ABSTRACT
The transform of Gabor is a powerful means of representation of an image. It can be used in an effective way for the analysis, the segmentation and the compression of image. However, the principal problem during its use is that no simple method allowing calculation of the coefficients of Gabor. In this study, like proposed an algorithm based on the FFT; however also show that there are a clear reduction of complexity and thus an increased convergence. At least, a whole of examples are given illustrating present approach.
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DOI: 10.3923/jas.2006.94.97
URL: https://scialert.net/abstract/?doi=jas.2006.94.97
INTRODUCTION
The transform of Gabor was in the beginning formulated by Gabor[1]. Gabor suggested making local the analysis of Fourier, while making use of windows. The whole of these transforms of Fourier located, form the transform of Gabor of a signal. Thus, it provides a local frequential analysis[1]. The coefficients of Gabor locate the frequential distribution of a signal or image. They also permit to represent or to approach better a signal or image. Bastiaans[2] determines the analytical expressions of the coefficients of Gabor. A network of neurons was presented by Daugman[3] finding the coefficients optimal of Gabor of an image. It proves that this method does not allow a satisfactory rebuilding of the image of origin. Gertner and Zeevi[4] use the transform of Zak to calculate the coefficients of an image and the auxiliary function window. This technique is very much used but it presents a too significant load of calculation. Yao[5] used a matrix of Toplitz for the calculation of the coefficients; this requires carrying out inversions of matrix. Teuner and Hosticka[6] present an adaptive filter with a complex algorithm of the least square LMS. But the filter is stable under certain conditions.
Our proposal in this article is to present a new approach for the calculation of the coefficients of Gabor. Like Wang et al., calculation is done via the FFT, however the approach which we carry out, who combines the discrete transform of Fourier[7] with the application of the auxiliary function window[2] allows a significant reduction of the complexity of the algorithm.
TRANSFORM OF GABOR
We seek to develop a sequence finished by a discrete representation of Gabor. That is say to:
(1) |
Where, the coefficients of Gabor are:
(2) |
m, n are integer and N.W = 2π the condition of sampling criticizes[8].
The expression h(k-mN)ejnWk represents the function window of Gabor of order (m, n).
g*(k) is the complex combined of the auxiliary function window deduced from the condition of biorthogonality[8] following:
(3) |
Where, h*(k) the complex combined of h(k) and δk constant of Kronecker defined by:
δk = 1 for k = 0 and δk = 0 for k ≠ 0.
The evaluation of g(k) consists in finding solutions in the condition of biorthogonality (3).
ALGORITHM OF THE DISCRETE TRANSFORM OF GABOR WITH ONE DIMENSION
The numerical implementation requires the truncation of the K-addition in (2) and (3).
M and N are integer satisfying the equality:
N1 = N.M and x(k) a discrete sequence length N1. The discrete representation of Gabor x(k) is:
(4) |
(5) |
Where, the auxiliary function g(k) is the solution of the following equation:
(6) |
Wexler and Raz[9] transform the equation (6) in form matrix:
(7) |
With
And H is a matrix N1*N1 whose elements are blocks of Hankel N*N defined by the functions window with order (m, n).
G is thus the solution of (7) defined by:
(8) |
When the auxiliary function is evaluated, we calculate the coefficients of Gabor by applying the following algorithm:
The discrete transform of Fourier of the sequence x(k)g*(k-mN) is:
(9) |
While comparing (5) and (9), we notice that Amr = amn when r = n.M for n = 0,1, N-1.
To calculate the coefficients of Gabor of a finished sequence x(k), it is enough to calculate the discrete transform of Fourier of the sequence x(k)g*(k-mN) at the points r = nM with n = 0,1, .N-1.
If NOp represent the number of operations carried out by the FFT, the number of operations carried out by this algorithm is M*NOp+M*N1 multiplications. However the number of operations carried out by the algorithm proposed by Wang et al.[10] is M*NOp+M*N1 multiplications+(M-1)*N1 additions
ALGORITHM OF THE DISCRETE TRANSFORM OF GABOR WITH TWO DIMENSIONS
That is to say a digital image of dimensionsN1l.
The discrete representation of Gabor of x(k,l) is:
(10) |
The coefficients of Gabor are calculated by the following expression:
(11) |
where, mk, ml, nk, nl and Nk, Nl, Mk, Ml are positive integers with:
NIk = Mk.Nk, N1l = Ml.Nl |
The functions h(k,l) and g(k,l) must satisfy the following equality:
(12) |
If the function 'window' is separable, h(k,l) = hk(k)hl(l), the auxiliary function is also separable.
As in the preceding case, we apply the algorithm of the discrete transform of Fourier to 2 dimensions to rebuild the image of origin.
The number of operations carried out to calculate these coefficients is Mk*Ml*NOp2 FFT with two dimensions + Mk*Ml*N1k*N1l multiplications. Knowing that NOp2 is the number of operations carried out by the FFT with two dimensions. On the other hand the algorithm presented[10] requires a number of operation equal to Mk*Ml*NOp2 FFT with two dimensions + Mk*Ml*N1k*N1l multiplications + (Mk+Ml-1)*Nlk*N1l additions.
RESULTS
We calculate initially the auxiliary function corresponding to the rectangular window. We notice that they are identical (Fig. 1).
Fig. 1: | Profile of rectangular window and its auxiliary h (k) and g (k) for M = 2, N = 32 and Nl= 64 |
Fig. 2: | Profile of Gaussian window and its auxiliary h (k) and g (k) for M = 16, N = 16 and Nl = 256 |
Fig. 3: | Profile of Gaussian window with two dimension |
Fig. 4: | Profile of auxiliary Gaussian window g(k, l) |
Fig. 5: | Image fruit |
Fig. 6: | Image fruit rebuilt |
Fig. 7: | Image with the coefficients of Gabor |
If the function 'window' of Gabor is Gaussian:
(13) |
Auxiliary function is[2]:
(14) |
Where, Ko = 1.854076 is a constant of standardization (Fig. 2). The function window and its auxiliary are supposed to be separable.
h(k,l) = h(k).h(l) and g(k,l) = g(k).g(l) |
Figure 3 and 4 show the profile of these functions. Figure 5 and 6, respectively show the image of origin and the image rebuilt with the coefficients of Gabor.
In Fig. 7 we show that the image of origin can be rebuilt with some coefficients of Gabor. We thus notice, whom the coefficients of Gabor which we calculated allow a faithful rebuilding of the image of origin.
CONCLUSION
This study calculated the coefficients of Gabor using the direct formulation of the auxiliary function. We could reduce the complexity of calculations. A comparison with study of Wang et al.[10] finished showing the interest of the algorithm presented in this research.
REFERENCES
- Bastiaans, M., 1981. A sampling theorem for the complex spectrogram and Gabor expansion of a signal into Gaussian elementary signals. Opt. Eng., 20: 591-598.
Direct Link - Daugman, J., 1988. Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression. IEEE Trans. Acoust. Speech, Signal Process., 36: 1169-1179.
CrossRef - Yao, J., 1994. Complete Gabor transform for signal representation. IEEE Trans. Image Process., 2: 152-159.
CrossRef - Daugman, J., 1985. Incertainty relationship for resolution in space, spatial frequency and orientation J. Opt., 2: 1160-1169.
CrossRef - Wang, L., C. Chen and W. Lin, 1994. An eficient algorithm to compute the complete set of discrete gabor coefficients. IEEE Trans. Image Process., 3: 87-92.
CrossRef - Teuner, A. and B.J. Hosticka, 1993. Adaptative gabor transformation for image processing. IEEE Trans. Image Process., 2: 112-117.
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