combined bispectrum-filtering technique for signal shape ... - Ticsp

COMBINED BISPECTRUM-FILTERING TECHNIQUE FOR SIGNAL SHAPE ESTIMATION WITH DCT-BASED ADAPTIVE FILTER Dmitry V. Fevralev1, Vladimir V. Lukin1, Alexander V. Totsky1, Jaakko Astola2, Karen Egiazarian2 1

Department of Transmitters, Receivers and Signal Processing, National Aerospace University, Chkalova Str. 17, 61070 Kharkov, Ukraine, Telephone/Fax: +38 (057) 3151186, E-mails: [email protected], [email protected], [email protected] 2 Tampere University of Technology, Signal Processing Laboratory, P. O. Box 553, FIN-33101, Tampere, Finland, Telephone: +358 3 3115 2923, +358 3 3115 3860, Fax: +358 3 3115 3087, E-mails: [email protected]; [email protected] ABSTRACT The paper introduces a combined approach to an unknown signal shape reconstruction realized with smoothing of complex-valued bispectrum by adaptive filter based on discrete cosine transform (DCT) and estimation of local variance within each block. The advantages of the proposed approach are demonstrated by computer simulations in comparison to the traditional bispectrum-based signal waveform reconstruction. Testing of the proposed combined bispectrum-filtering technique is performed for several signal waveforms and for various input signal-tonoise ratios (SNRs). Some other combined bispectrumfiltering techniques are considered. 1. INTRODUCTION In practice, estimation of signal parameters is quite often carried out in conditions of a limited a priori knowledge about statistical properties of noise. Moreover, situations of low input SNR are rather frequently met in real life. This provokes the hindering of important information that is often encoded in the signal shape. Such problems are typical for various signal processing applications like radars [1, 2], sonars [3], optical and radio astronomy [4], digital image processing [5], digital signal processing in medical diagnostic systems [6], etc. In particular, radar range profiles are usually distorted by interferences caused by both heterogeneity of electromagnetic wave propagation channel and internal noises of receiving radar front-end devices [1, 2]. Moreover, random time translations (shifts) of received radar echo responses usually appear due to the random nature of target motion and fluctuations of radio physical parameters occurring in electromagnetic wave propagation channel. These random time translations do not allow carrying out coherent averaging of radar data for obtaining high resolution radar range profiles. In this case, quite good results can be obtained with bispectrum-based strategy applied at the secondary stage of received signal processing. The most important advantages of bispectrum-based signal reconstruction technique are rather high noise immunity (for a noise with symmetrical probability density function), invariance property to signal translations and ability of preserving phase Fourier

spectrum of a signal. The latter property provides an opportunity to reconstruct an unknown signal shape for further extraction of important information about radar range profiles observed in noise. At the same time, the performance of bispectrumbased signal reconstruction system sufficiently depends on the output SNR [4] that can be too low for reliable estimation of signal parameters. In such a case, some extra procedures of signal processing can be added in bispectrum-based framework for improving output SNR [2, and 7–13]. For example, the filtering of bispectrum estimate was proposed in [9, 11]. It has been demonstrated in [9] that filtering of real and imaginary parts of bispectrum estimate is more efficient procedure comparing to the filtering of magnitude and phase bispectra. Further efforts in this direction [10, 11] have shown that some techniques of filtering the real and imaginary parts of bispectrum provide significant improving of output SNR in comparison with traditional bispectrum-based signal reconstruction method [4]. Unfortunately, in the cited papers, only one particular test signal case was considered and non-adaptive filtering algorithms have been implemented. Other practical types of input signals have not been under consideration and it has not been clarified what filtering techniques are optimal or, at least, quasioptimal for the considered application. The goal of the paper is twofold. First, we perform additional performance study of combined bispectrumfiltering technique for different types of signals. The second goal is to design novel adaptive and robust techniques for filtering of 2-D processes able to operate in conditions of a limited a priori knowledge about interference characteristics in the bispectral domain. The proposed combined bispectrum-filtering technique is based on the 2-D DCT, evaluation of the optimal parameters of the corresponding filters, estimation of the bounds that can be reached in the improvement of the output SNR and analysis of the conditions under that maximum SNR exists. 2. COMBINED BISPECTRUM-FILTERING TECHNIQUE Let us assume that a set of independent M realizations is observed at the input of digital signal reconstruction system in the form of additive mixture of some deterministic

signal of a priori unknown waveform s(i) (i = 0, 1,…,I–1) and Gaussian noise. It is also supposed that our signal can be randomly translated in time domain from one realization to other but the signal shape does not vary in the observed set of realizations. Then, the m-th realization x(m)(i) (m = 0, 1,…,M) can be expressed as x (m) (i) = s(i − τ(m) ) + n (m) (i) , (1) (m)

where n (i) is the m-th realization of additive white Gaussian noise (AWGN) with zero mean and unknown variance σ(m)2 ; τ(m) is the random time shift. G Generally, the combined bispectrum-filtering signal reconstruction technique includes the following signal processing steps [10-12]: 1) Calculation of M sample bispectrum estimates   (BE) B(m) (p,q) by direct method [4] in the form of triple X

product of Fourier transforms of time sequences (1) as   (m) (p, q) = X  (m) (p) ⋅ X  (m) (q) ⋅ X  (m) (-p - q) , B (2) X (m)  where X (...) is the direct discrete Fourier transform of x (m) (i) ; p=0,1,…,I-1 and q=0,1,…,I-1 are the indices of independent frequencies in the bi-frequency plane. 2) Accumulation of the sample estimates (2) for   q) ensemble averaged over all M realiobtaining BE B(p,

  q)} parts. The second approach assumes sepaIm{B(p, rate filtering of the magnitude and phase bispectra. It was demonstrated that the first approach provided sufficiently better results. Because of this, only the separate filtering    q)} and Im{B(p,  q)} will be considered beof Re{B(p, low. However, the following question arises - what filters are reasonable to employ in this case? The point is that   q) is neither of additive nor of the noise leaking to B(p, multiplicative nature in strict sense. Saying more exactly, this noise possesses signal (bispectrum) depending properties [8]. Note that noise leaks into bispectrum domain from the input noisy signal and is subjected to several nonlinear transformations. Characteristics of the noise in the bispectrum domain depend not only on the variance of input noise but on the shape of signal Fourier spectrum.   q)} and The graphs of the functions Re{B(p,   q)} corrupted by noise are plotted in Figures 1 Im{B(p, and 2, respectively, for a pulse signal of rectangular shape and pulse length of ∆t=7. Only a quarter of the bifrequency plane is shown in Figures 1 and 2 due to the well-known symmetry property of bispectrum [1].

zations.

  q) to obtain a smoothed estimate 3) Filtering B(p,

  filt (p, q) (this procedure is absent in traditional techB nique [4]). 4) Performing the recovery of the magnitude    | Sbisp (r) | and phase ϕbisp (r) Fourier spectrum estimates   filt (p, q) = (r = 0,1,…I-1) from the estimate B    filt (p, q) | ⋅e jγ (p,q) by recursive algorithm [4] as =| B      filt (p, q) | /(| S (p) | ⋅ | S (q) |) , | S bisp (p + q) |=| B bisp bisp     ϕbisp (p + q) = ϕbisp (p) + ϕbisp (q) − γ (p, q) ;  5) Signal shape s(i) recovery by inverse Fourier  transform of the Fourier spectrum S bisp (r) obtained from

bispectrum. As shown in [9], the significant improvement can be obtained for output SNR by additional filtering of esti  ( p, q ) in the aforementioned processing step 3). mate B This filtering procedure permits obtaining the estimates   filt (p, q) smoothed and they have to be as close as posB sible to the bispectrum of original (noise-free) signal. Moreover, several different approaches exist for obtaining   filt (p, q) . estimates B Two approaches to smoothing the complex-valued   q) are considered in [9]. The first one is the function B(p,   q)} and imaginary separate filtering of the real Re{B(p,

x 10

5

3 2 1 60

0

40 20

40

20 60

  q)} for ∆t = 7, SNR inp =0.35. Fig. 1. Re{B(p,

x 10

4

3 2 1 0 -1 -2

60 40 20

40

20 60

  q)} for ∆t = 7, SNR inp =0.35. Fig. 2. Im{B(p,

   q)} and Im{B(p,  q)} As seen, the functions Re{B(p, are quite noisy. And here we deal with atypical situation, i.e. with noise properties that are not often met in theory [13, 14]. For majority of frequencies in bispectral domain, induced noise possesses non-Gaussian probability density function (PDF) with zero mean and unknown variance [13]. Such properties of noise make problems in selecting   q)} a proper filter for smoothing 2-D functions Re{B(p,   q)} . One more item is that the behaviour of and Im{B(p, information component of these functions is a priori unknown. Thus, it is difficult to recommend some non-linear non-adaptive filter [14] and to properly select its parameters (scanning window size, trimming factors for αtrimmed filters, etc.) In turn, among various types of 2-D filters, nowadays the filters based on the orthogonal discrete cosine or wavelet transforms [15, 16] are considered to be the most effective. Note that the filters based on the orthogonal transforms (DCT and DWT) have been designed originally for suppression of AWGN with a priori known and constant variance [15]. However, several modifications of filters have been designed later for the cases when nonGaussian or signal dependent noise is present [16-18]. It has been demonstrated in [17, 18] that the filter based on the orthogonal transforms can be used successfully in the case when probability density function is not Gaussian in the strict sense but it is approximately of Gaussian shape. Moreover, the efficiency of 1-D filtering of real and imaginary parts of complex-valued Fourier spectrum recovered from bispectrum with DCT-based filter was demonstrated in [12]. All the above-stated peculiarities allow to suppose that 2-D DCT-based filtering can be    q)} and Im{B(p,  q)} . efficient for smoothing Re{B(p, It is known [16, 18] that DCT- based filtering is carried out for square shape blocks. Processing both with overlapping or non-overlapping blocks is possible. The best performance of the algorithm can be obtained for full overlapping of blocks. But this requires more time for computations. Full overlapping means that each consequent block is shifted by only one sample with respect to the previous one. Respectively, the worst performance but the minimal time consumption due to sufficient decreasing of summation and multiplication operations are observed for non-overlapping blocks. This mode corresponds to the block mutual shift equal to the block size. However, the so-called blocking artifacts can arise in the case of non-overlapping block filtering. One restriction of DCT-based filters is that they need a priori knowledge of noise variance (or standard deviation) value for calculation of threshold value [15, 16, 18]. However, noise variance supposed to be unknown a priori both in the considered model (1) and, respectively, in bispectrum domain as well. Therefore, it is necessary to estimate noise variance.

There exist two possible ways. According to the first one, the averaged variance of the observed process is defined for total process and used for calculation of the constant-valued threshold for all blocks. The second alternative way is to estimate local variance inside each block and to calculate on its basis the local threshold value for the given block. It was demonstrated in [12] that DCT-based filtering with adaptively adjusted threshold is more efficient procedure. Let us modify the 1-D adaptive DCT-based filtering technique to the case of processing 2-D functions. Similarly to 1-D signal case, let us estimate the standard deviation (SD) for each block as  (6) σp,q = 1, 483 ⋅ med{| Wpq (x, y) |} , where med {…} is the median value of the sample; Wpq(x,y) denote the DCT coefficients for the block with N N N coordinates (p0=p– b +1, p– b +2,… p + b , q0=q – 2 2 2 Nb Nb Nb +1, q– +2,… q + ); Nb is the block side size 2 2 2 (each block contains NbxNb samples). Thus, the proposed filtering procedure includes the following steps:   q) - calculation of DCT spectral samples of B(p, -

-

containing NbxNb samples in each block; estimation of the local SD according (6);  calculation of threshold β ⋅ σ p,q , where β defines smoothing properties of the filter. Usually the latter value is within the interval 2…4 [15-18], note that noise filtering improves with β increasing but it leads to worse detail preserving; zeroing of the samples those absolute values are  smaller than the threshold β ⋅ σ p,q ; carrying out inverse DCT; joint processing (averaging) of block outputs for all pixels if overlapping blocks are used.

3. ANALYSIS OF COMPUTER SIMULATION RESULTS

The efficiency of the proposed bispectrum-filtering technique has been studied by computer simulations. The following set of parameters was computed: а) the output variance of fluctuations averaged by total number of experiments that is calculated as 2 1 K 2 (7) σout = ∑ σout k , K k =1 where σ2outk = min t

1 I −1  1 I −1  [(sk (i) − s(i − t)) − ∑ (sk (i) − s(i − t))]2 and ∑ I i =0 I i =0

 sk (i) is the estimate of reconstructed signal obtained in kth experiment (k= 1,… ,K) and K is the number of experiments; t is the shift introduced in order to take into account the well-known translation invariance property of bispectrum (t= 1, … ,I-1) [4];

b) the value SNRout averaged by total number of experiments and calculated at the output of the signal re2 construction system as SNR out = Ps σout , where Ps is the signal power; c) the parameter ε = SNR out SNR inp that shows how efficiently noise is suppressed at the output with respect to SNRinp observed at the system input. Three test signals of the same type (denoted below as ##1, 2, and 3) containing two positive-valued pulses of rectangular shape and amplitudes of 2 and 6 were used for computer simulation (see Figures 3a, 4a, and 5a). The pulse lengths were selected equal to ∆t1,2,3 = 3, 7, and 11 samples for the test signals ##1, 2, and 3, respectively. The powers of these signal were equal Ps1 = 0.46, Ps2 = 1.05 and Ps3 =1.60, respectively. The interval between two pulses was fixed and it was equal to 5 samples. The test signal has been randomly shifted from one realization to the next one with maximum deviation equal to τmax = 24 samples. The total length of each sequence is I = 256 samples. During each statistical experiment, M independent realizations were processed and each experiment was repeated K=30 times. Particular noisy realizations x(m)(i) are represented in Figures 3b, 4b, and 5b for the test signals ##1, 2, and 3 and the corresponding SNRinp values are presented.

s(i) 6

s(i) 6 4 2 0

50 100 150 200 250 i a

x(i) 6 4 2 0 -2 50 100 150 200 250 i b Fig. 4. The test signal #2 (a) (∆t = 7) and its particular realization xm(i) (b) for SNR inp =1.05.

s(i) 6

4

4 2

2 0

50 100 150 200 250 i a

x(i) 6

0

50 100 150 200 250 i a

4

x(i) 6

2

4

0

2

-2

0

50 100 150 200 250 i b Fig. 3. The test signal #1 (a) (∆t = 3) and its particular realization xm(i) (b) for SNR inp =0.46.

-2 50 100 150 200 250 i b Fig. 5. The test signal #3 (a) (∆t = 11) and its particular realization xm(i) (b) for SNR inp =1.60.

The performance of the proposed combined bispectrum-filtering technique was estimated and compared to traditional bispectrum-based signal reconstruction technique [4], as well as with combined non-adaptive bispectrum-filtering technique using median and α-trimmed filters [17]. The following four different methods were studied in our computer simulations: 1) the traditional bispectrum-based signal reconstruction technique [4] (Technique #1); 2) the combined bispectrum-filtering signal reconstruc  q) by the stantion technique with filtering of B(p, dard median filter with the sliding window size of 5x5 [14] (Technique #2). 3) the combined bispectrum-filtering technique with   q) by the α-trimmed filter with the filtering of B(p, sliding window size of 5x5 and trimming of 5 maximum and 5 minimum values [14] (Technique #3). 4) the proposed technique with DCT-based filtering of   q) (Technique #4). B(p, For the latter Technique the block size was 8x8 pixels and three different values of the parameter β have been considered. Filtering was carried out both with partial and full block overlapping. Note that the best results have been obtained for full overlapping of blocks. In case of partial overlapping with the shift equal to Nb/2, the provided ε values were smaller by 3…15 % than in case of full overlapping. So, below we present only data for DCT-based filtering with full overlapping. The results of computer simulations are given in Tables 1 – 3. Table 1. The values ε obtained for the test signal #1 SNR inp Technique 1 2 3 β =2.7 β =3.2 4 β =3.7

1.53

0.92

0.46

0.23

0.15

9.07 26.11 13.74 11.22 11.71 12.02

12.80 33.04 17.78 15.19 19.11 14.34

16.80 34.45 27.42 24.93 27.99 28.19

17.07 39.54 38.58 42.67 45.77 51.10

14.17 46.15 62.72 68.55 66.38 89.25

Table 2. The values ε obtained for the test signal #2 SNR inp Technique 1 2 3 β =2.7 β =3.2 4 β =3.7

3.50

2.10

1.05

0.53

0.35

9.42 5.99 5.91 10.19 11.06 11.54

11.25 9.80 10.34 14.34 14.05 13.93

16.30 17.73 20.46 27.17 27.84 28.39

14.75 28.36 34.09 48.25 60.85 54.95

13.50 32.38 37.97 62.44 72.23 64.35

Table 3. The values ε obtained for the test signal #3 SNR inp Technique 1 2 3 β =2.7 β =3.2 4 β =3.7

3.20

1.60

0.80

0.53

0.32

17.61 18.88 13.53 19.72 20.82 21.53

16.12 27.04 22.20 30.50 30.94 32.08

15.84 35.92 33.65 44.42 42.92 44.61

13.57 38.26 38.08 50.00 54.95 53.71

10.35 33.87 36.60 47.16 51.80 53.51

Comparative analysis of the results given in Tables 1–3 has demonstrated several interesting dependencies. As seen, the proposed Technique #4 outperforms the conventional Technique #1 for all the considered cases (for all test signals, values of SNR inp , and analyzed values of the parameter β ). The non-adaptive Techniques ## 2 and 3 comparing to the Technique #1 can provide both improving and worsening the SNRout (increasing or decreasing of ε). This depends on the test signal parameters and SNR inp . For example, for the test signal #1 the Techniques # 2 occurs to be very effective for SNR inp =1.53. But the same Technique # 2 is characterized by the minimal ε among the considered bispectrum-filtering methods for SNR inp =0.15 (see Table 1). Moreover, the use of nonadaptive Techniques ## 2 and 3 can even lead to ε decreasing in comparison to traditional bispectrum Technique # 1. For instance, this happens for the test signal #2 when SNR inp are rather large (3-50 and 2.10, see Table 2). These results confirm the existence of problems in selection of proper non-adaptive filters to be applied within combined bispectrum-filtering framework. Therefore, practical use of the Techniques ##2 and 3 is limited. The Technique #4 outperforms the Techniques 2 and 3 in most practically important cases. The exclusion is the signal #1 for which the Techniques ##2 and 3 can provide larger values of ε for relatively large input SNR inp . The latter peculiarity can be explained by smoothness of the processed 2-D function. Note that the proposed adaptive Technique #4 is the most effective (among the considered techniques) in cases of low SNR inp for all test signals. The optimal value of the parameter β is within interval from 3.2 to 3.7. Visual comparison of the real and imaginary parts of the BE before (Figures 1 and 2) and after (Figures 6 and 7) filtering allows getting imagination about the efficiency of the proposed Technique #4. Moreover, such estimation of the performance is suitable in visual comparison of the signal shapes reconstructed by Techniques ##1, 2 and 4. It is evident that the use of the proposed adaptive DCT-based filter provides noise suppression in the real and imaginary parts of BE.

large distortions but, at the same time, it provides better noise suppression than Technique #1. x 10

 7 s(i)

5

6

2

5

1.5

4

1

3

0.5 60

0

2

40 20

1

20

40

60

0

  q)filt } processed by adaptive DCT-based Fig. 6. Re{B(p,

filter for ∆t = 7, SNR inp = 0.35, β = 3.7.

-1

50

100

150

200

250 i

 Fig. 9. The reconstructed signal s(i) obtained by the

Technique #3, ∆t = 7, SNR inp = 0.35.

 7 s(i) x 10

4

6

3

5

2

4

1

3

0

2

-1

1

60

-2

0

40

20

20

40

-1

60

  q)filt } processed by adaptive DCT-based Fig. 7. Im{B(p,

filter for ∆t = 7, SNR inp = 0.35, β = 3.7.

 7 s(i) 6 5 4 3 2 1 0 -1 50

100

150

200

250

 Fig. 8. The reconstructed signal s(i) obtained by the

i

Technique #1, ∆t = 7, SNR inp = 0.35. Significant residual fluctuations are observed in the signal reconstructed by traditional bispectrum Technique #1 (see Fig. 8). Technique #3 (see Fig. 9) causes rather

50

100

150

200

250 i

 Fig. 10. The reconstructed signal s(i) obtained by the

Technique #4, ∆t = 7, SNR inp = 0.35, β = 3.7. In turn, the proposed Technique #4 provides only negligible noise presence and small distortions in the reconstructed signal (see Fig. 10). The presented results demonstrate good performance of the proposed technique within the total range of the considered input SNRs and test signals. This technique provides improvement of the SNRout by 8 dB comparing with traditional technique in the most important cases of low input SNRs. As it was supposed, the use of the non-adaptive algorithms seems to be unsuitable in general. However, in some cases they are able to give good results (see, for example, the data in Table 1 obtained by Technique 2). In turn, adaptive procedures are able to change their parameters depending on the situation at hand (behavior of the processed 2-D functions and noise level). The comparison of the values of ε has been also carried out for the proposed Technique #4 and the techniques earlier developed in [12]. The results are approximately of the same value for rather large input SNRs. However, the proposed Technique #4 provides the values ε that are larger by approximately 10…20% than adaptive technique

considered in [12]. Recall that the difference is in using either 2-D DCT-based filtering for the Technique #4 and 1-D DCT-based filtering for the methods in [12]. Certainly, 2-D DCT-based filtering requires larger processing time. But within entire framework of combined bispectrum-filtering processing it does not lead to considerable increasing of total computation time. 4. ANALYSIS OF OTHER METHODS BASED ON ADAPTIVE FILTERING

Numerical simulation results presented in the previous Section show that the use of adaptive filtering within combined bispectrum-filtering approach leads to obvious benefits. However, the use of the designed local adaptive DCT-based filters is not the only opportunity. Thus, in this Section we consider and carry out brief performance analysis for three other techniques. First, in our paper [19] a method of adaptive filtering based on the so-called Z-parameter have been designed. This method (further Technique # 5) is based on the use of preliminary nonlinear filter, calculation of Z-parameter for each sliding window position, and hard-switching between two or several nonlinear filter outputs according to results of Z-parameter comparison to the corresponding threshold. One advantage of this filter is that noise type (additive, multiplicative, mixed) should not be known in advance. For the considered application, we used the αtrimmed filter with the sliding window size of 5x5 and trimming of 5 maximum and 5 minimum values as a preliminary filter. Besides, this filter as well as the standard 5x5 median filter were used as component filters (switching between their outputs was performed). The threshold was equal to 0.4. Second, we considered an opportunity of applying the standard sigma filter [20] where for each given sliding window position local standard deviation is estimated according to (6) (see Section 2). Then, using this estimate, 2σ-neighborhood is determined and averaging of pixel values that belong to this neighborhood for given sliding window position is carried out. This method is referred below as Technique # 6. Third, it is possible to apply a two-stage filtering procedure. At the first stage, Technique # 4 (β=3.2) is applied and then its output is subjected to post-processing by the 3x3 center weighted median filter with the center weight 3. The goal of such post-processing is to remove spiky values retained by DCT-based filter. This approach is noted as Technique # 7. The obtained simulation results for the Techniques # 5, 6, and 7 for all three test signals are presented below in Tables 4, 5, and 6, respectively. Table 4. The values ε obtained for the test signal #1 SNR inp Technique 5 6 7

1.53

0.92

0.46

0.23

0.15

20.34 14.26 10.97

22.77 14.06 17.99

29.31 26.27 36.41

42.87 30.73 40.31

49.92 35.13 61.96

Table 5. The values ε obtained for the test signal #2 SNR inp Technique 5 6 7

3.50

2.10

1.05

0.53

0.35

33.55 12.71 11.61

38.12 19.57 14.64

40.45 34.26 28.71

45.17 29.14 60.84

43.82 28.14 73.16

Table 6. The values ε obtained for the test signal #3 SNR inp Technique 5 6 7

3.20

1.60

0.80

0.53

0.32

17.20 9.83 21.05

24.78 21.92 31.39

39.13 29.77 41.93

46.08 27.80 55.37

39.52 21.61 52.51

Comparing the data in Tables 4, 5, and 6 to the corresponding data in Tables 1, 2, and 3 as well as analyzing these data within each Table, it is possible to conclude the following. The Technique # 6 is able to produce some performance improvement in comparison to the traditional bispectrum method (Technique # 1) but among the considered combined bispectrum-filtering techniques the Technique # 6 is surely not the best. The use of adaptive nonlinear filter (Technique # 5) instead of non-adaptive ones (Technique # 2 and Technique # 3) can be expedient. The Technique # 5 either provides the largest values of ε (like, e.g., for the test signals #2 and #3, see data in Tables 5 and 6 and compare them to the corresponding data in Tables 2 and 3) or appropriately large values of ε. The use of the Technique # 7 instead of the Technique # 4 does not seem reasonable. Really, in majority of the considered situations the Technique # 7 produces practically the same values of ε as the Technique # 4 does. Since the Technique # 7 is slightly more complicated than the Technique # 4, there are no reasons of applying the Technique # 7. 5. CONCLUSIONS

The results of investigations show that the proposed approach that uses the adaptive smoothing of BE by 2-D DCT-based filtering allows significant improving the SNR value obtained at the output of signal reconstruction system in comparison to the traditional bispectrum-based technique. Performance analysis of several combined bispectrum-filtering techniques has been performed and it offers data for thorough comparisons. The developed approach (Technique # 4) provides the best signal reconstruction system performance comparing to the non-adaptive techniques in most cases. It is shown that the maximum improvement has been obtained for small input SNR that seems to be very important in many practical applications. The novelty of the proposed technique is in the local estimation of standard deviations inside each block as well as in the corresponding calculation of hard threshold for DCT-based filter.

It seems reasonable to apply the proposed technique for secondary data processing in radar range profiles classification and target recognition for high-resolution radars. However, this is not the only application where the designed filtering techniques can be efficiently used. One more area worth investigating is the processing of radar images formed by side look aperture radars for which mixed multiplicative and additive noises with unknown variances can be present. REFERENCES

[1] Zhang X.D., Shi Y, and Bao Z., “A new feature vector using selected bispectra for signal classification with application in radar target recognition”, IEEE Trans. on Signal Processing, September 2001, vol. 49, No 9, pp. 1875 – 1885. [2] Totsky A.V., Lukin V.V., Kurbatov I.V., Astola J.T., Egiazarian K.O., “Combined bispectrum-filtering techniques for radar output signal reconstruction in ATR applications”, Proceedings of International Conference "Automatic Target Recognition XIII"; Ed. Firooz A. Sadjadi; Orlando (USA), April 2003, SPIE vol. 5094, pp. 301-312. [3] Sasaki K., Sato T, Nakamura Y., “Holographic Passive Sonar”, IEEE Trans. Sonic Ultrasonic, SU-24, May 1997, pp. 193-200. [4] Bartelt H., Lohmann A.W., Wirnitzer B., “Phase and amplitude recovery from bispectra”, Applied Optic, 1984, vol. 23, pp. 3121-3129. [5] A. Totsky, V. Lukin, J. Astola, I. Kurbatov, K. Egiazarian, “Bispectrum based reconstruction technique by tapering pre-distortion of image rows”, Proceedings of the Seventh All-Ukrainian Conference on Signal/Image Processing and Pattern Recognition, Kiev, Ukraine, Oct. 2004, pp. 229-232. [6] Nakamura M., “Waveform estimation from noisy signal with variable signal delay using bispectrum averaging”, IEEE Trans. On Biomedical Engineering, 1993, 40, No 4, pp. 769-782. [7] Totsky A., Krylov O., Kurbatov I., Lukin V., Astola J., Egiazarian K., “Statistical investigations of bispectral and image restoration for Gaussian and non-Gaussian noise environments”, Proc. International TICSP, Workshop on Spectral Methods and Multirate Signal Processing SMMSP’2001, Pula, Croatia, June 2001, pp. 231-241. [8] Totsky A.V., Roenko A.A., Lukin V.V., Zelensky A.A., Astola J.T., Egiazarian K.O., “Combined bispectrum-median reconstruction of 1-D signal waveform”, Proceedings of the 2004 International TICSP Workshop on Spectral Methods and Multirate Signal Processing, SMMSP’2004, Vienna (Austria), September 2004, pp. 8793. [9] Totsky A.V., Kurbatov I.V., Lukin V.V., Zelensky A.A., “Use of 2-D Filtering of Bispectrum Estimations for 1-D Signal Reconstruction in Mixed Noise Environment”, Proc. of Intern. TICSP Workshop on Spectral Methods and Multirate Signal Processing, SMMSP'2002, TICSP Series No 17, Toulouse, France, Sept 2002, pp. 171-178.

[10] Lukin V., Totsky A., Kurekin A., Kurbatov I., Astola J., Egiazarian K., “Signal Waveform reconstruction from noisy bispectrum estimations pre-processed by vector filters”, Proceedings of the Seventh Internatrional Symp. On Signal Processing and Applications, Paris (France), 2003, July, 1-4, Vol. 2, pp. 169-172. [11] Totsky A.V., Kurbatov I.V., Lukin V.V., Zelensky A.A., Astola J., Egiazarian K., “Estimation of Unknown Signal Shape by Bispectrum-Filtering Techniques for Gaussian and non-Gaussian Noise Environments”, Uspekhi sovremennoy radioelectroniki (Zarubezhnaya Radioelectronica, v. 1), Moscow, 2005, № 4, pp. 43-54. [12] Fevralev D.V., Lukin V.V., Totsky A.V., Egiazarian K., Astola J., “Signal shape reconstruction by DCT-based filtering of fourier spectrum recovered from bispectrum data”, SMMSP’2005, Riga (Latvia), June 2005, pp. 87-93. [13] Roenko A.A., Lukin V.V., Totsky A.V., “Using of robust estimates for bispectrum-based signal processing”, Radioelektronni i computerni systemy, Issued by National Aerospace University, Kharkov, Ukraine, 2004, Vol. 4 (8), pp. 5-13 (in Russian). [14] Astola J., Kuosmanen P., Fundamentals of nonlinear digital filtering, CRC Press LLC, Boca Raton, USA, 1997. [15] Donoho D.L., Johnstone I.M., “Adapting to unknown smoothness by wavelet shrinkage”, Journal of American Statistical Association, 1995, No 11, Vol. 90, pp. 12001224. [16] Oktem R., “Transform Domain Algorithms for Image Compression and Denoising”, Thesis for the degree of Doctor of Technology, Tampere University of Technology (Tampere, Finland), 2000. – 142 p. [17] Gotchev A., “Spline and Wavelet Based Techniques for Signal and Image Processing”, Thesis for the degree of Doctor of Technology, Tampere University of Technology (Tampere, Finland), 2003. – 171 p. [18] K. Egiazarian, V.Melnik, V. Lukin, J. Astola, “Local Transform Based Denoising for Radar Image Processing”, Proceedings of IS&T/SPIE International Conference on Nonlinear Image Processing and Pattern Analysis XII, San Jose, USA, Jan. 2001, SPIE Vol. 4304, pp. 170-178. [19] V. Lukin, V. Melnik, A. Pogrebniak, A. Zelensky, J. Astola, K. Saarinen, “Digital adaptive robust algorithms for radar image filtering”, Journal of Electronic Imaging, 1996, Vol. 5(3), pp. 410-421. [20] J.-S. Lee, “Digital image smoothing and the sigma filter”, Computer Vision, Graphics and Image Processing, 1983, Vol. 24, pp. 255-269.