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.
TopIntroduction
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.
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
(1) The norm of
u is defined as the Euclidean norm
(2) in order to relate to the energy of the signal.
The inner product of two signals u and v is defined as
(3) where
vi* denotes conjugate of the
vi.