- Research Article
- Open Access
A New Denoising System for SONAR Images
© Alexandru Isar et al. 2009
- Received: 1 November 2008
- Accepted: 21 July 2009
- Published: 13 October 2009
The SONAR images are perturbed by speckle noise. The use of speckle reduction filters is necessary to optimize the image exploitation procedures. This paper presents a new denoising method in the wavelet domain, which tends to reduce the speckle, preserving the structural features and textural information of the scene. Shift-invariance associated with good directional selectivity is important for the use of a wavelet transform (WT) in many fields of image processing. Generally, complex wavelet transforms, for example, the Double Tree Complex Wavelet Transform (DT-CWT) have these useful properties. In this paper, we propose the use of the DT-CWT in association with Maximum A Posteriori (MAP) filters. Such systems carry out different quality denoising in regions with different homogeneity degree. We propose a solution for the reduction of this unwanted effect based on diversity enhancement. The corresponding denoising algorithm is simple and fast. Some simulation results prove the performance obtained.
- Discrete Wavelet Transform
- Wavelet Coefficient
- Synthetic Aperture Radar Image
- Image Denoising
- Speckle Noise
The first goal of this paper is the additive noise denoising kernel in Figure 1.
computation of a forward WT,
filtering with a nonlinear filter,
computation of the corresponding inverse wavelet transform (IWT).
They use the Discrete Wavelet Transform (DWT) and the soft-thresholding filter. They do not make any explicit hypothesis on the noise-free image. So, this method can be considered nonparametric. Their unique statistical hypothesis refers to the noise, considered additive white and Gaussian, (AWGN). The soft-thresholding filter is used to put to zero all the wavelet coefficients with the absolute value smaller than a threshold. This threshold is selected to minimize the min-max approximation error.
The soft-thresholding filter was enhanced in , where the same hypotheses concerning the noise-free image and the noise were considered.
Some relatively recent research has addressed the development of statistical models of wavelet coefficients of natural images and application of these models to image denoising [3–9]. The corresponding signal processing treatments can be considered as parametric denoising methods. An appealing particularity of the WTs is the interscale dependence. If at a given scale a coefficient is large, its correspondent at the next scale (having the same spatial coordinates) will be also large. The wavelet coefficients statistical models which exploit the dependence between coefficients give better results compared to the ones using an independent assumption [5, 6, 8, 9]. The denoising is performed in [5, 6] with the aid of maximum a posteriori (MAP) filters.
If we use the denotation for the wavelet coefficients of the noise-free image and for the wavelet coefficients of the noise then it can be written, The MAP estimation of , , realized using the observation is given by the following MAP filter equation:
where represents the probability density function (pdf) of . Generally, the above equation has no analytical solution. There are some exceptions. For example if both and are zero mean Gaussian distributed, with variances and then the MAP filter becomes the very well known zero-order Wiener filter . If the coordinates of the current pixel are ( ), then the input-output relation of the local zero-order Wiener filter is
The model of natural images is given by heavy tailed distributions, . So, the utilization of zero-order Wiener filters in image denoising applications cannot provide the best performance. This drawback can be partially compensated by a better estimation of the local variance of the noise-free image, realized with the aid of a two-stage denoising system. The first stage treats the acquired image providing a pilot for the second stage [10–13]. A very nice contribution of  is the idea of directional windows. The rectangular windows used for the estimation of the local variance of the clean image are replaced by elliptical windows oriented following the preferential direction of the current detail subimage. This is a first example of exploiting the intrascale dependence of the wavelet coefficients, mentioned in . Of course the intrascale dependence of the wavelet coefficients is an intrinsic property of a WT, but it can be exploited only if there is a model at hand to describe it.
If is Gaussian distributed and has a Laplacian distribution (this is a heavy tailed one), then the MAP filter becomes an adaptive soft thresholding filter . The local variant of this MAP filter is described by
The zero-order Wiener filter and the adaptive soft-thresholding filter are two examples of marginal MAP filters. If the models of the clean image and of the noise are bivariate distributions then the MAP filter can take into account the interscale dependence of the wavelet coefficients. This is the case of the bishrink filter . In this case the coefficient represents the parent of the coefficient ( is the wavelet coefficient at the same position as , but at the next coarser scale). Then, , and the vectors , , and can be constructed. We can write: . The noise is assumed i.i.d. Gaussian:
The model of the noise-free image proposed in  is
another heavy tailed distribution. The input-output relation of the bishrink filter is
This estimator requires prior knowledge of the noise variance and of the marginal variance of the clean image for each wavelet coefficient. To estimate the noise variance from the noisy wavelet coefficients, a robust median estimator from the finest scale wavelet coefficients is used :
In  the marginal variance of the th coefficient is estimated using neighboring coefficients in the region , a squared shaped window centered on this coefficient with size . To make this estimation one gets where represents the marginal variance of noisy observations and . For the estimation of the marginal variance of noisy observations, in  the following relation is proposed:
where is the size of the neighborhood . Then can be estimated as:
In , a similar technique is used, but the bivariate a priori pdf of the clean image of type is considered. Unfortunately in this case the MAP filter equation can not be solved analytically, some numerical methods being required. The advantage of an analytical solution of the MAP filter equation lies in a fast implementation (the numerical methods are avoided) and in the possibility to perform a sensitivity analysis.
The DWT has some drawbacks : the lack of shift invariance and the poor directional selectivity. These disadvantages can be diminished using a complex wavelet transform, like, for example, the DT-CWT . The MAP filters constructed in [5, 6] act in the field of the DT-CWT.
In  a method for removing noise from digital images is described, based on a statistical model of the coefficients of overcomplete multiscale oriented basis. This decomposition is named steerable pyramid. Following it, the image is subdivided into subbands using filters that are polar-separable in the Fourier domain. In scale, the subbands have octave bandwidth with a functional form constrained by a recursive system diagram. In orientation, the functional form is chosen so that the set of filters at a given scale span a rotation-invariant subspace. This decomposition can be considered as a WT with shift invariance and very good directional selectivity. Neighborhoods of coefficients at adjacent positions (intrascale dependence) and scales (interscale dependence) are modeled as the product of two independent random variables: a Gaussian vector and a hidden positive scalar multiplier. The latter modulates the local variance of the coefficients in the neighborhood, and is thus able to account for the empirically observed correlation between the coefficients amplitudes. Under this model, named Gaussian scale mixture (GSM), the Bayesian least squares estimate (BLS) of each coefficient reduces to a weighted average of the local linear estimate over all possible values of the hidden multiplier variable.
In , three novel wavelet domain denoising methods for subband-adaptive, spatially-adaptive and multivalued image denoising are developed. The core of this approach is the estimation of the probability that a given coefficient contains a significant noise-free component, which is called signal of interest. The WTs used in  are the DWT and the UDWT.
The aim of this paper is to correct the behavior of the bishrink filter in the homogeneous regions of very noisy images.
All the denoising methods already presented have some drawbacks. A potential solution to correct these behaviors is to fuse multiple denoising schemes. Using different denoising schemes, we can consider the results as different estimates of the image. Different schemes show dissimilar types of artifacts. Through linear combination of the results, in  the norm of the error to find the optimum coefficients in a least-square-error sense is minimized. The wavelet transform, the contourlet transform, and the adaptive 2-D Wiener filtering are used in  as blocks of denoising schemes. Averaging of the results is also proposed as a special case of linear combination in  and show that it yields near-optimal performance. Its major disadvantage is the oversmoothing. This fusion technique can be improved if the mean computation is delocalized. Nonlocal (NL) means algorithms are proposed in . The NL-means algorithm tries to take advantage of the high degree of redundancy of any natural image. Every small window in a natural image has many similar windows in the same image. In a very general sense, one can define as "neighborhood of a pixel " any set of pixels in the image so that a window around looks like a window around . All pixels in that neighborhood can be used for predicting the value at . Given a discrete noisy image the estimated value is computed as a weighted average of all the pixels in the image, , where the weights depend on the similarity between the pixels and and satisfy the usual conditions and In order to compute the similarity between the image pixels, a neighborhood system on I is defined. While producing state-of-the-art denoising results, this method is computationally impractical . Its high computational complexity is due to the cost of weights calculation for all pixels in the image during the process of denoising. For every pixel being processed, the whole image is searched, and differences between corresponding neighborhoods are computed. The complexity is then quadratic in the number of image pixels. In  the computational complexity of the algorithm proposed in  is addressed in a different fashion. The basic idea proposed in  is to pre-classify the image blocks according to fundamental characteristics, such as their average gray values and gradient orientation. This is performed in a first path in , and while denoising in the second path, only blocks with similar characteristics are used to compute the weights. Accessing these blocks can be efficiently implemented with simple lookup tables. The basic idea is then to combine ideas from , namely, weighted average based on neighborhoods similarity, with concepts that are classical in information theory and were introduced in image denoising context. Images with much finer texture and details will not benefit that much from the denoising; while reducing most of the noise, this type of processing will inevitably degrade important image features . The first problem is to distinguish between good and bad candidates for denoising. Many natural images exhibit a mosaic of piecewise smooth and texture patches. This type of image structure calls for position (spatial-) varying filtering operation. Textured regions are characterized by high local variance. In order to preserve the detailed structure of such regions, the level of filtering should be reduced over these regions. The basic concept amounts to a reduction in the extent of filtering in regions where signal power exceeds that of the noise. So, the solution proposed in  supposes the anisotropic treatment of the acquired image taking into account the local variance values of its regions. This procedure can be seen like an NL-means algorithm where the classification of the image blocks is done on the basis of their local variance. The algorithm proposed in this paper is of the same kind but it has two stages architecture. The first stage performs a first denoising followed by a classification of the denoised image blocks based on their local variances. The second stage fuses multiple denoising schemes. The fusion is done with the aid of the classes provided by the first stage with an NL-means like algorithm. But this algorithm does not operate on a single image. It operates on the set of results of all the denoising schemes to be fused. So in our case the delocalization is realized by the diversification of the denoising schemes. We average some pixels having the same coordinates in different results of different denoising schemes. The running time of the algorithm proposed in  is linear in the number of image pixels. The algorithm in  is also fast. The algorithm proposed in this paper is even faster. Our algorithm is explained as follows.
First we prove that the performance of the bishrink filter degrades with the increasing of and with the decreasing of . Next we propose a new strategy for the correction of those degradations. It is based on architecture in two-stage. In the first stage a variant of the genuine denoising algorithm proposed in  is applied obtaining a first result. Computing the standard deviation of each pixel of the first result, the pilot image is obtained. Its pixels are classified in regions according to their values. This is equivalent with the image blocks preclassification proposed in . Our classification criterion is the same as that proposed in .
The set of coordinates of the pixels belonging to one of these regions will represent one of the masks used in the second stage. At the basis of the construction of the second stage lies the idea of diversification. Using different diversification mechanisms, denoising schemes are obtained. Their output images are called partial results. These results are synthesized with the aid of the masks generated at the end of the first stage. The synthesis is achieved by NL-averaging.
The structure of this paper is the following. In Section 2 a sensitivity analysis of the bishrink filter is presented and some of its drawbacks are identified. Then a solution to reduce these drawbacks is proposed and analyzed. In Section 3 all the details of the proposed denoising algorithm are given. Section 4 is dedicated to the presentation of the simulation results and to some comparisons with the best available wavelet based image denoising results conceived to illustrate the effectiveness of the proposed algorithm. The paper's conclusion is formulated in Section 5.
The estimator described by (5)–(10) is named bishrink filter and is applied in the field of the DT-CWT. The sensitivity of the bishrink filter with the estimation of the noise standard deviation ( ) can be computed with the relation:
The input-output relation of the bishrink filter (7) can be put in the following form:
So, it can be written:
The absolute value of this sensitivity is an increasing function of . When the value of the estimation of the noise standard deviation is higher than the performance of the bishrink filter is poorer.
Another very important parameter of the bishrink filter is the local estimation of the marginal variance of the noise-free image . The sensitivity of the estimation with is given by
The regions with high homogeneity in the Barbara image correspond to the dark regions in the image of local variances. All the pixels belonging to a perfect homogeneous region have the same value. So, their local variances are equal with zero. The values of the pixels belonging to a textured region oscillate in space and they have not null local variances. Finally, the pixels belonging to an edge have the higher local variances. So, the bishrink filter treats the edges very well, the estimation of the textured regions must be corrected and the worst treatment corresponds to the homogeneous regions. The denoising quality of pixels with slightly different will be very different in the homogeneous regions. The sensitivity increases with the increasing of . So, the degradation of the homogeneous and textured zones of the noise-free image is amplified by the increasing of the noise standard deviation. Consequently, the most difficult regime of the bishrink filter corresponds to the treatment of homogeneous regions of very noisy images.
Similar sensitivity analyses can be accomplished for the zero-order Wiener filter or for the adaptive soft-thresholding filter, concluding that their worst behavior corresponds to the homogeneous regions of their noise-free input image component. Secondly, the local variance of a pixel gives some information about the frequency content of the region to which the considered pixel belongs. If the pixels of a given region have low local variances, then the considered region contains low frequencies. If these pixels have high local variances then the considered region contains high frequencies. The aim of this paper is to reduce the distortion produced by a denoising system based on the DT-CWT and the bishrink filter as a consequence of the sensitivities and .
This picture represents a homogeneous region of the Lena image affected by AWGN with , denoised with the method proposed in . Some visible artifacts can be observed.
The frequency content of a class corresponding to a higher value of local variance is richer than the frequency content of a class that corresponds to a smaller value of local variance. So, for the fusion of a class corresponding to a smaller value of local variance, an increased number of partial results are necessary. The fusion procedure uses a different number of partial results, from class to class, because these classes have different uncertainties. It is based on an NL-means like algorithm.
Using the masks generated at the end of the first stage, we identify the corresponding classes in each partial result. Each one contains only the pixels with the coordinates specified by the corresponding mask. The amount of noise reduction and the oversmoothing degree in the interior of a class increase with the increasing of the number of partial results used. The fusion procedure proposed prevents the oversmoothing using a different number of partial results in regions with different local variances of the noise-free component.
Our previous simulations suggest a value of six for . So, we have six classes and six denoising systems in Figure 4 ( ). For the first class we have only one weight ( and ). For the second class we have two weights ( and ) and so on. For the sixth class, all the weights of the system in Figure 4 are not nulls ( ).
Other fusion techniques, like median filtering or maximum's detection can be also imagined. A very interesting fusion technique, based on the use of the multiwavelet DWT, is proposed in .
Any of the three variants of the bishrink filter proposed in this paper has better performance than the local zero-order Wiener filter. The DT-CWT is superior to the DWT or the UDWT in denoising applications. So, the performance obtained using the proposed denoising method is superior to the performance reported in  or in .
3.1. The Proposed Implementation
The first block of the first stage implements a DT-CWT and the third one the corresponding inverse transform (IDT-CWT). So, a first result is obtained. The pilot image is generated by the segmentation of done by the block Segm. The segmentation is realized by the comparison of the local standard deviation of each pixel of the first result with some thresholds. This way the data contained in each class is uniform. Registering the coordinates of the pixels belonging to each class, six masks are generated.
The second stage of the denoising system in Figure 5 is represented in blue. To realize the diversification required in the second stage of the proposed algorithm, two types of WT, DT-CWT_A and DT-CWT_B, are computed, obtaining the wavelet coefficients and . Next, three variants of bishrink filter: -the genuine one, -the adaptive bishrink filter with global estimation of the local variance, and -the mixed bishrink filter, are applied in the field of each DT-CWT. Six sequences of estimations of the wavelet coefficients: , , , , , and are obtained. For each one the inverse WT, IDT-CWT, is computed, obtaining six partial results: , , , , , and . This way the redundancy was increased because the actual volume of data is six times higher than the initial volume of data.
With the aid of the six masks generated at the end of the first stage, the six classes of each partial result are identified. Using the class selectors CS1–CS6, the partial results are individually treated. Each mask is used by the corresponding class selector. These systems select the pixels of their input image with the coordinates belonging to the correspondent mask. CS1 is associated with the class which contains the higher values of the local standard deviation and treats the image . It generates the first class of the final result and contributes to the generation of the classes of the final result. CS2 corresponds to the next class of and treats the image , participating to the construction of the classes of the final result. CS3 corresponds to the next class of and treats the image . It contributes to the construction of the classes of the final result and so on. Finally CS6 is associated to the remaining class of and treats the image . It participates to the construction of the sixth class of the final result (that contains the smaller values of the local variance). By NL-averaging (an NL-means like methodology), the six classes of the final result are obtained. The first class of the final result, , is identical with the first class of the image and represents the output of CS1. The second class of the final result, , is obtained averaging the pixels of the outputs of CS1 and CS2 and so on. For the last class of the final result , containing soft textures and homogeneous zones, all the pixels belonging to the outputs of CS1, CS2,…, and CS6 are averaged. Assembling these classes by concatenation, the final estimation is obtained. In the following, the construction of each block in Figure 5 is presented in detail.
3.2. The Diversification Mechanisms
The first diversification mechanism refers to the construction of the DT-CWT. Since an image usually consists of several regions of different smoothness, the sparsity of its representation in a single wavelet domain is limited. This naturally motivates using multiple wavelet transforms to denoise. This procedure is used, for example, in . Besov balls are convex sets of images whose Besov norms are bounded from above by their radii. Projecting an image onto a Besov ball of proper radius corresponds to a type of wavelet shrinkage for image denoising. By defining Besov balls in multiple wavelet domains and projecting onto their intersection using the projection onto convex sets (POCSs) algorithm, an estimate is obtained in , which effectively combine estimates from multiple wavelet domains.
There are two kinds of filters used for the computation of the DT-CWT: for the first decomposition level and for the other levels . The first diversification mechanism is realized through the selection of two types of filters for the first level. The first one is selected from the (9,7)-tap Antonini filters pair and the second one corresponds to the pair of Farras nearly symmetric filters for orthogonal 2-channel perfect reconstruction filter bank, . The idea of diversification by using multiple mother wavelets was also exploited in [19, 21], where the bishrink filter was associated with DWT. The same WT was used in . The synthesis of the final result was carried out in  by simple averaging and in [20, 21] by variational frameworks.
and then the variances:
Finally, the relation (10) can be applied. In the case of the bishrink filter with global estimation of the local variance, the detail wavelet coefficients produced by the first tree of the DT-CWT computation block are indexed with and the detail wavelet coefficients produced by the other tree are indexed with . Applying in order the relations (15), (16), and (10) for the two trees implementing each of the DT-CWTs, the local parameters: , , , , and are computed in each neighborhood . Then the global estimation of the marginal standard deviation can be done:
It can be observed that the bishrink filter with global estimation of the local variance conserves better the extreme values of the clean component of the input image.
The filter F3 is the mixed bishrink filter, proposed in . After three iterations of each DWT representing one tree of a DT-CWT, the pdf of wavelet coefficients can be considered Gaussian. The mixed bishrink filter acts for the first three iterations of each DWT as a bishrink filter with global estimation of local variance, for the forth iteration it acts as a local adaptive Wiener filter and for the fifth iteration (the last one) it acts as a hard thresholding filter, , with the threshold equal with 3 .
3.3. The Classification
The image is segmented in classes whose elements have a value of the local variance, belonging to one of six possible intervals, , where and . An example is presented in Figure 2. The Barbara image perturbed by AWGN with was denoised obtaining the partial result . The six classes of this partial result are represented too.
The class selector CS in Figure 5 selects the class associated to the interval .
The preliminary tests already mentioned also suggest the following values for the bounds of the intervals : , and
We present three types of simulation results: for AWGN, for synthesized speckle noise, and for real SONAR images.
Firstly we compare the proposed additive noise denoising kernel to other effective systems in the literature, namely, the interscale orthonormal wavelet thresholding denoising system proposed in , the multiwavelet approach from , the genuine bishrink filter proposed in , the processor based on the S S family of distributions presented in , the BLS-GSM system proposed in , and the denoising system based on the estimation of the probability of the presence of a signal of interest proposed in . The comparison was done using four images: Peppers, Lena, Boat and Barbara, all having the same size pixels.
First, we compared the performance in terms of output PSNRs. Next, we analyzed the visual aspect of the results. Let and denote the clean and the denoised images. The root mean square (rms) of the approximation error is computed by
where is the number of pixels. The PSNR in dB is given by
The Ra values of results for Lena test image and different noise levels (σ n ) of (A) clean image, (B) noisy image, (C) system in , and (D) proposed system.
In each experiment, the enhancement of realized through the proposed denoising system is higher than the enhancement of Ra realized through the denoising system in . So our goal to accomplish a better treatment of the homogeneous regions is objectively verified.
It is interesting to evaluate the various denoising methods from a practical point of view: the computation time. With the simple univariate method proposed in , the whole denoising process (including four iterations of an orthonormal wavelet transform) lasts approximately 1.6 seconds using a Power Mac G5 workstation with 1.8-GHz PowerPC 970 CPU. With the interscale-dependent thresholding function proposed in  the whole denoising task takes about 2.7 seconds.
In the same paper the computation times for some of the other denoising methods presented in Table 1 are appreciated. For example for the redundant BLS-GSM with estimation window of size the computation time for denoising images is appreciated at 311.8 seconds. The computation time of the ProbShrink algorithm described in , for estimation window with size is appreciated in  at 6.6 seconds. In  is noted that the All Cops implementation of the denoising algorithm based on the association of the DT-CWT with the genuine bishrink filter takes 25 seconds on a 450 MHz Pentium II. The All Cops program for the proposed algorithm takes 41 seconds on a 2.4 GHz Pentium IV. So, the classifications established following the computation speed criterion and the output PSNR criterion for the denoising methods analyzed on the basis of Table 1 are inverse. All the previous considerations were made for images of size .
4.2. Speckle Synthesized
From the PSNR point of view our method has the best performance among those compared in Table 3.
A comparison between the speckle reduction methods described in , (A) raw image, (B) results obtained using the association LLMSE-UWD, (C) results obtained using the association MAP-UWD, and the proposed method (D).
It can be observed that the proposed method makes a good treatment of edges and of homogeneous regions. Its drawback is the textures treatment, some of the fine textures of the clean component of the acquired image being erased by the denoising. A better analysis of the visual aspect of the proposed method can be carried out if it is applied to the test image proposed in , which consist of a collection of six synthetic and real subimages. The subimages 1, 5, and 6 represent real scenes. The subimage 2 contains three types of synthesized textures and the subimage 4 contains some synthesized homogeneous regions and contours. The subimage 3 represents a zone of the Lena image. In this case the speckle noise is synthesized following the procedure presented in . The methods from [21, 26] are compared with the proposed denoising method.
4.3. Real SONAR Images
Figure 7 shows the original SONAR image "Swansea" and the results obtained with the method in  and the proposed method. The visual analysis of the filtered image proves the correctness of our assumptions. Indeed the result of the proposed method has a better visual aspect, the result in  being slightly over-smoothed. An objective measure of the homogeneity degree of a region was proposed for SAR images and it is named enhancement of the Equivalent Number of Looks (ENLs). It is defined by the ratio of the square of the mean and the variance of the pixels situated in the considered region. The enhancement of the ENL of a denoising method in a homogeneous region is defined by the ratio of the ENLs of the considered region computed before and after the application of the method.
The performance obtained for homogeneous regions through the proposed denoising method is certificated by the important enhancement of the ENL obtained considering a region of pixels.
This paper presents an effective image denoising algorithm for SONAR images. It is based on a new additive noise denoising kernel that improves the treatment of homogeneous zones for very noisy images. Inspired from , our algorithm uses one of the best WTs and a very good MAP filter. It outperforms the visual aspect and the PSNR enhancement performances of other denoising methods. The diversity enhancement technique proposed, based on the segmentation of the local standard deviation image obtained at the output of the first stage, is more general. It can be applied with minor modifications to other denoising architectures. This is an ad-hoc method and we have not all the rigorous theoretical justification yet.
We presented our simulation results and compared them with published results in order to illustrate the effectiveness of the proposed algorithm. The comparisons suggest that the results obtained are competitive with the best results reported in the literature for SONAR images denoising.
One of our future research works will be the inclusion in the proposed algorithm of the intrascale dependence of wavelet coefficients information. We believe that the idea of directional estimation windows proposed in  is a good candidate for this task. Another possible solution for the inclusion of the intrascale dependence is the consideration of the phase information provided by the DT-CWT in a similar manner to the one recently proposed by Kingsbury , in association with the BLS-GSM algorithm.
The research of new diversification mechanisms and of new synthesis techniques, like, for example, that proposed in , will represent other future directions for our team.
Finally we will provide a good theoretical explanation for the selection of the thresholds of the segmentation algorithm.
We have established a collaboration with the specialists from the French Sea Institute, IFREMER, from Brest, with respect to the denoising of SONAR images. We acknowledge the importance of many hours spent in meetings with Xavier Lurton and Jean-Marie Augustin dedicated to our introduction in SONAR technology. The authors would like to thank the anonymous reviewers for the very competent and efficient aid they have given. The research reported in this paper was developed in the framework of a grant of the Romanian Research Council (CNCSIS) with the title "Using Wavelets Theory for Decision Making" no. 349/13.01'09.
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