i kN u i ( ) ( ) + = . (1) The norm of u is defined as the Euclidean norm u = æ è ççç ö ø ÷÷÷ = å ui i N 2