Modified Gabor Wavelets for Image Decomposition and Perfect Reconstruction

Modified Gabor Wavelets for Image Decomposition and Perfect Reconstruction

Reza Fazel-Rezai (University of North Dakota, USA) and Witold Kinsner (University of Manitoba, Canada)
DOI: 10.4018/jcini.2009062302
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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.
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Since its first formulation in 1984 (Grossmann, 1984), the wavelet transform has become a common tool in signal processing, in that, it describes a signal at different levels of detail in a compact and readily interpretable form (Daubechies, 1992). Wavelet theory provides a unified framework for a number of techniques which had been developed independently for various signal processing applications (Rioul, 1991). Different views of signal theory include multiresolution signal processing as used in computer vision, subband coding as developed for speech and image compression, and wavelet series expansion as developed in applied mathematics (Mallat, 1999).

In general, wavelets can be categorized in two types: real-valued and complex-valued. There are some benefits in using complex-valued wavelets. Gabor wavelet is one of the most widely used complex wavelets. More than a half of century ago, Gabor developed a system for 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.

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