Gabor transform

We describe the Zak-Gabor transform for a single stripe and a chosen fixed scale parameter D. Let M be a number of translations of g, and K be the number of frequency shifts where we always take M = P/K. The Gabor elementary functions (GEF), are defined as shifted and modulated versions of g, namely... 

C2] Chinen, Troy T. and Todd R. Reed, A Performance Analysis of Fast Gabor Transform Methods, Graphical Models And Image Processing, Vol. 59, No. 3, pp. 117-127, May 1997... 

CWT might be interpreted as a generalization of the Gabor transformation, which in turn is the time resolved extension of Fourier transformation. For a detailed discussion of CWT basics please refer to the comprehensive literature [2, 4, 8]. 

Given the limitations of the FT, some approximations are needed to handle nonstationary signals. The discrete FT (DFT) and the short-time FT (STFT, a.k.a. the Gabor transform) are two alternative transformation methods that address this issue. 3 jn the mid-20th century, Jean Ville pointed out that... 

The second exponential function in Eq. (29) is used in ordinary Fourier transforms, where no window is applied. In the Gabor wavelet transform a window around / = t is superposed by the first part of Eq. (29) (i.e., by the error function). On the other hand, in the classical Gabor transform, not the Gabor wavelet transform, a windowed Fourier transform with a fixed width window is applied, as illustrated in Fig. 39b. 

Upper left panel of Fig. 6 shows the evolution of the average energy of the field free system E = Tr[(Tr V r-))ps t)] during the process and the upper right panel gives Tr[pl(t)] which is a measure of the decoherence of the quantum system. Lower panels of Fig. 6 show the Gabor transform of the field indicating when the frequencies appear in time... 

E = Tr[ Tr. + V(r ))ps(f)] of the field free system, right upper panel, measure of the purity of the system Tr[pj t)]. Lower panels. Gabor transform (Eq. 18) of the field showing when the frequencies operate during the process... 

The final chapter in Volume 69 is concerned with the synthesis, stereochemistry, and transformation of cyclopentane-, cyclohexane-, cyclohep-tane-, and cyclooctane-fused 1,3-oxazines, 1,3-thiazines, and pyrimidines and is authored by Professors Ferenc Ftilop, Gabor Bernath, and Kalevi Pihlaja from the Universities of Szeged in Hungary and Turku in Finland. This is a field which has shown rapid development over the last dozen years because of the increased availability of spectroscopic and other analytical methods allowing definition of the precise steric chemistry of these compounds. 

Zak-Gabor coefficients calculation and transformation into a features vector 0.06... 

Gabor s expansion coefficients can be recovered from the product form (8). In fact, application of the discrete Zak transform (5) to both sides of (3) yields... 

C3] Bastiaans, Martin J., Gabor s Expansion and the Zak Transform for Continuous-Time and Discrete-Time Signals, w Josh Zeevi and Ronald Coifman (ed.), Signal and Image Representation in Combined Spaces, pp. 1—43, Academic Press, Inc., 1995... 

Wavelet analysis takes Gabor s idea one step further it defines a windowing transform technique with variably sized window regions. The continuous wavelet transform of the sequence h(t) is defined by Equation 10.23... 

R. Carmona, W.-L. Hwang, and B Torresani. Practical Time-Frequency Analysis. Gabor and Wavelet Transforms with an Implementation in S. Acar demic Press, 1998. 

A robust detection system should be able to provide both frequency and spatial information. While frequency serves to identify the presence of a defect, spatial information is necessary to identify the location of defect. It is well known that the Fourier transform suffers from a lack of spatial information while Gabor filtering methods are very computational expensive since the 2D convolution between the image and filter must be carried out in a sliding window throughout the entire image. These limitations could be eliminated by wavelets. 

Gabor wavelet transform, illustrated in Fig. 39a, is one of the continuous transforms, in which the following wavelet, F((z — x)/d), is used ... 

Gabor wavelet Signal from an acceleration sensor Continuous transform with a Gabor wavelet [cf. Eq. (29)] applied to a commercial scale AFBC Fukayama et al. (2000)... 

Zhu, J., Vai, M.I. and Mak, P.U. (2004) Gabor wavelets transform and extended nearest feature space classifier for face recognition. In Proceedings of the Third International Conference on Image and Graphics, Hong Kong, pp. 246-249. 

Each couple of obtained contrast-enhanced images Imi, Imi and wi, Iw2 is separately preprocessed by a 2D Matched Gaussian Kernel Filtering [10] and a 2D Gabor Wavelet Transformation [16]. 

D-Gabor Wavelet Transformation. The contrast in retinal images is herein enhanced by means of a 2D Gabor wavelet transformation [16]. A continuous wavelet transformation, P, (b, 6, s), is determined by the scalar product of a generic image with the transformed wavelet y/h.e,s... 

There are other variants of Gabor functions some use different envelope functions, or vary the envelope width at different positions. They also have various other names, such as short-time or windowed Fourier transforms. Most technical details concerning these functions have been omitted here, but can be found in a basic text. ... 

---