J. Vis. Commun. Image R. 22 (2011) 401–410

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J. Vis. Commun. Image R.

journal homepage: www.elsevier .com/ locate/ jvc i

Removing interference components in time–frequency representations usingmorphological operators

Soledad Gómez a,1, Valery Naranjo b, Ramón Miralles a,⇑,1

a Instituto de Telecomunicaciones y Aplicaciones Multimedia (iTEAM), Dpto. de Comunicaciones, Universidad Politécnica de Valencia, Camino de Vera S/N, 46022 Valencia, Spainb Instituto de Bioingeniería y Tecnología orientada al Ser Humano, Universidad Politécnica de Valencia, Camino de Vera S/N, 46022 Valencia, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 July 2010Accepted 16 March 2011Available online 26 March 2011

Keywords:Time–frequency representationsMorphological image processingElimination of cross-componentsUnderwater signal processingNon-stationary signalsWigner-Ville transformGeodesic operatorMorphologic operator

1047-3203/$ - see front matter � 2011 Elsevier Inc. Adoi:10.1016/j.jvcir.2011.03.007

⇑ Corresponding author.E-mail addresses: sogogar@upvnet.upv.es (S. Góme

Naranjo), rmiralle@dcom.upv.es (R. Miralles).1 This work was supported by the National R&D Pr

02975 (Spain), FEDER programme and Generalitat Vale

Time–frequency representations have been of great interest in the analysis and classification of non-sta-tionary signals. The use of highly selective transformation techniques is a valuable tool for obtainingaccurate information for studies of this type. The Wigner-Ville distribution has high time and frequencyselectivity in addition to meeting some interesting mathematical properties. However, due to the bi-lin-earity of the transform, interference terms emerge when the transform is applied over multi-componentsignals. In this paper, we propose a technique to remove cross-components from the Wigner-Ville trans-form using image processing algorithms. The proposed method exploits the advantages of non-linearmorphological filters, using a spectrogram to obtain an adequate marker for the morphological process-ing of the Wigner-Ville transform. Unlike traditional smoothing techniques, this algorithm providescross-term attenuations while preserving time–frequency resolutions. Moreover, it could also be appliedto distributions with different interference geometries. The method has been applied to a set of differenttime–frequency transforms, with promising results.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction The WVD satisfies some interesting mathematical properties

Time–frequency representations [1,2] are essential when work-ing with non-stationary signals such as ultrasonic or acoustic sig-nals. The most straightforward technique for representing asignal in the time–frequency plane is the short time Fourier trans-form (STFT). This technique consists of prewindowing the signalunder test x(t) around a particular time t and calculating its Fouriertransform. The spectrogram, the plot of short-time spectra over atime–frequency plane, is commonly used for the analysis of non-stationary signals in the time–frequency domain [3]. Nevertheless,as a result of the Heisenberg–Gabor principle [4], the spectrogramhas a trade-off between time and frequency resolutions. Whenusing the spectrogram, the results are dependent on the choiceof the window length, which limits the number of applicationswhere this can be applied.

Other types of time–frequency representations are based on theWigner-Ville distribution (WVD). The WVD is defined as:

Wxðt; mÞ ¼Z 1

�1xðt þ u=2Þx�ðt � u=2Þe�j2pmudu ð1Þ

ll rights reserved.

z), vnaranjo@dcom.upv.es (V.

ogram under Grant TEC2008-nciana CMAP 340.

[5,2,6], such as marginal properties or preservation of time and fre-quency shifts. Moreover, the WVD simultaneously provides highselectivity in time and frequency while the spectrogram has atrade-off between time and frequency resolutions. Despite this,the WVD has a drawback because it produces a great number ofcross-components when it is applied over multi-component sig-nals. Taking the above into consideration, the main objective of thisstudy is to achieve a time–frequency representation, withoutcross-terms (such as the spectrogram), which can be as selectiveas possible in time–frequency planes (such as the WVD).

This paper presents an innovative method for the elimination ofcross-components in the WVD. In addition to cross-term attenua-tion, our algorithm preserves time–frequency resolutions. To dothis, it incorporates image-processing algorithms based on mathe-matical morphology.

A review of different techniques for the reduction of cross-com-ponents from the WVD is presented in Section 2. The basis tounderstand the presented research and the proposed method,which is based on morphological filters, are explained in Sections3 and 4, respectively. Finally, some results and conclusions are dis-cussed in Sections 5 and 6.

2. State of the art

The importance of achieving a non-artefacted image of thespectrogram is crucial because this kind of representations has

402 S. Gómez et al. / J. Vis. Commun. Image R. 22 (2011) 401–410

been analyzed from the point of view of image processing in agreat deal of applications, such as estimating signal parameters[7] or even separating different signal components [8].

The interference structure of the WVD and its properties havebeen the subject of extensive studies [6,9]. Research on eliminatingcross-components from the WVD has been of enormous benefitwhen working with multi-component signals. Since interferenceterms are oscillatory, previous studies were related to attenuateinterference components by means of a smoothing operation.These studies lie in the convolution of the WVD with a 2-Dsmoothing kernel [9,10]. The Cohen class of shift-invariant trans-formations provides an important framework for WVD smoothingalthough the design of a satisfactory kernel is a complex task. Manyresearch papers have been published on this topic with the objec-tive of achieving cross-term attenuation in the WVD, while pre-serving time–frequency resolutions and all the desiredmathematical properties. As an example, we can refer to [11],where the most suitable kernel is selected by means of processingthe corresponding image of the ambiguity function. Some otherproposals related to kernel design can be seen in [12,13]. The maindifficulty of these techniques is the selection of the most appropri-ate kernel for each type of signal.

The approach presented in [14] reduces the cross-componentsby combining the Fourier–Bessel expansion with the WVD. TheFourier–Bessel expansion decomposes a multi-component signalinto a number of mono-component signals, and the WVD is appliedover each mono-component signal. It is a powerful tool to elimi-nate interferences from the WVD. However, the method has onerestriction – it assumes that signal components are well-separatedin the frequency domain. If the signal components are not well-separated, it is likely that the method will not correctly performthe frequency components separation, as will be discussed in Sec-tion 5. An improvement of this method is presented in [15]. Thisimprovement will be also tested and compared with our methodin Section 5.

Alternative approaches to remove cross-components from theWVD distribution are based on the use of image-processing algo-rithms [16,17]. However, these methods assume a priori knowl-edge of the interference geometry.

A new method to remove interferences from the WVD is pre-sented in this paper. It starts from our earlier contribution [18]and represents an improvement on the obtained results. We over-come the main drawback of [18] since that method was not appro-priate to reduce interferences when cross-terms are touchingsignal terms. In this paper, we introduce another morphologicaloperator to remove interferences from the WVD even when inter-ferences overlap signal terms. New simulations have also been car-ried out and discussed to demonstrate the application of theproposed method not only to remove interferences from theWVD but also to remove them from distributions with differentinterference geometries. The results from [14,15] are comparedto those obtained using our method.

3. Method basis

The basic idea of the presented method is to apply morpholog-ical filters to reduce interference components (cross-terms) fromimages corresponding to noisy time–frequency representations.We want to recover only those signal terms from the WVD thatare also present in the spectrogram distribution, thereby preserv-ing the resolution of the WVD. The method uses the image corre-sponding to the spectrogram as a marker for a morphologicalreconstruction carried out over the WVD, which we are seekingto denoise. Essentially, the method uses the opening by k-recon-

struction operator (or for simplicity k-reconstruction), which is de-scribed below.

3.1. Definition of k-reconstruction

The k-reconstruction operator belongs to a family of transfor-mations called geodesic operators [19–22]. A geodesic operator in-volves two input images: the marker and the reference. Amorphological transformation is applied to the marker and the re-sult is forced to remain above or below the reference. Let f and g(two grayscale images) be the reference and the marker images,respectively, which are both defined in the same domain as:

f ðxÞ : E! T ;

where (x) 2 E is the pixel position. In the case of valued discreteimages, T ¼ ftmin; tmin þ 1; . . . ; tmax} (in general T � Z or R, or anycompact subset of Z or R) is an ordered set of gray-levels. Typically,in digital 8-bit images we have tmin = 0 and tmax = 255.

Let us define the unitary geodesic k-dilation of the marker gwith respect to the reference f, dð1Þf ;k ðgÞ, as the point-wise minimumbetween the reference and the unitary non-flat dilation, dð1Þk , of themarker, which is:

dð1Þf ;k ðgÞ ¼ dð1Þk ðgðxÞÞ ^ f ðxÞ:

The unitary k-dilation, dð1Þk , represents the dilation with a uni-tary non-flat structuring function b(x), with bðxÞ 2 FfE; T g beinga weighting function defined as:

bðxÞ ¼�k; x 2 B;

�1; x R B:

�

Thus, the unitary k-dilation, dð1Þk , will be defined by theexpression:

dð1Þk ðf ÞðxÞ ¼ f ðyÞ : f ðyÞ ¼ sup½f ðzÞ � k�; z 2 Bxf g _ f ðxÞ:

The geodesic k-dilation of size n of the marker g with respect tothe reference f is obtained by performing n successive geodesic k-dilations of g with respect to f:

dðnÞk;f ðgÞ ¼ dð1Þf ;k dðn�1Þf ;k ðgÞ

h iwith dð0Þf ;k ðgÞ ¼ g.

The k-reconstruction [23] of the reference image f from the mar-ker g is defined as the geodesic k-dilation of g with respect to f untilstability:

creck ðf ðxÞ; gðxÞÞ ¼ dðkÞf ;kðgðxÞÞ;

where k is dðkÞf ;kðgÞ ¼ dðkþ1Þf ;k ðgÞ.

Using the k-reconstruction operator, only the reference imagepixels that actually touch the marker will be reconstructed withthe maximum value of the reference. The remainder of the con-nected areas touched by the marker will be reconstructed withan intensity level that will decrease with a slope equal to k [24,23].

3.2. Use and behavior of k-reconstruction

The results of applying the k-reconstruction operator are ex-plained in Figs. 1 and 2 using a simulated signal to illustrate theperformance of the operator. Test images have been used to illus-trate the example. These images are composed by straight linesand zigzag lines. We consider the zigzag lines to be the undesiredlines and the straight lines those to be preserved. This exampledemonstrates how we are able to reduce zigzag lines using the k-reconstruction operator, and how, when the undesired lines over-lap the desired ones, the use of different values of k provides satis-factorily differentiated results.

Fig. 1. (a) Reference image, (b) marker, (c) result of k-reconstruction with k = 0, (d)result of k-reconstruction with k = 15.

S. Gómez et al. / J. Vis. Commun. Image R. 22 (2011) 401–410 403

In Fig. 1, the undesired zigzag lines do not overlap the desiredstraight lines. In contrast, in Fig. 2, the undesired zigzag lines over-lap the desired straight lines.

Focussing on Fig. 1, our aim is to prove how the operator is ableto reconstruct only the straight lines (desired lines). Image Fig. 1(b)is used as marker for the morphological operation carried out overimage Fig. 1(a). By using the k-reconstruction operator with differ-ent values of k (k = 0 and k = 15), we obtain images Fig. 1(c) and (d),respectively. From here, we can point out how the k-reconstructionoperation provides identical solutions independently of the value

Fig. 2. (a) Reference image, (b) marker, (c) result of k-reconstruction with k = 0, (d)

of k and efficiently removes the undesired lines. Note that in thisexample the zigzag lines do not overlap the desired ones.

Focussing on Fig. 2, our aim is also to reconstruct only thestraight lines. Note how, in this case, the undesired zigzag linesoverlap the desired straight lines. Image Fig. 2(b) is used as markerfor the morphological operation carried out over the reference im-age Fig. 2(a). By using the k-reconstruction operator with values ofk = 0, k = 15, and k = 30, we obtain images Fig. 2(c)–(e), respec-tively. Note that if a value of k = 0 is used, the operator does notprovide good results because the zigzag lines that touch the mar-ker have also been reconstructed. Higher values of k provide a goodsolution for removing these undesired components. Note that if kincreases, the intensity of the interferences decreases.

By considering these two examples, we can draw the followingconclusions: when signals and cross-components are not touching,the operator achieves satisfactory results independent of the valueof k. However, when the signal and cross-terms are touching, the k-reconstruction operator still achieves satisfactory results, but theselection of an appropriate value of k will improve the result. Theselection of the correct marker will also improve the final resultsas will be justified in the next section.

The morphological reconstruction operator was presented in[18] as a useful tool to reduce cross-components from interfereddiagrams. The method proposed in [18] performs satisfactory solu-tions only when cross-terms do not touch signal terms. The stan-dard reconstruction operator, as introduced in [18], represents aparticular case of the k-reconstruction operator for values of k = 0.

In the next section, a general scheme describing the completemethod is presented. The method has been called k-ReconstructionCross-Component Removal (k-RCCR).

4. General scheme of k-RCCR

The basis of the k-RCCR method is outlined in Fig. 3. The centralidea of the method is to recover the signal information from thosetime–frequency representations suffering from interference terms.The method performs a morphological reconstruction operation. Abinary image is obtained from the spectrogram, and it is used as amarker in order to ensure the inclusion of only signal terms in the

result of k-reconstruction with k = 15, (e) result of k-reconstruction with k = 30.

Fig. 3. General scheme of the k-RCCR method.

404 S. Gómez et al. / J. Vis. Commun. Image R. 22 (2011) 401–410

marker. To improve accuracy, the marker must be pre-processed.For example, in those applications where time and frequency res-olutions are simultaneously required, the method should be ap-plied using the skeleton of the spectrogram as a marker and theimage corresponding to a WVD as a reference.

The skeleton of the spectrogram is obtained by performing thehomotopic skeleton associated to the iteration of the thinningtransformation [19,22]. This algorithm is based on the Hit-or-misstransform, which removes points of a binary set according to a fi-nite family of templates but preserving the homotopy of the set.Note also that the approach does not require any parameter. Thisskeleton is a signal that contains the minimum quantity of infor-mation that is sufficient to locate each point of the support ofthe signal in frequency and time domains [19,20,25]. The skeletonis a good choice of marker since the spectrogram has a poor time–frequency resolution, but its components are located at the sameposition as that of the desired components that we wish to recon-struct from the WVD.

The method performs its operation as follows:

� The spectrogram of the input signal is computed, and the resultis called gi(x).� The noisy time–frequency transformation, a Wigner-Ville trans-

form, is also computed from the input signal. The image corre-sponding to the distribution that displays interference isreferred to as f(x).� The image gi(x) is pre-processed using a homotopy preserving

skeleton algorithm [22], thus resulting in gskel(x). The methodremoves cross-components from f(x) using gskel(x) as a markerfor the k-reconstruction operation. In comparison with thespectrogram, the output of the scheme – the reconstructedimage fr(x) – is an image with improved time–frequency resolu-tion, without cross-components.

The selection of k plays an important role in improving the re-sults. When we apply the proposed method to a diagram showinginterference, the higher the value of k, then the higher the value ofinterferences that can be reduced by the method. However, if k in-creases strongly we could also eliminate signal information. Theselection of the accurate value of k is now discussed.

4.1. Selection of parameter k

An error function has been formulated in order to evaluate theresults found with our method for different values of k. The koptimum

is the value of k that minimizes the error function given by Eq. (2).

As stated above, the skeleton of a signal contains the minimumquantity of information needed to locate the signal in both dimen-sions. In order to evaluate the error function, MSE minimization isemployed. For every value of k we compute the difference in en-ergy between the skeleton of the original signal, gskel(x), and theskeleton computed from the reconstructed diagram, f skel

rkiðxÞ, using

a value of k = ki (see Eq. (2)).

errorðkiÞ ¼P

xgskelðxÞ �P

xf skelrkiðxÞ

��� ���2Pxf skel

r ðxÞ : ð2Þ

When the diagram is reconstructed using k = koptimum, the skele-ton of the spectrogram and the skeleton corresponding to thereconstructed diagram are likely to be very similar. This is due tothe fact that, when processing the WVD with the appropriate valueof k, the interference terms will disappear and as such do not con-tribute to the skeleton. In addition, the signal terms will remain intheir original position contributing to the skeleton in the same wayas the signal components from the spectrogram. When the WVD isreconstructed with a value of k lower than the koptimum, k < koptimum,several interference components that touch the signal terms willremain following reconstruction. These unremoved componentswill contribute to the skeleton image, thereby increasing the valueof the error function. On the other hand, when the WVD is recon-structed with a value of k higher than the koptimum, k > koptimum, thecross-terms are removed and do not contribute to the skeleton ofthe reconstructed image. However, by using k > koptimum, desiredinformation of the signal, which disappears from the skeleton ofthe reconstructed diagram, will be lost, thus increasing the errorfunction.

In Section 5.1, an algorithm for obtaining the koptimum is dis-cussed. The results obtained using the k-RCCR method are also pre-sented. The method operation has been compared with themethods proposed in [14,15] and also with classical methods, suchas the filtering using 2-D Gaussian kernels. Furthermore, the cor-rect performance of the method when it is applied to other typesof transforms has been demonstrated.

5. Results and discussion

The operating method is applied to a set of synthetic test signalsx1(n), x2(n), and x3(n), and to two real signals x4(n) and x5(n).

To clarify the formulation of the synthetic signals let us intro-duce how to formulate the linear frequency-modulated and sinu-soidal frequency-modulated signals.

Fig. 4. Evaluating the error function for different values of k.

S. Gómez et al. / J. Vis. Commun. Image R. 22 (2011) 401–410 405

On the one hand, with f0 being the normalized initial frequencyand f1 the normalized final frequency, we can compute the N pointsrecord of a linear frequency-modulated signal as:

f ðnÞ ¼ f0 � n� N2

� �þ f1 � f0

2ðN � 1Þ � ðn� 1Þ2 � N2� 1

� �2" #

;

n ¼ 1;2; . . . ;N;xðnÞ ¼ exp j � 2 � p � f ðnÞð Þ:

ð3Þ

On the other hand, let us call f0 the normalized initial frequency,f1 the normalized final frequency and Np the sinusoidal period. Wedefine the following parameter /

/ ¼ acos 1=4� f1 þ f0

2

� �� 2f1 � f0

� �

We can compute the N points record of a sinusoidal frequency-modulated signal as:

U ¼ p � ðf0 þ f1Þ � nþf1 � f0

2� Np sin

2p � nNp

þ /

� �� sinð/Þ

� �;

n ¼ �N=2þ 1;�N=2þ 2; . . . ;N=2;xðnÞ ¼ expðj �UÞ:

ð4Þ

With this basic formulation, a brief outline of these signals ispresented bellow:

� x1(n): it is composed by two linear frequency-modulated signals(see Eq. (3)) whose components are well-separated in the fre-quency domain. The parameters for the first component werefixed as: f0 = 0 and f1 = 0.2. The parameters for the second com-ponent were selected as: f0 = 0.35 and f1 = 0.45. For both compo-nents N = 128 points were computed and x1(n) is obtainedadding the two components.� x2(n): it is composed by two linear frequency-modulated signals

(see Eq. (3)) whose components are not well-separated in thefrequency domain. The parameters for the first component werefixed as: f0 = 0 and f1 = 0.4. The parameters for the second com-ponent were selected as: f0 = 0.22, and f1 = 0.5. For both compo-nents N = 128 points were computed and x2(n) is obtainedadding the two components.� x3(n): it is composed by two sinusoidal frequency-modulated

signals (see Eq. (4)) whose components are not well-separatedin the frequency plane. The first sinusoid has the followingparameters f0 = 0.25, f1 = 0.35, and Np = 100. The second sinu-soid has parameters f0 = 0.35, f1 = 0.45, and Np = 100. For bothcomponents N = 128 points were computed and x3(n) isobtained adding the two components.� x4(n) and x5(n): two underwater signals. The first signal is a real

signal from a beluga whale and the second from a tugboat.

The automatic classification of these signals is a typical applica-tion of our developments.

The spectrogram and Wigner-Ville distributions of the discretetime signals presented hereafter were computed using the MAT-LAB time–frequency toolbox [26]. Being N is the number of pointsof the signal under test, the spectrogram of this signal is computedfor time instants from 1 to N, using N frequency bins and a N

4-pointHamming analysis window normalized to unit energy. On theother hand, the Wigner-Ville distribution was computed for thesame time instants and using the same number of frequency bins.In this paper, the effect of the type and the size of the analysis win-dow used to compute the spectrogram is not considered since theskeleton of the spectrogram contains the minimum quantity ofinformation to locate each component. Therefore, the skeleton willnot change when different windows are used. Moreover, the skel-

eton will have the same geometry independently of the originalresolution of the computed spectrogram.

5.1. Estimation of parameter k

Before discussing the results obtained from the method, it isimportant to calculate the value of the selected k. The method de-scribed in Section 4.1 has been used to numerically evaluate the er-ror function (see Eq. (2)) using the reconstructed diagram of signalx3(n), x4(n), and x5(n) for several values of ki in the interval from 0to 100. Remember that if the parameter k = 0 the k-reconstructionoperator performs the same operation as the reconstruction oper-ator, and the interferences which touch the desired signals are notremoved as was demonstrated in [18].

Fig. 4 shows the error evolution for different values of k. The ‘‘⁄’’line corresponds to the evaluation of the error function for theWVD diagram of an underwater signal received from a tugboat,x5(n); the ‘‘h’’ line corresponds to the evaluation of the error func-tion for the WV diagram of an underwater signal of the belugawhale, x4(n); the ‘‘�’’ line corresponds to the reconstruction of theWVD of the sinusoidal signal x3(n).

Moreover, several simulations over a varied set of signals, of asimilar nature to the set herein presented, corroborate the earlierresults. These simulations provide values of koptimum near to 25 inall cases, and this value is in accordance with the value that wehave empirically set as optimum. From this, we can assume andgeneralize the value of koptimum = 25 as a representative choice forthese kinds of signals.

Next, we provide some representative examples applying themethod over the set of test and real signals. Firstly, in Section5.2, we compare our method results with those obtained usingthe method proposed in [14,15]. Both methods are applied overthe set of test signals. Secondly, in the same section, we have in-cluded a new example where total overlap between signal andinterferences is managed; the results obtained with the proposedmethod are compared to those obtained with techniques of filter-ing using 2-D Gaussian kernels. Then, in Section 5.3, the k-RCCRmethod is tested over the set of real signals. Finally, in Section5.4, the method is also tested when applying it over other typesof frequency distributions.

5.2. Method comparison

The method proposed in [14] has been tested to compare theirresults with those obtained using our method. The former method

406 S. Gómez et al. / J. Vis. Commun. Image R. 22 (2011) 401–410

reduces cross-components by combining the Fourier–Bessel (FB)expansion with the WVD. The Fourier–Bessel expansion decom-poses a multi-component signal into a number of mono-compo-nent signals [27], and the WVD is applied over each mono-component signal. In the same way as proposed in [18], this is apowerful tool to eliminate interferences from the WVD when sig-nal components are well-resolved. If interferences overlap or touchsignal components, the method cannot be expected to correctlyperform the separation of frequency components. Hence, theWVD will return a diagram showing interferences. The authors in[14] have recently presented an improvement to their method inorder to solve this problem. This technique combines the time-or-der representation based on short-time Fourier–Bessel expansionand the Wigner-Ville distribution (WVD) [15] which noticeablyimproves the results of the previous method in some cases, aswe can corroborate in the examples.

We present three representative examples to discuss thesepoints: Figs. 5–7 represent the results obtained when processingsignals x1(n), x2(n), and x3(n), respectively.

Fig. 5 represents the results of the method proposed in [14] incomparison with the k-RCCR method. Figs. 6 and 7 represent theresults of the method proposed in [14] and the results obtainedwith its improvement [15] in comparison with the k-RCCR method.

In Figs. 5(a), 6(a), and 7(a) we represent the Fourier–Besselcoefficients of the three input signals, x1(n), x2(n), and x3(n), respec-tively. Note how the two components from each signal can be cor-rectly separated. Once the two main components from the signalsare separated, we can retrieve each signal component separatelyand compute one WVD from every component. Figs. 5(c) and(d), 6(c) and (d), and 7(c) and (d) show the WVD from each signalretrieved. Fig. 5(e) represents the sum of the two images Fig. 5(c)and (d). In the same way, Fig. 6(e) represents the sum of the twoimages Figs. 6(c) and (d) and 7(e) represents the sum of thetwo images Fig. 7(c) and (d). From Fig. 5(e), we can emphasize

Fig. 5. (a) FB coefficients of x1(n); (b) spectrogram of signal x1(n), gi1(x); (c) WVD of the retd; (f) gskel

1 ðxÞ; (g) WV transform of x1(n), f1(x); (h) reconstructed image f WV1r ðxÞ.

the outstanding results of the method [14] applied to x1(n) whosecomponents are well-separated in the frequency domain. On theother hand, when signal components are not well-separated (seeFigs. 6(e) and 7(e)), the method [14] was unable to achieve goodresults since it cannot correctly separate their Fourier–Bessel com-ponents. Figs. 6(f)–(h) and 7(f)–(h) show the result obtained usingthe improved method [15]. Specifically, Figs. 6(f) and (g) and 7(f)and (g) represent the WVD of each signal component, retrievedfrom x2(n) and x3(n), respectively, using the short-time FB expan-sion. Fig. 6(h) shows the sum of images Fig. 6(f) and (g). In thesame way, Fig. 7(h) shows the sum of images Fig. 7(f) and (g). InFig. 6(h), we can observe how in the case of a linear frequency-modulated signal (x2(n)) the interferences are noticeably reducedusing the improved method. In the case of non-linear modulatedsignals (Fig. 7), the improvement achieves a good quality of the sig-nal terms but it does not reduce the interferences.

Figs. 5(b), (f), (g), and (h), 6(b), (i), (j), and (k), and 7(b), (i), (j),and (k) show the images used and obtained using the k-RCCRmethod. In Figs. 5(b), 6(b), and 7(b) we have the spectrogramof the three input signals x1(n), x2(n), and x3(n), that is gi1(x),gi2(x), and gi3(x), respectively. Figs. 5(f), 6(i), and 7(i) representthe skeleton of images gi1(x), gi2(x), and gi3(x), respectively, thatis gskel

1 ðxÞ; gskel2 ðxÞ and gskel

3 ðxÞ (the markers). Figs. 5(g), 6(j), and7(j) represent f1(x), f2(x), and f3(x), respectively, the referenceimages for the morphological operator, these being the images thatcorrespond to the WVD of x1(n), x2(n), and x3(n). Finally, Figs. 5(h),6(k), and 7(k) represent the results of applying our method (usingk = 25), which are the reconstructed images f WV1

r ðxÞ; f WV2r ðxÞ, and

f WV3r ðxÞ. The three reconstructed images contain only the desired

information from the WVD, preserving time–frequency resolutionsand avoiding any cross-components. We can highlight the satisfac-tory results obtained over the three test signals independently ofthe component locations, even when interferences touch signalterms.

rieved signal 1; (d) WVD of the retrieved signal 2; (e) composition of images of c and

Fig. 6. (a) FB coefficients of x2(n); (b) SP of signal x2(n), gi2(x); (c) WVD of the retrieved signal 1 using FB coefficients; (d) WVD of the retrieved signal 2 using FB coefficients;(e) composition of images of c and d; (f) WVD of the retrieved signal 1 using short-time FB coefficients; (g) WVD of the retrieved signal 2 using short-time FB coefficients; (h)composition of images of f and g; (i) marker gskel

1 ðxÞ; (j) WV transform of x2(n), f2(x); (k) f WV2r ðxÞ.

S. Gómez et al. / J. Vis. Commun. Image R. 22 (2011) 401–410 407

Finally, we illustrate the operation of our method in comparisonwith classical techniques. We show an experiment where totaloverlap between signal and interference components is tested.

Specifically, we now compare the k-RCCR results with those ob-tained using WVD 2-D Gaussian kernels. As stated in Section 2,many research papers have been published on this topic with theselection of the correct kernel being the main drawback of thesetechniques since it depends of the type of signal.

In the following example, we use a three component signalcomposed by three harmonically related chirps (instantaneous fre-quencies of f, 2f, and 3f). Here the cross-terms between the firstand third component will land on top of the second componentso as to test the ability of our algorithm to manage complete over-lapping components. Also, filtering with 2-D Gaussian kernels isrecognized to work well with problems of this kind.

The tested chirp signal is composed by three linear frequency-modulated signals. The first frequency-modulated component is adiscrete-time signal with N = 256 samples, having initial normal-ized frequency of f 1

0 ¼ 0:01 and final normalized frequencyf 11 ¼ 0:1. The second component is a signal with the same number

of points, which have initial normalized frequency of f 20 ¼ 2f 1

0 andfinal normalized frequency of f 2

1 ¼ 2f 11 . Finally, the third linear fre-

quency-modulated signal contains the same number of points andhas initial normalized frequency of f 3

0 ¼ 3f 10 and final normalized

frequency of f 31 ¼ 3f 1

1 . The three components of the signal have atime reference for the phase of N

2.Fig. 8 shows the results of filtering the three-component chirp

signal using a 2-D Gaussian kernel, with a temporal smoothingwindow that we call G and a frequency smoothing window thatwe call H. Fig. 8(a) and (b) show the spectrogram and Wigner-Villedistributions of the signal, respectively. Fig. 8(c) shows the resultsof filtering using a 63-point time smoothing Gaussian window anda 63-point frequency smoothing Gaussian window. Fig. 8(d) showsthe results using 127-point G window and 63-point H window. Fi-nally, Fig. 8(c) shows the results of filtering using a 63-point timesmoothing Gaussian window and a 127-point frequency smooth-ing Gaussian window.

Fig. 9 shows the results of applying the k-RCCR method to thesame three-component chirp signal. Fig. 9(a) shows the markerfor the morphological operation, and Fig. 9(b) shows the recon-structed image.

Considering the results shown in Figs. 8 and 9, we can con-clude that the 2-D filtering techniques remove interference termsbut reduce the WVD resolution. As the length of the windows in-crease, the resolution also increase, but it never reaches the WVDresolution. The k-RCCR method achieves a high resolution of com-ponents located at f and 3f frequencies removing the interfer-ences. The drawback is that the component locates at frequency

Fig. 7. (a) FB coefficients of x3(n); (b) SP of x3(n); (c) WVD of the retrieved component 1 using FB coefficients; (d) WVD of the retrieved component 2 using FB coefficients;(e) composition of images of b and c; (f) WVD of the retrieved component 1 using short-time FB coefficients; (g) WVD of the retrieved component 2 using short-time FBcoefficients; (h) composition of images of f and g; (i) marker g3(x); (j) WV transform of signal x3(n); (k) f WV3

r ðxÞ.

Fig. 8. (a) Spectrogram of the three-component chirp signal. (b) WVD of the three-component chirp signal. (c) Result of the filtering using 63-point G window and 63-point Hwindow. (d) Result of the filtering using 127-point G window and 63-point H window. (e) Result of the filtering using 63-point G window and 127-point H window.

408 S. Gómez et al. / J. Vis. Commun. Image R. 22 (2011) 401–410

2f (the case of total overlap) can not be completely recoveredusing the k-RCCR method. Other morphological operators can be

used to improve this situation, and this research will be done infuture work.

Fig. 9. Results of applying our method to a three-component chirp signal. (a)marker; (b) reconstructed image.

Fig. 11. (a) Spectrogram of x5(n); (b) marker gskel5 ðxÞ; (c) WVD of x5(n);

(d) reconstructed image f WV5r ðxÞ.

S. Gómez et al. / J. Vis. Commun. Image R. 22 (2011) 401–410 409

5.3. Application to real signals

In this section, we discuss the operating method applied to twounderwater signals: a signal from a beluga whale, x4(n), and a sig-nal from a tubboat, x5(n).

Figs. 10(a) and 11(a) are the spectrogram of x4(n) and x5(n),respectively. Figs. 10(b) and 11(b) are the skeletons of the spectro-gram that are used as markers, gskel

4 ðxÞ and gskel5 ðxÞ. Figs. 10(c) and

11(c) represent the WVD of x4(n) and x5(n), respectively. Thereconstructed images, f WV4

r ðxÞ and f WV5r ðxÞ, are represented in Figs.

10(d) and 11(d), respectively.From here, we can conclude that our method achieves great re-

sults when it is applied over real signals. Moreover, Fig. 11(d) notonly highlights the satisfactory results of the method over real sig-nals, it also demonstrates the usefulness of the k-reconstruction incomparison with the reconstruction operator (k-reconstructionwith k = 0). In [18], the same signal, x5(n), was used to illustratethe drawback of the reconstruction operator when it is used toreconstruct diagrams whose interference terms overlap the desiredsignal.

5.4. Using other time–frequency distributions

In this section, the results of applying the proposed method toother time–frequency distributions are shown, namely: the Marge-nau–Hill (MH) and the Choi–Williams (CW) distributions. Bothtransforms have different interference geometries in comparisonwith the WVD [1].

Fig. 10. (a) Spectrogram of signal x4(n); (b) marker gskel4 ðxÞ; (c) WVD of x4(n);

(d) f WV4r ðxÞ.

Once again, in this example, the input signals are x3(n) andx4(n). As such, the images used as markers for the k-reconstructionoperation are the same as those used in the previous subsections,mainly images gskel

3 ðxÞ (Fig. 7(b)) and gskel4 ðxÞ (Fig. 10(b)).

Figs. 12(a) and 13(a) display the images corresponding to theChoi–Williams transform of both input signals, respectively. Theseimages have been used as references for the k-reconstruction oper-ator. Figs. 12(b) and 13(b) represent f CW1

r ðxÞ and f CW2r ðxÞ, which

are the reconstructed images from Figs. 12(a) and 13(a) usingthe proposed method.

In the same way, Figs. 12(c) and 13(c) represent the imagesthat correspond to the Margenau–Hill transform of signals x3(n)and x4(n) and are used as references. Figs. 12(d) and 13(d) repre-sent f MH1

r ðxÞ and f MH2r ðxÞ, the reconstructed images.

The correct operation of the method and its application to othertransforms, that have different interference geometries, is corrob-orated by Figs. 12(b) and (d) and 13(b) and (d). It can be observed

Fig. 12. (a) CW distribution of signal x3(n); (b) reconstructed image f CW1r ðxÞ; (c) MH

distribution of signal x3(n); (d) reconstructed image f MH1r ðxÞ.

Fig. 13. (a) CW transform of x4(n); (b) f CW2r ðxÞ, (c) MH transform of x4(n);

(d) f MH2r ðxÞ.

410 S. Gómez et al. / J. Vis. Commun. Image R. 22 (2011) 401–410

how the method strongly reduces the cross-components, therebypreserving time–frequency resolutions.

To conclude, we point out that our method achieves great re-sults when applied over real signals regardless of the interferencegeometry and regardless of the type of transform applied.

6. Conclusions

A new method for the removal of cross-components from time–frequency transformations has been presented. The proposedmethod uses morphological operators to reduce interference com-ponents from time–frequency images. It has been tested not onlyover test signals, but also over real examples of underwater signals.In comparison with other techniques in the same field, our methodovercomes the main drawback, providing a correct operation evenwhen signal and interference terms overlap. Moreover, it can beapplied to distributions with different interference geometry, andit preserves time–frequency resolutions.

The method achieves outstanding results in removing interfer-ence components not only from the WVD, but also from othertypes of transforms such as the Margenau–Hill (MH) or the Choi–Williams (CW) distributions. Its great performance when it is ap-plied over transforms with different interference geometries hasalso been demonstrated.

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