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The use of a two-dimensional Hilbert transform for Wigner analysis of 2-dimensional real signals
Signal Processing 19 (1990) 205-220 Elsevier
205
THE USE OF A TWO-DIMENSIONAL HILBERT TRANSFORM ANALYSIS OF 2-DIMENSIONAL REAL SIGNALS
FOR WIGNER
Yue Min ZHU, Fran~oise PEYRIN (Member EURASIP) and Robert GOUTI'E (Member EURASIP) Laboratoire de Traitement du Signal et UItrasons, URA C N R S 1216, Avenue Albert Einstein, 69621 Villeurbanne Cedex, France Received 9 January 1989
Abstract. A method for Wigner analysis of 2-dimensional real signals, based on a 2-dimensional Hilbert transform is proposed. The method allows to obtain a 2-dimensional Wigner distribution which is more easily exploitable than the Wigner distribution of the 2-dimensional real signal. This is shown both by theoretical demonstrations and by examples on simulated and physical images. Zusammenfassung. Ein Verfahren zur Wigner-Analyse zweidimensionaler reeller Signale auf der Basis einer zweidimensionalen Hilbert-Transformation wird vorgeschlagen. Die Methode liefert eine zweidimensionale Wigner-Verteilung, die leichter auszuwerten ist als die Wigner-Verteilung des reellen zweidimensionalen Signals. Dies wird sowohl theoretisch als auch anhand yon Beispielen simulierter und realer Bilder gezeigt. R6sum6. Nous avons propos6 une m6thode pour l'analyse des signaux r6els bidimensionnels par la transformation de Wigner, en s'appuyant sur une transform6e de Hilbert bidimensionnelle. Cette m&hode permet d'obtenir une transformation de Wigner bidimensionelle plus facile ~ exploiter que la TW du signal r6el bidimensionnel. Ceci est montr6 aussi bien par une d6monstration th6orique que par des exemples sur des images simul6es et physique.
Keywords. Wigner distribution, image analysis, two-dimensional Hilbert transform, two-dimensional analytic signal.
1. Introduction
The Wigner distribution (WD) is now recognized to be a powerful tool for analysis of nonstationary signals [2, 3, 5-7, 10-12]. Recent studies have shown that this signal transform is also interesting for two-dimensional (2-D) image analysis applications [1, 4, 8, 9, 15-17]. Indeed, in this case, the WD offers a space/spatial-frequency representation of the image, allowing a precise analysis of its nonstationarities (texture for example). In the case of a one-dimensional (I-D) signal, it is well-known that the WDs of real signals present interference terms due to the interaction between positive and negative frequencies. These terms are troublesome for analysis of the results, and can be eliminated by applying the WD to the 0165-1684/90/$3.50 O 1990, Elsevier Science Publishers B.V.
analytic signal associated with the real signal, rather than to the real signal itself [3, 10]. In its applications to 2-D signals, the WD raises, as will be shown in the following, the similar interference problem due to the interaction between positive and netative spatial frequencies. In order to avoid this problem and following the same idea as in the 1-D case, it is possible to introduce a so-called '2-D analytic signal', and apply the WD to the associated 2-D analytic signal instead of to the 2-D real signal itself. However, in the 2-D case, the 2-D analytic signal is not unique [13]. There then exist many possibilities of solving the interference problem using the concept of a 2-D analytic signal. In this paper, we propose a method for Wigner analysis of 2-D real signals, based on a 2-D Hilbert trans-
Y.M. Zhu et al. / Hilbert transform Jor Wigner analysis of 2-D signals
206
form. First, we introduce a 2-D discrete Hilbert transform, proposed by Read and Treitel in the context of 2-D recursive filter stabilization [14] to construct a 2-D analytic signal. We then derive the relation between the WDs of the 2-D real signal and its associated 2-D analytic signal. Finally, examples on both simulated and physical images are given to illustrate advantages brought by the 2-D analytic signal WD for the analysis of 2-D real signals.
This leads to the following spectral relation:
~Y(f( v) =j(1 - 2H ( v) )ff'f( v), where H(v) defined by H(v)=
(7)
denotes the Heaviside function
10 if v > 0 , ' if v = 0 , if v < 0 .
(8)
Thus, we obtain the spectrum of the analytic signal given by 2. Background on 1-dimensional analytic signal
~z(v) = 2H(v)o%f(v).
Let f ( t ) be a temporal real signal. Its associated analytic signal denoted by z(t) is the complex signal defined by
z(t) = f ( t ) + j g ( t ) ,
(1)
g(t) = I p . v 7r
(2)
with [" f(~') dr, dR t - z
where P.V means the Cauchy principal value. In definition (1) it can be seen that the analytic signal is directly obtained from the real signal by adding an imaginary part g(t). This imaginary part is known as the Hilbert transform (HT) of f ( t ) and will be denoted Y(f(t) in the following. The HT can also be expressed as a convolution o f f ( t ) with a kernel k(t)
~ f ( t ) = ( f * k)(t),
(3)
where "*" means convolution and k is the kernel given by 1 k(t) = P.V. - - . 7rt
(4)
The spectrum of k(t) is given by
J;k(v) = - j sign(v),
(5)
where ~ denotes the Fourier operator and sign(. ) is the sign function defined by sign(t)={-1, 1, Signal Processing
if t < 0 , if t > 0 .
(6)
(9)
The analytic signal then possesses the spectral property to have no negative frequencies while it contains all the spectral information on the real signal. By duality, it can be shown that the real and imaginary parts of the FT of a causal real signal form a Hilbert transform pair. This property will be exploited in [14]. From the discussion above, we underline the point that the analytic signal can be completely determined using the HT. Based on these remarks, we shall in the following section construct our 2-D analytic signal by using a discrete 2-D Hilbert transform.
3. 2-dimensional Hilbert transform and 2-dimensional analytic signal
In this section, we develop the definition of a 2-D analytic signal based on the 2-D discrete HT proposed by Read and Treitel [14]. For this purpose, we first give the formal definition of a 2-D analytic signal and then determine it after deriving, from Read and Treitel's discrete definition, the Hilbert transform for 2-D continuous-variable signals.
3.1. Formal definition of a 2-D analytic signal Let f(x, y) be a 2-D real signal. By analogy with the I-D case we define its associated 2-D analytic
Y.M. Zhu et aL / Hilbert transform for Wigner analysis of2-D signals
signal as a 2-D complex signal z(x, y), such that
z(x, y) =f(x, y) +jg(x, y),
(10)
where g(x,y) is a real function assumed to be related to f(x, y) by a two-dimensional convolution with an appropriate kernel k(x, y): g(x, y) = ( f * k)(x, y).
(11)
Taking the Fourier transform of both sides of (10) and in view of (11), we obtain
fz(u, v)= (1 + j ~ k ( u , v))~f(u, v).
(13)
14)
Relation (14) gives the frequential representation of the 2-D analytic signal. From (10) and (14), it can easily be shown that
fig(u, v)=j(1 -2re(u, v))o~f(u, v).
f lk(u, v)= -J;k(u, v).
(18)
o~k(u, v)= -~j[sign(u) + sign(v)].
(19)
So,
Taking the inverse FT of both sides of (19) gives the kernel function k(x, y):
k(x,y)=~(8(x) P.V 1-!-+~(y)P.V~x). ~ry (20) To determine re(u, v), we can insert (13) in (19), getting
re(u, v)= 4~[sign(u) + sign(v) + 2],
relation (12) can be written as
~z(u, v)= 2re(u, v)~f(u, v).
This function being pure imaginary and odd, k is hence real and odd. Thus we have
(12)
Defining a filter function re(u, v) by
m(u, v)= ~(1 +jffk(u, v)),
207
(15)
In the following we derive the kernel k(x, y) and the filter function by transposing the Read and Treitel 2-D discrete HT to the continuous case.
(21)
namely, i
m(u,v)=
if u > 0 , v > 0 , if u < 0 , v < 0 , elsewhere.
(22)
The filter re(u, v) is depicted in Fig. 1. It can be seen that the thus obtained 2-D analytic signal z(x,y) is a complex signal whose FT vanishes in quadrant III and is twice (resp. half) the FT of the 2-D real signal in quadrant I (resp. in quadrants II and IV).
3.2. 2-dimensional continuous Hilbert transform derived from Read and Treitel discrete Hilbert transform The definition of the 2-D discrete HT proposed by Read and Treitel and its reformulation are recalled in Appendix A. Transposing the discrete definition given by (A7) in Appendix A into the continuous case, we obtain
Ik(u,v)=
if u < 0 , v < 0 , elsewhere.
(16)
Relation (16) can be further written in the following form:
o~ ~k(u, v)=lj[sign(u)+sign(v)].
(17)
011)
Ill
0
l 2
(II)~ v 1/2 u Fig. 1. Characteristic of the filter m(u, vL Vol. 19, No. 3, March 1990
Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
208
In the next section, we apply the above obtained 2-D analytic signal to the WD and study its effect on the interference terms.
Indeed, the 2-D real signal f(x, y) being the real part of its associated analytic signal can be written as
f(x, y) = ½[z(x, y) + z * (x, y)]. 4. Wigner distribution of the 2-dimensional analytic signal
Using the bilinearity property of the WD [3] yields
4.1. Definition of 2-dimensional Wigner distribution I f f ( x , y) represents a 2-D signal (real or complex), its WD is a function of 4 variables, two spatial variables x, y and two spatial frequency variables u, v, defined by [9]:
Wt(x , y , u , v )
=
I~ 2 f ( x
+~c~,y+~fl)l
,
f , (X-~oq 1 y--½[3) X exp{--2jrr(ua + v/3)} d a d/3,
(23)
where " * " denotes complex conjugation. The WD can also be equivalently defined in terms of the FT F(u, v) o f f ( x , y) by
fJR F(u+~); 2
(25)
Wt(x,y, u, v)=¼[Wz(x,y, u, v) + Wz(x, y, - u , - v ) + 2 Re(Wz.z*(X,y, u, v))].
(26)
Relation (26) shows that the WD of a 2-D real signal is composed of two symmetrical terms W~(x,y, u, v) and W._(x,y,-u,-v) to which is added the interference term Re( W..: * ( x, y, u, v ) ) which does not bring any information in itself. To reduce its influence, we shall use the above introduced 2-D analytic signal.
4.2. Wigner distribution of the 2-dimensional analytic signal The 2-D analytic signal defined by (10-14) can be rewritten as follows:
v+~)
z(x, y) = [(6 + j k ) * f ] ( x , y),
(27)
F*(u-½~1, v-½~) exp{Zj'rr(x~7 +y~:)} dr/d~. (24) The 2-D WD presents many interesting properties for 2-D signal processing applications. The first of them is its reality. The WD of any 2-D real or complex signal is always a real function, which offers great facilities of manipulation. Despite of its reality, the WD still contains phase information on the 2-D signal since it is a reversible transform. The WD has also properties with respect to operations such as translation, modulation, scaling, multiplication and convolution. In addition, the WD posseses particular properties such as support conservation and energetic characteristics. However, the behaviour of the WD is sometimes difficult to interpret because of problems such as interference terms due to the interaction between positive and negative spatial frequencies. Signal Processing
where 6(x, y) is a 2-D Dirac delta function, and k(x, y) has been defined by (20). The WD of the convolution of two signals being equal to the convolution, with respect to the spatial variables, of the two signals WDs [3], the WD of z is then given by
Wz(x, y, u, v) = fR 2 W s ( x - x', y - y', u, v) x W~+jk(X', y', u, V) dx' dy'.
(28)
The obtention of the 2-D analytic signal WD then returns to evaluate the WD Ws+jk of the 2-D distribution function ( 6 + j k ) . The calculation of the latter being tedious, we give only the result, the intermediate derivation steps of which can be found in Appendix B. The WD of the 2-D analytic signal z(x,y) is
Y.M. Zhu et al. / Hilbert transformfor Wigner analysis of 2-D signals given by
W~(x, y, u, v) = (½+H(u)) I Wt(x-x',y, u, v) Jn ×
sin(4~rx'u)
dx'
7rX ~
+(½+I-I(v))f.
Wf(x, y - y ' , u, v)
×
sin(4~ry'v) . , ~ry' oy
+ fR2 W j ( x - x ' , y - y ' , u , v ) sin(4wx'u) sin(4~ry'v). , ox dy'
27r2x'y '
+
~2
W j ( x - x ' , y - y ' , u, v) x
cos(4wx'u) cos(4~ry'v)
2 2x,y ,
dx'dy'
+½Wr(x, y , u, v), (29) where the Heaviside function H ( . ) is defined by (8). Relation (29) shows that Wz(x, y, u, v) involves three types of filters: --A one-dimensional ideal low-pass filter with a (4v Sin(4~rtv)~ cut-off frequency 2v\ 41rtv /" two-dimensional ideal low-pass filter with cut-off frequencies 2u and 2v in the (u, v) plane (16uv Sin(47rxu) sin(4xryv)']
--A
4~rxu
4wyv ]"
two-dimensional ideal high-pass filter with cut-off frequencies 2u and 2v in the (u, v) plane 16uv cos(47rxu) cos(4~ryv)~
--A
4rrxu
47ryv
/"
Because of the filtering effects of these filters, and for great u, v values, the four first integral terms in (29) become approximately first term ~ ± Wr, second term ~ + IVy, third term = ± ~1 Wr, fourth term ~ 0.
209
The sign + l or - 1 in the above terms is taken according to the sign of u o r / a n d v. Hence, the 2-D analytic signal WD W~(x, y, u, v) behaves for all ( x , y ) 6 R 2 in the (u, v) plane as follows: quadrantI : W~4Wr, quadrant II : Wz~ Wl, quadrant III: Wz ~ 0 , quadrant IV: Wz ~ WI. We can then observe that - - Wz cancels in quadrant III. - - W~ has the same amplitude as the 2-D real signal WD in quadrants II and IV. - - In quadrant I, Wz is four times greater than the 2-D real signal WD. These remarks mean that the influence of the redundant information (recall that in the 2-D real signal WD, there always exists the identity of quadrants I and III, and of quadrants II and IV of the (u, v) plane), and of the interference terms which occur near (u, v) = (0, 0), is considerably reduced in the 2-D analytic signal WD. In the following section, we shall give some examples for illustrating the above discussed theoretical results.
5. A p p l i c a t i o n
examples
As mentioned previously, the WD gives a 4-D space/spatial-frequency representation of a 2-D image. For a fixed spatial point, this 4-D representation reduces to a 2-D function which can be considered as a local spectrum corresponding to that point of the image, so applying the WD to successive spatial points of an image allows to track its spectral variations in space. Of course, in practice, it is not always necessary to apply the WD to all the points of the image, since, on one hand, information given by the WD at a number of points may be sufficient, and on the other hand in some applications, we are only concerned with some points of interest. In our present problem, as can be seen in the foregoing formulations, the interference being due to the interaction between Vol, 19, No. 3, March 1990
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KM. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
positive and negative spatial frequencies, all the local spectra obtained from the WD of a real image can contain interference terms. Thus, for illustrating the interference p h e n o m e n o n , it is sufficient to consider the WD only at some particular image points (2-D slices of the 2-D WD), the conclusion remaining general for all other image points. In order to verify the theoretical performance of 2-D analytic signal WDs and to illustrate more clearly their advantages over 2-D real signal WDs, we first present some simulation studies by using two simulated test 2-D images whose characteristics are a priori known. Then, an application to a physical image is considered.
U
y
x
i!li m ~ 1 1 D JlD D qRD b qP m
-
i
Q
Signal Processing
•
a
i I
~ll I Ill
i
i I¢
i
II
Ii
m m I t | • 0 |l|l!
I I
i i • i * II !1 Itlt l|lltltlllll| i t i I I e 4 O il | II 6 D • | • • Q I | • W D D ~ O 4 t P e I Ba I g Q i i ~ Q • W B 4 ~ 4~ ~m g e W II ~ S I C l l l l
. - . - " 7.i : :i : : i: : :i : :i : :i ~
E X A M P L E A. We first consider a simple case where the simulated test image contains solely one component. In this case, the interference terms come from only the interaction between positive and negative spatial frequencies. Such an image of size 128 × 128 is represented in Fig. 2a. As discussed above, for illustrating its complete 2-D WD, in principle we have to display at the same time 16 384 local spectra of size 128 × 128. Obviously, such a presentation is not realistic and neither necessary. For that reason, we choose two representative image points to illustrate the interference phenomenon. Figure 3 shows the WDs of the real image at two image points (32, 32) and (96, 96). At first viewing, these 2-D WD slices present some symmetrical structure with respect to the origin of the frequency plane, and the four quadrants, in each of which a greater and brighter spot is present, are identical. The four greater white spots represent in fact the spectral information related to the modulation characteristics of the image (as can be seen in Fig. 2a). In particular, their position in the frequency plane tells the variation degree of the grey levels of the image in that region around the point in question. Thus, tracking in space their frequential position changement allows to get an insight into how the image's spectral content varies with spatial position. This can be observed in Fig. 3 where the fact that the four greater white spots in Fig. 3b are further from the
m
i I I llO BI OR is a a 6 qD I IF Q I ~ ~ ~ • tlb /~ ~ u i qP 9 0 B ~ ~ m B i
ID
Ib i I
I i i l I 10111
I
I
II,
II ii
I
~II • i Ik I If
m
ii
8 8 1 1 1 1
dl ~ 6 i I
i i I ii I iP
iii I
II
(a)
(b) Fig. 2. Simulated test image with one component and its 2-D Hilbert transform. (a) Simulated test image. (b) Its 2-D HUbert transform. origin than those in Fig. 3a agrees perfectly with the theoretical prediction. Note however that the above analysis and interpretation are not always easily assessed because of the fact, as we can observe in Fig. 3, that the useful information (the four white spots) are drowned by the parasite
Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2,D signals
Ca)
(b) Fig. 3, 2-D WD slices of the real image illustrated in Fig. 2a, (a) 2-D WD slice corresponding to the point (32, 32). (b) 2-D WD slice corresponding to the point (96, 96).
interference terms due to the interaction between positive and negative spatial frequencies. Furthermore, it can be seen that only one of the four white spots is sufficient to represent the 2-D WD slice because of the symmetry characteristic of the latter one.
211
For reducing these interference terms and redundant information, we now calculate, instead of the WD of the real image, that of its associated analytic image (2-D analytic signal). In Fig. 2b, we have given the 2-D Hilbert transform of the real image, obtained by using Read and Treitel's formula, The 2-D WD slices of the thus obtained analytic image (always at the same points as in Fig. 3) are shown in Fig. 4, from which it can be seen that (i) the 2-D slice of the analytic image Wz is zero in quadrant III; (ii) the interference terms around the origin are largely reduced; (iii) the redundant white spots situated at quadrants II and IV are attenuated; and (iv) the informational spot in quadrant I is best represented. The simulation results are thus perfectly conform to the theoretical analysis. In short, the informational white spot representing the spectral characteristic of the image is more apparent in the 2-D WD slices of the analytic image.
EXAMPLE B. We now consider a more complex case where the siinulated test image is composed of two sinusoidal images, the one being a pure cosine image which has a constant frequency, and the other being a so-called chirp image whose spectral content varies with spatial position. Such a 128 x 128 image is represented in Fig. 5a. It is known that in this case, the WD of the image presents interference terms due to the interaction between the two components. At the same time, according to our foregoing discussion, it also presents the interference terms coming from the interaction between positive and negative spatial frequencies. The WDs of the above real image at two image points (33, 33) and (47, 47) are given in Fig. 6, in which we notice, around the origin of the frequency plane, the presence of the interference terms coming from the interaction between positive and negative spatial frequencies. In either quadrant I or III, we remark two greater and brighter spots, between which are present some bright annulus representing the interference terms coming from the interaction between the two components. These two white Vol. 19, NO. 3, March 1990
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Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
(a)
(b)
(b)
Fig. 4. 2-D WD slices of the analytic image associated with the real image illustrated in Fig. 2a. (a) 2-D WD slice corresponding to the point (32, 32). (b) 2-D WD slice corresponding to the point (96, 96).
Fig. 5. Simulated test image with two components and its 2-D Hilbert transform. (a) Simulated test image. (b) Its 2-D Hilbert transform.
spots are in fact the two f r e q u e n c y c o m p o n e n t s r e p r e s e n t i n g r e s p e c t i v e l y the two i m a g e c o m p o n e n t s . W e also o b s e r v e , in the two local s p e c t r a from the W D , t h a t one o f the two f r e q u e n c y c o m p o n e n t s k e e p s a l w a y s the s a m e p o s i t i o n in the f r e q u e n c y p l a n e when p a s s i n g f r o m the local spec-
t r u m at p o i n t (33, 33) to that at (47, 47), this o n e r e p r e s e n t i n g the cosine i m a g e c o m p o n e n t , a n d that the other, in contrast, d i s p l a c e s t o w a r d h i g h e r frequenties, it r e p r e s e n t i n g the c h i r p i m a g e c o m p o n e n t . This is precisely in a g r e e m e n t with the t h e o r e t i c a l p r e d i c t i o n , a n d shows again the ability
Signal Processing
Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
(a)
(b)
213
(a)
(b)
Fig. 6. 2-D WD slices of the real image illustrated in Fig. 5a. (a) 2-D WD slice corresponding to the point (33, 33). (b) 2-D WD slice corresponding to the point (47, 47).
Fig. 7. Smoothed versions of the same 2-D WD slices illustrated in Fig. 6. (a) Smoothed 2-D WD slice corresponding to the point (33, 33). (b) Smoothed 2-D WD slice corresponding to the point (47, 47).
of the WD to track in space the spectral variations in an image. However, it can be seen that the above spectral analysis by means of the 2-D WD slices is not easily achieved because of the presence of the interference terms. To cope with this problem, we can seek to process, as usually done until now by
most of the researchers, some smoothing operations to attenuate the interference terms. But, as we shall show in the following, smoothing operations alone would not be sufficient to minimize the influence of the interference terms and to eliminate redundant information. For illustrating this, we give in Fig. 7 the smoothed version of the VoL 19, No. 3. March 1990
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Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
same 2-D WD slices as illustrated in Fig. 6. The smoothing procedure was the following: first choose a 7 x 7 smoothing neighbourhood centered at the considered point, then compute the 49 2-D WD slices within the zone, finally averaging all the obtained 2-D WD slices produces the smoothed local spectrum corresponding to the central point. C o m p a r e d with Fig. 6, interference terms in Fig. 7 have been largely reduced. This is particularly evident in Fig. 7b. However, the interference terms around the origin of the frequency plane have not been completely eliminated, and the symmetry redundancy remains still. Now, if we use both the 2-D analytic image and the smoothing procedure, the legibility of the local spectra can be considerably improved. In Fig. 5b, we have represented the 2-D Hilbert transform of the original test image. Figure 8 shows the 2-D slices of the 2-D analytic image WD, in which we observe, compared with Fig. 6, the total disappearance of the interference terms around the origin of the frequency plane as well as the redundant symmetry structure. A more apparent result is again found in Fig. 9 where we show the smoothed versions of the same 2-D WD slices as illustrated in Fig. 8. This example shows that the 2-D analytic image allows to efficiently eliminate the interference terms due to the interaction between positive and negative spatial frequencies, and that after that, further reduction of residual interference terms can be accomplished by using other smoothing techniques.
im
(a)
(b) EXAMPLE
C. As a last example, we take a phy-
sical image of size 512x512 shown in Fig. 10. Generally, such kind of images cannot be considered as m o n o c o m p o n e n t images. This means that its WD can present interference terms which may both come from the interaction between positive and negative spatial frequencies, and from the interaction between the various components in the image. In this case, as shown in the precedent example, the use of the 2-D analytic image should be the first thing to do in all tentatives of reducing them. Signal Processing
Fig. 8. 2-D WD slices of the analytic image associated with the real image illustrated in Fig. 5a. (a) 2-D WD slice corresponding to the point (33, 33). (b) 2-D WD slice corresponding to the point (47, 47).
We have, in this example, chosen two regions (128 x 128 pixels) of interest over which the WD was applied. The first one concerns a sub-image around the point (134, 168), and the second a sub-image around the point (262, 135) (illustrated respectively on the upper and lower left corner of
Y.M. Zhu et at / Hilbert transform for Wigner analysis of 2-D signals
(a)
(b) Fig. 9. Smoothed versions of the same 2-D slices illustrated in Fig. 8. (a) Smoothed 2-D WD slice corresponding to the point (33, 33). (b) Smoothed 2-D WD slice corresponding to the point (47, 47). Fig. 10b). A l m o s t all the p o i n t s o f significant a m p l i t u d e in the o b t a i n e d 128 x 128 2-D W D slices b e i n g c o n c e n t r a t e d in a s m a l l e r r e g i o n a r o u n d the origin, we r e p r e s e n t , in all the f o l l o w i n g figures, o n l y a 64 x 64 v e r s i o n a r o u n d the origin o f the original results (at the s a m e r e s o l u t i o n ) in o r d e r to a c h i e v e a b e t t e r visual a s s e s s m e n t o f the difference b e t w e e n the different local s p e c t r a f r o m the W D .
215
(a)
(b) Fig. 10. Original women image, its sub-images and the 2-D Hilbert transforms of the sub-images. (a) Women image. (b) Left half plane: sub-images; right half plane: 2-D Hilbert transforms of the sub-images.
F i g u r e 11 shows the 2-D W D slices (64 × 64) o f the w o m e n s u b - i m a g e s c a l c u l a t e d at the two a b o v e points. As b e f o r e , we first give in Fig. 12 the s m o o t h i n g results o f the 2-D W D slices i l l u s t r a t e d in Fig. 11. It is seen that there is little difference b e t w e e n the 2-D W D slices a n d their s m o o t h e d versions, e x c e p t that the latters have b e e n slightly Vol. 19, NO. 3, March 1990
Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
216
(a)
(a)
(b)
(b)
Fig. 11. 2-D W D slices of the women sub-images. (a) 2-D W D slice corresponding to the point (134, 168). (b) 2-D W D slice
Fig. 12. Smoothed versions of the same 2-D WD slices illustrated in Fig. 11. (a) Smoothed 2-D WD slice corresponding
corresponding to the point (262, 135).
to the point (134, 168). (b) Smoothed 2-D WD slice corresponding to the point (262, 135).
attenuated at high frequencies. Coarsely, as expected, the first local spectrum (Fig. 1 la) corresponding to the image point (134, 168) is characterized by a set of dominant spatial frequencies located almost along the v axis of the frequency plane. This is due to the fact that the first sub-image (illustrated on the u p p e r left corner of Fig. 10b)
presents more abrupt grey level variations in the y direction than in the x direction. In contrast, the second local spectrum (Fig. 1 lb) corresponding to the image point (262, 135) differs from the above one. The distribution of its dominant spatial frequencies is around the origin of the frequency plane. Sometimes, these frequencies are more
Signal Processing
Y.M. Zhu et aL / Hilbert transform for Wigner analysis of 2-D signals
(a)
(a)
(b)
(b)
217
Fig. 13. 2-D WD slices of the analytic images associated with the women sub-images in Fig. 10b. (a) 2-D WD slice corresponding to the point (134, 168). (b) 2-D WD slice corresponding to the point (262, 135).
Fig. 14. Smoothed versions of the same 2-D WD slices illustrated in Fig. 13. (a) Smoothed 2-D WD slice corresponding to the point (134, 168). (b) Smoothed 2-D WD slice corresponding to the point (262,135).
n u m e r o u s in q u a d r a n t s I a n d I I I t h a n in q u a d r a n t s II a n d IV. This set o f c h a r a c t e r i s t i c f r e q u e n c i e s is r e s p o n s i b l e for the r e l a t i v e l y h o m o g e n e o u s varia t i o n s t r u c t u r e o f the grey levels o f the s u b - i m a g e ( i l l u s t r a t e d o n the l o w e r left c o r n e r o f Fig. 10b).
have s h o w n in p r e c e d e n t s i m u l a t e d e x a m p l e s , the i n t e r f e r e n c e terms d u e to the i n t e r a c t i o n b e t w e e n positive a n d negative s p a t i a l frequencies can g r e a t l y m a s k useful i n f o r m a t i o n . In Fig. 13, we have r e p r e s e n t e d the 2-D W D slices o f the a n a l y t i c i m a g e s a s s o c i a t e d with the two w o m e n s u b - i m a g e s , the i n v o l v e d 2-D H i l b e r t t r a n s f o r m s ( i l l u s t r a t e d on
H o w e v e r , the f u r t h e r p r e c i s e a n d exact a n a l y s i s o f the a b o v e results m a y b e difficult, since, as we
Vol. 19. No. 3. M a r c h 1990
218
Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
the right half plane of Fig. 10b) of the sub-images having been calculated by using Read and Treitel's formula. Comparing Fig. 13 with Fig. 11, we can observe the significant difference between the WDs of the real image and its associated analytic image. Indeed, careful examinations show that some dominant components, which did not appear in the local spectra from the WDs of the real images, are now revealed in those from the WDs of the analytic image. We can also remark the changement of the relative brightness of the dominant components in Fig. 13. In short, characteristic frequencies seem to be better represented in Fig. 13 than in Fig. 11. Sometimes, further improvement in local spectrum feature visibility can be achieved by employing the same smoothing procedure used above. The thus obtained results are shown in Fig. 14. Notice here that the combined use of the 2-D analytic signal and the smoothing procedure provides valuable results of primary interest. Of course, further work is needed before one can realize, for example, image feature extraction as well as pattern recognition using the WD based spectral analysis.
Appendix A. Read and Treitel 2-dimensional discrete Hilbert transform The definition of the 2-D discrete Hilbert transform (HT) proposed by Read and Treitel was introduced in the following manner [14]. Let f ( n , , n2) be a finite and causal discrete pulse verifying f ( n , , n2) = 0
for n, >! N,/2 or n2 >! N2/2
where n, varies over the discrete set {0, 1. . . . , N, 1} and n2 varies over the discrete set {0, 1 , . . . , N 2 - 1}. The discrete Fourier transform (DFT) of this pulse can be expressed as
F(k,, k2)= PR(k,, k2)+jPI(k,, k2),
Signal Processing
(A.2)
where PR(k,, k2) and PI(k,, k2) are respectively the real part and imaginary part of the complex spectrum F( k,, kz). Read and Treitel have defined a 2-D discrete Hilbert transform (DHT) by considering that PR(k,, k2) and PI(kl, k2) form a 2-D HT pair. By inverse DFT and because f ( n , , n2) is a real causal discrete pulse, they obtained DFT
6. Conclusion The WD offers a new approach for analysis of the local spectral characteristics of 2-D signals (or images). In practice however, the WD can produce interference terms which are often troublesome for interpretation of the results. In particular, in its applications to 2-D real signals, the influence of the interference terms coming from the interaction between positive and negative spatial frequencies may be not negligible. For reducing it, we have, in this paper, proposed a practical method by using a 2-D analytic signal based on a known 2-D discrete Hilbert transform. The proposed 2-D analytic signal WD exhibits an important reduction of interference terms, and has been shown to be more easily exploitable than the 2-D real signal WD.
(A.1)
'
PI(n,, n2) = - j [ s g n ( n , , n2)
+ b d y ( n , , n2)] D F T - ' PR(n,, n2) ,
(A.3)
with sgn(n,, n2) 0 < n, <½N,, 0 < n2<½N2,
f 1, 0,
½Nl
~l N 2 < 1"12< N 2 ,
elsewhere, (A.4)
and
{"
bdy(n,, n2)
n,=0, 0< n2<½N2, n,=0, ½N2
-1, 0,
n2=0, 0 < n, <½N1, nz = 0, 1N, < nl < N I , elsewhere.
(a.5)
Y.M. Zhu et al. / Hilbert transformfor Wigneranalysis of 2-D signals Now, if we call k(n~, n2) the kernel of the 2-D D H T according to the notation in Section 2, (A.3) can be written as
219
tigating the four following integrals: fR 2 sign(u + r/) sign(u - r/)
D F T 1 PI(nl, n2)
x exp{4j~r(xr/+y¢)} d r / d sc
= D F T l k(n~, n2) D F T -~ PR(nl, n2), 6(Y) sin(4~rxIu I) - J6(x, y), 2"rrx
-
(A.6) where
L
2 sign(u + r/) sign(v - ~:)
D F T 1 k(n~, n2) = - j ( s g n ( n , , n2) + b d y ( n ~ , n2)).
(A.7)
x exp{4j~r(xr/+ys¢)} d r / d sc 1
A p p e n d i x B. D e r i v a t i o n o f
=
Wa+jk
The Wigner distribution (WD) W~+jkof the 2-D distribution function (~ + j k ) is derived as follows: According to the bilinearity property of the WD, it can be written
47r2xy exp(--4j~r(xu --yv)),
~2 sign(v + ~:) sign(u -- rl) x exp{4jw(xr/+ys~)} d r / d sc 1
4~r2xy exp(4jrr(xu - yv)),
Wa+jk(X, y, u, v) = Wa+ W i t + 2 Re[ W~jk].
(B.1)
x exp{4j~r(xrl +ys¢)} d r / d s¢
It can easily be shown that
W~(x, y, u, v) = 6(x, y).
fu2 sign(v + sc) sign(v - ~:)
(B.2)
To calculate Wjk, we use the frequential definition (24). In this case, replacing the integrand function in (24) by ~ k defined in (19) yields
8(x) sin(4~y]v]) l a ( x , y ) . 2wy 4 Thus, we get
Wjk(X, y, U, V)
Wjk(X, y, U, V) = IR 2 [sign(u + rl) + s i g n ( v + ~)] • [sign(u - r/) + sign(v - ~:)] exp{4j'rr(xr/+ y~:)} d rI d~
sin(4~ylv]) = 8(y) sin(4~rxlu[) - - g(x) 27rx 2~ry
-~
cos 4,rr(xu-yv) 2~r2xy
1 ~6(x, y).
(B.4)
We now evaluate the term W~ak : = f~2 [sign(u + rl) sign(u - r/) W~ak(X, y, U, V) + sign(u + r/) sign(v - ~) = JR 2 sign(u -- r/) sign(v --~:) + sign(v + ~:) sign(u - 7/) x exp{4j~r(xr/+ y~:)} d r / d sc
+ sign(v + ~) sign(v - ~)] x exp{4j~r(xr I + ys¢)} d r / d s e.
(B.3)
The integrals in (B.3) can be evaluated by inves-
.{8(y) . 8(x) . = -J ~2-rrx exp(4j-rrxu) + 2-~-y exp(4j~ryv)).
(s.5) Vol. 19, No. 3, March 1990
220
Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
Consequently, 2 R e [ Wa,jk] = 8 ( y ) sin(4~rxu) + ~ ( x ) s i n ( 4 r r y v ) . ~rx ~ry
(B.6)
It results f r o m (B.2), (B.4) a n d (B.6) t h a t
Wa+jk(x, y, u, v) = I s ( x , 2 y ) q cos 4rr(xu - y u ) 2~rZxy + a ( y ) sin(4crxu)(½sign(u) + 1) ~x +
~(x) ~ry
s i n ( 4 ~ y v ) ( ½ s i g n ( v ) + 1).
(B.7)
References [1] R. Bamler and H. Gliinder, "The Wigner distribution function of two-dimensional signals. Coherent-optical generation and display", Optica Acta, Vol. 30, 1983, pp. 1789-1803. [2] G.F. Boudreaux-Bartels and T.W. Parks, "Time-varying filtering and signal estimation using Wigner distribution synthesis techniques", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-34, June 1986, pp. 442-451. [3] T.A.C.M. Claasen and W.F.G. Mecklenbraiiker, "The Wigner distribution--A tool for time-frequency signal analysis", Philips J. Res., Vol. 35, 1980, pp. 217-250, 276300, 372-383. [4] G. Cristobal, J. Bescos, J. Santamaria and J. Montes, "Wigner distribution representation of digital images", Pattern Recognition Letters, Vol. 5, 1987, pp. 215-221. [5] B. Escudi6, "'Repr6sentation en temps et fr6quence des signaux d'6nergie finie: analyse et observation des signaux', Annales des T~l~communications, Vol. 34, No. 3-4, 1979, pp. 101-111.
Signal Processing
[6] P. Flandrin, "Some feature of time-frequency representations of multicomponent signals", Proc. Internat. Conf. on Acoust. Speech Signal Process., 1984, pp. 41B4.1-B4.4. [7] P. Flandrin and B. Escudi6, "Time and frequency representation of finite'energy signals: A physical property as a result of an Hilbertian condition", Signal Process., Vol. 2, No. 2, 1980, pp. 93-100. [8] L. Jacobson and H. Wechsler, "A theory for invariant object recognition in the frontparallel plane", IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. PAM1-6, No. 3, May 1984, pp. 325-331. [9] L. Jacobson and H. Wechsler, "The Wigner distribution and its usefulness for 2D image processing", Proc. 6th Internat. Joint Conf. on Patter Recognition, Munich, October 1982, pp. 19-22. [10] C.P. Janse and A.J.M. Kaizer, "Time-frequency distributions of loudspeakers: the application of the Wigner distribution", J. Audio. Engrg. Soc., Vol. 31, No. 4, 1983, pp. 198-223. [11] W. Martin and P. Flandrin, "Wigner-Ville spectral analysis of nonstationary processes", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-33, December 1985, pp. 1461-1470. [12] F. Peyrin and R. Prost, "A untied definition for the discrete-time, discrete frequency and discrete time-frequency Wigner distribution", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-34, August 1986, pp. 858-867. [13] F. Peyrin, Y.M. Zhu and R. Goutte, "Extension of the notion of analytic signal for multidimensional signals. Applications to images", in: I.T. Young et al., eds., Signal Processing 111: Theories and Applications, North-Holland, Amsterdam, 1986, pp. 677-680. [14] P.R. Read and S. Treitel, "The stabilization of twodimensional recursive filters via the discrete Hilbert transform", IEEE Trans. Geosci. Electron., 1973, Vol. GE-I1, pp. 153-207. [15] T. Reed and H. Wechsler, "Tracking of nonstationaries for texture fields", Signal Process., Vol. 14, 1988, pp. 95-102. [16] Y.M. Zhu, F. Peyrin and R. Goutte, "Transformation de Wigner-Ville: description d'un nouvel outil de traitement du signal et des images", Ann. Tdl~communications, Vol. 42, No. 3-4, 1987, pp. 105-118. [17] Y.M. Zhu, F. Peyrin and R. Goutte, "Utilisation de la transformation de Wigner-Ville pour le traitement num6rique des signaux et des images: mise en oeuvre et interpretation", llbme CoIL GRETSI, Nice, 1987, pp. 21-24.
205
THE USE OF A TWO-DIMENSIONAL HILBERT TRANSFORM ANALYSIS OF 2-DIMENSIONAL REAL SIGNALS
FOR WIGNER
Yue Min ZHU, Fran~oise PEYRIN (Member EURASIP) and Robert GOUTI'E (Member EURASIP) Laboratoire de Traitement du Signal et UItrasons, URA C N R S 1216, Avenue Albert Einstein, 69621 Villeurbanne Cedex, France Received 9 January 1989
Abstract. A method for Wigner analysis of 2-dimensional real signals, based on a 2-dimensional Hilbert transform is proposed. The method allows to obtain a 2-dimensional Wigner distribution which is more easily exploitable than the Wigner distribution of the 2-dimensional real signal. This is shown both by theoretical demonstrations and by examples on simulated and physical images. Zusammenfassung. Ein Verfahren zur Wigner-Analyse zweidimensionaler reeller Signale auf der Basis einer zweidimensionalen Hilbert-Transformation wird vorgeschlagen. Die Methode liefert eine zweidimensionale Wigner-Verteilung, die leichter auszuwerten ist als die Wigner-Verteilung des reellen zweidimensionalen Signals. Dies wird sowohl theoretisch als auch anhand yon Beispielen simulierter und realer Bilder gezeigt. R6sum6. Nous avons propos6 une m6thode pour l'analyse des signaux r6els bidimensionnels par la transformation de Wigner, en s'appuyant sur une transform6e de Hilbert bidimensionnelle. Cette m&hode permet d'obtenir une transformation de Wigner bidimensionelle plus facile ~ exploiter que la TW du signal r6el bidimensionnel. Ceci est montr6 aussi bien par une d6monstration th6orique que par des exemples sur des images simul6es et physique.
Keywords. Wigner distribution, image analysis, two-dimensional Hilbert transform, two-dimensional analytic signal.
1. Introduction
The Wigner distribution (WD) is now recognized to be a powerful tool for analysis of nonstationary signals [2, 3, 5-7, 10-12]. Recent studies have shown that this signal transform is also interesting for two-dimensional (2-D) image analysis applications [1, 4, 8, 9, 15-17]. Indeed, in this case, the WD offers a space/spatial-frequency representation of the image, allowing a precise analysis of its nonstationarities (texture for example). In the case of a one-dimensional (I-D) signal, it is well-known that the WDs of real signals present interference terms due to the interaction between positive and negative frequencies. These terms are troublesome for analysis of the results, and can be eliminated by applying the WD to the 0165-1684/90/$3.50 O 1990, Elsevier Science Publishers B.V.
analytic signal associated with the real signal, rather than to the real signal itself [3, 10]. In its applications to 2-D signals, the WD raises, as will be shown in the following, the similar interference problem due to the interaction between positive and netative spatial frequencies. In order to avoid this problem and following the same idea as in the 1-D case, it is possible to introduce a so-called '2-D analytic signal', and apply the WD to the associated 2-D analytic signal instead of to the 2-D real signal itself. However, in the 2-D case, the 2-D analytic signal is not unique [13]. There then exist many possibilities of solving the interference problem using the concept of a 2-D analytic signal. In this paper, we propose a method for Wigner analysis of 2-D real signals, based on a 2-D Hilbert trans-
Y.M. Zhu et al. / Hilbert transform Jor Wigner analysis of 2-D signals
206
form. First, we introduce a 2-D discrete Hilbert transform, proposed by Read and Treitel in the context of 2-D recursive filter stabilization [14] to construct a 2-D analytic signal. We then derive the relation between the WDs of the 2-D real signal and its associated 2-D analytic signal. Finally, examples on both simulated and physical images are given to illustrate advantages brought by the 2-D analytic signal WD for the analysis of 2-D real signals.
This leads to the following spectral relation:
~Y(f( v) =j(1 - 2H ( v) )ff'f( v), where H(v) defined by H(v)=
(7)
denotes the Heaviside function
10 if v > 0 , ' if v = 0 , if v < 0 .
(8)
Thus, we obtain the spectrum of the analytic signal given by 2. Background on 1-dimensional analytic signal
~z(v) = 2H(v)o%f(v).
Let f ( t ) be a temporal real signal. Its associated analytic signal denoted by z(t) is the complex signal defined by
z(t) = f ( t ) + j g ( t ) ,
(1)
g(t) = I p . v 7r
(2)
with [" f(~') dr, dR t - z
where P.V means the Cauchy principal value. In definition (1) it can be seen that the analytic signal is directly obtained from the real signal by adding an imaginary part g(t). This imaginary part is known as the Hilbert transform (HT) of f ( t ) and will be denoted Y(f(t) in the following. The HT can also be expressed as a convolution o f f ( t ) with a kernel k(t)
~ f ( t ) = ( f * k)(t),
(3)
where "*" means convolution and k is the kernel given by 1 k(t) = P.V. - - . 7rt
(4)
The spectrum of k(t) is given by
J;k(v) = - j sign(v),
(5)
where ~ denotes the Fourier operator and sign(. ) is the sign function defined by sign(t)={-1, 1, Signal Processing
if t < 0 , if t > 0 .
(6)
(9)
The analytic signal then possesses the spectral property to have no negative frequencies while it contains all the spectral information on the real signal. By duality, it can be shown that the real and imaginary parts of the FT of a causal real signal form a Hilbert transform pair. This property will be exploited in [14]. From the discussion above, we underline the point that the analytic signal can be completely determined using the HT. Based on these remarks, we shall in the following section construct our 2-D analytic signal by using a discrete 2-D Hilbert transform.
3. 2-dimensional Hilbert transform and 2-dimensional analytic signal
In this section, we develop the definition of a 2-D analytic signal based on the 2-D discrete HT proposed by Read and Treitel [14]. For this purpose, we first give the formal definition of a 2-D analytic signal and then determine it after deriving, from Read and Treitel's discrete definition, the Hilbert transform for 2-D continuous-variable signals.
3.1. Formal definition of a 2-D analytic signal Let f(x, y) be a 2-D real signal. By analogy with the I-D case we define its associated 2-D analytic
Y.M. Zhu et aL / Hilbert transform for Wigner analysis of2-D signals
signal as a 2-D complex signal z(x, y), such that
z(x, y) =f(x, y) +jg(x, y),
(10)
where g(x,y) is a real function assumed to be related to f(x, y) by a two-dimensional convolution with an appropriate kernel k(x, y): g(x, y) = ( f * k)(x, y).
(11)
Taking the Fourier transform of both sides of (10) and in view of (11), we obtain
fz(u, v)= (1 + j ~ k ( u , v))~f(u, v).
(13)
14)
Relation (14) gives the frequential representation of the 2-D analytic signal. From (10) and (14), it can easily be shown that
fig(u, v)=j(1 -2re(u, v))o~f(u, v).
f lk(u, v)= -J;k(u, v).
(18)
o~k(u, v)= -~j[sign(u) + sign(v)].
(19)
So,
Taking the inverse FT of both sides of (19) gives the kernel function k(x, y):
k(x,y)=~(8(x) P.V 1-!-+~(y)P.V~x). ~ry (20) To determine re(u, v), we can insert (13) in (19), getting
re(u, v)= 4~[sign(u) + sign(v) + 2],
relation (12) can be written as
~z(u, v)= 2re(u, v)~f(u, v).
This function being pure imaginary and odd, k is hence real and odd. Thus we have
(12)
Defining a filter function re(u, v) by
m(u, v)= ~(1 +jffk(u, v)),
207
(15)
In the following we derive the kernel k(x, y) and the filter function by transposing the Read and Treitel 2-D discrete HT to the continuous case.
(21)
namely, i
m(u,v)=
if u > 0 , v > 0 , if u < 0 , v < 0 , elsewhere.
(22)
The filter re(u, v) is depicted in Fig. 1. It can be seen that the thus obtained 2-D analytic signal z(x,y) is a complex signal whose FT vanishes in quadrant III and is twice (resp. half) the FT of the 2-D real signal in quadrant I (resp. in quadrants II and IV).
3.2. 2-dimensional continuous Hilbert transform derived from Read and Treitel discrete Hilbert transform The definition of the 2-D discrete HT proposed by Read and Treitel and its reformulation are recalled in Appendix A. Transposing the discrete definition given by (A7) in Appendix A into the continuous case, we obtain
Ik(u,v)=
if u < 0 , v < 0 , elsewhere.
(16)
Relation (16) can be further written in the following form:
o~ ~k(u, v)=lj[sign(u)+sign(v)].
(17)
011)
Ill
0
l 2
(II)~ v 1/2 u Fig. 1. Characteristic of the filter m(u, vL Vol. 19, No. 3, March 1990
Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
208
In the next section, we apply the above obtained 2-D analytic signal to the WD and study its effect on the interference terms.
Indeed, the 2-D real signal f(x, y) being the real part of its associated analytic signal can be written as
f(x, y) = ½[z(x, y) + z * (x, y)]. 4. Wigner distribution of the 2-dimensional analytic signal
Using the bilinearity property of the WD [3] yields
4.1. Definition of 2-dimensional Wigner distribution I f f ( x , y) represents a 2-D signal (real or complex), its WD is a function of 4 variables, two spatial variables x, y and two spatial frequency variables u, v, defined by [9]:
Wt(x , y , u , v )
=
I~ 2 f ( x
+~c~,y+~fl)l
,
f , (X-~oq 1 y--½[3) X exp{--2jrr(ua + v/3)} d a d/3,
(23)
where " * " denotes complex conjugation. The WD can also be equivalently defined in terms of the FT F(u, v) o f f ( x , y) by
fJR F(u+~); 2
(25)
Wt(x,y, u, v)=¼[Wz(x,y, u, v) + Wz(x, y, - u , - v ) + 2 Re(Wz.z*(X,y, u, v))].
(26)
Relation (26) shows that the WD of a 2-D real signal is composed of two symmetrical terms W~(x,y, u, v) and W._(x,y,-u,-v) to which is added the interference term Re( W..: * ( x, y, u, v ) ) which does not bring any information in itself. To reduce its influence, we shall use the above introduced 2-D analytic signal.
4.2. Wigner distribution of the 2-dimensional analytic signal The 2-D analytic signal defined by (10-14) can be rewritten as follows:
v+~)
z(x, y) = [(6 + j k ) * f ] ( x , y),
(27)
F*(u-½~1, v-½~) exp{Zj'rr(x~7 +y~:)} dr/d~. (24) The 2-D WD presents many interesting properties for 2-D signal processing applications. The first of them is its reality. The WD of any 2-D real or complex signal is always a real function, which offers great facilities of manipulation. Despite of its reality, the WD still contains phase information on the 2-D signal since it is a reversible transform. The WD has also properties with respect to operations such as translation, modulation, scaling, multiplication and convolution. In addition, the WD posseses particular properties such as support conservation and energetic characteristics. However, the behaviour of the WD is sometimes difficult to interpret because of problems such as interference terms due to the interaction between positive and negative spatial frequencies. Signal Processing
where 6(x, y) is a 2-D Dirac delta function, and k(x, y) has been defined by (20). The WD of the convolution of two signals being equal to the convolution, with respect to the spatial variables, of the two signals WDs [3], the WD of z is then given by
Wz(x, y, u, v) = fR 2 W s ( x - x', y - y', u, v) x W~+jk(X', y', u, V) dx' dy'.
(28)
The obtention of the 2-D analytic signal WD then returns to evaluate the WD Ws+jk of the 2-D distribution function ( 6 + j k ) . The calculation of the latter being tedious, we give only the result, the intermediate derivation steps of which can be found in Appendix B. The WD of the 2-D analytic signal z(x,y) is
Y.M. Zhu et al. / Hilbert transformfor Wigner analysis of 2-D signals given by
W~(x, y, u, v) = (½+H(u)) I Wt(x-x',y, u, v) Jn ×
sin(4~rx'u)
dx'
7rX ~
+(½+I-I(v))f.
Wf(x, y - y ' , u, v)
×
sin(4~ry'v) . , ~ry' oy
+ fR2 W j ( x - x ' , y - y ' , u , v ) sin(4wx'u) sin(4~ry'v). , ox dy'
27r2x'y '
+
~2
W j ( x - x ' , y - y ' , u, v) x
cos(4wx'u) cos(4~ry'v)
2 2x,y ,
dx'dy'
+½Wr(x, y , u, v), (29) where the Heaviside function H ( . ) is defined by (8). Relation (29) shows that Wz(x, y, u, v) involves three types of filters: --A one-dimensional ideal low-pass filter with a (4v Sin(4~rtv)~ cut-off frequency 2v\ 41rtv /" two-dimensional ideal low-pass filter with cut-off frequencies 2u and 2v in the (u, v) plane (16uv Sin(47rxu) sin(4xryv)']
--A
4~rxu
4wyv ]"
two-dimensional ideal high-pass filter with cut-off frequencies 2u and 2v in the (u, v) plane 16uv cos(47rxu) cos(4~ryv)~
--A
4rrxu
47ryv
/"
Because of the filtering effects of these filters, and for great u, v values, the four first integral terms in (29) become approximately first term ~ ± Wr, second term ~ + IVy, third term = ± ~1 Wr, fourth term ~ 0.
209
The sign + l or - 1 in the above terms is taken according to the sign of u o r / a n d v. Hence, the 2-D analytic signal WD W~(x, y, u, v) behaves for all ( x , y ) 6 R 2 in the (u, v) plane as follows: quadrantI : W~4Wr, quadrant II : Wz~ Wl, quadrant III: Wz ~ 0 , quadrant IV: Wz ~ WI. We can then observe that - - Wz cancels in quadrant III. - - W~ has the same amplitude as the 2-D real signal WD in quadrants II and IV. - - In quadrant I, Wz is four times greater than the 2-D real signal WD. These remarks mean that the influence of the redundant information (recall that in the 2-D real signal WD, there always exists the identity of quadrants I and III, and of quadrants II and IV of the (u, v) plane), and of the interference terms which occur near (u, v) = (0, 0), is considerably reduced in the 2-D analytic signal WD. In the following section, we shall give some examples for illustrating the above discussed theoretical results.
5. A p p l i c a t i o n
examples
As mentioned previously, the WD gives a 4-D space/spatial-frequency representation of a 2-D image. For a fixed spatial point, this 4-D representation reduces to a 2-D function which can be considered as a local spectrum corresponding to that point of the image, so applying the WD to successive spatial points of an image allows to track its spectral variations in space. Of course, in practice, it is not always necessary to apply the WD to all the points of the image, since, on one hand, information given by the WD at a number of points may be sufficient, and on the other hand in some applications, we are only concerned with some points of interest. In our present problem, as can be seen in the foregoing formulations, the interference being due to the interaction between Vol, 19, No. 3, March 1990
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KM. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
positive and negative spatial frequencies, all the local spectra obtained from the WD of a real image can contain interference terms. Thus, for illustrating the interference p h e n o m e n o n , it is sufficient to consider the WD only at some particular image points (2-D slices of the 2-D WD), the conclusion remaining general for all other image points. In order to verify the theoretical performance of 2-D analytic signal WDs and to illustrate more clearly their advantages over 2-D real signal WDs, we first present some simulation studies by using two simulated test 2-D images whose characteristics are a priori known. Then, an application to a physical image is considered.
U
y
x
i!li m ~ 1 1 D JlD D qRD b qP m
-
i
Q
Signal Processing
•
a
i I
~ll I Ill
i
i I¢
i
II
Ii
m m I t | • 0 |l|l!
I I
i i • i * II !1 Itlt l|lltltlllll| i t i I I e 4 O il | II 6 D • | • • Q I | • W D D ~ O 4 t P e I Ba I g Q i i ~ Q • W B 4 ~ 4~ ~m g e W II ~ S I C l l l l
. - . - " 7.i : :i : : i: : :i : :i : :i ~
E X A M P L E A. We first consider a simple case where the simulated test image contains solely one component. In this case, the interference terms come from only the interaction between positive and negative spatial frequencies. Such an image of size 128 × 128 is represented in Fig. 2a. As discussed above, for illustrating its complete 2-D WD, in principle we have to display at the same time 16 384 local spectra of size 128 × 128. Obviously, such a presentation is not realistic and neither necessary. For that reason, we choose two representative image points to illustrate the interference phenomenon. Figure 3 shows the WDs of the real image at two image points (32, 32) and (96, 96). At first viewing, these 2-D WD slices present some symmetrical structure with respect to the origin of the frequency plane, and the four quadrants, in each of which a greater and brighter spot is present, are identical. The four greater white spots represent in fact the spectral information related to the modulation characteristics of the image (as can be seen in Fig. 2a). In particular, their position in the frequency plane tells the variation degree of the grey levels of the image in that region around the point in question. Thus, tracking in space their frequential position changement allows to get an insight into how the image's spectral content varies with spatial position. This can be observed in Fig. 3 where the fact that the four greater white spots in Fig. 3b are further from the
m
i I I llO BI OR is a a 6 qD I IF Q I ~ ~ ~ • tlb /~ ~ u i qP 9 0 B ~ ~ m B i
ID
Ib i I
I i i l I 10111
I
I
II,
II ii
I
~II • i Ik I If
m
ii
8 8 1 1 1 1
dl ~ 6 i I
i i I ii I iP
iii I
II
(a)
(b) Fig. 2. Simulated test image with one component and its 2-D Hilbert transform. (a) Simulated test image. (b) Its 2-D HUbert transform. origin than those in Fig. 3a agrees perfectly with the theoretical prediction. Note however that the above analysis and interpretation are not always easily assessed because of the fact, as we can observe in Fig. 3, that the useful information (the four white spots) are drowned by the parasite
Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2,D signals
Ca)
(b) Fig. 3, 2-D WD slices of the real image illustrated in Fig. 2a, (a) 2-D WD slice corresponding to the point (32, 32). (b) 2-D WD slice corresponding to the point (96, 96).
interference terms due to the interaction between positive and negative spatial frequencies. Furthermore, it can be seen that only one of the four white spots is sufficient to represent the 2-D WD slice because of the symmetry characteristic of the latter one.
211
For reducing these interference terms and redundant information, we now calculate, instead of the WD of the real image, that of its associated analytic image (2-D analytic signal). In Fig. 2b, we have given the 2-D Hilbert transform of the real image, obtained by using Read and Treitel's formula, The 2-D WD slices of the thus obtained analytic image (always at the same points as in Fig. 3) are shown in Fig. 4, from which it can be seen that (i) the 2-D slice of the analytic image Wz is zero in quadrant III; (ii) the interference terms around the origin are largely reduced; (iii) the redundant white spots situated at quadrants II and IV are attenuated; and (iv) the informational spot in quadrant I is best represented. The simulation results are thus perfectly conform to the theoretical analysis. In short, the informational white spot representing the spectral characteristic of the image is more apparent in the 2-D WD slices of the analytic image.
EXAMPLE B. We now consider a more complex case where the siinulated test image is composed of two sinusoidal images, the one being a pure cosine image which has a constant frequency, and the other being a so-called chirp image whose spectral content varies with spatial position. Such a 128 x 128 image is represented in Fig. 5a. It is known that in this case, the WD of the image presents interference terms due to the interaction between the two components. At the same time, according to our foregoing discussion, it also presents the interference terms coming from the interaction between positive and negative spatial frequencies. The WDs of the above real image at two image points (33, 33) and (47, 47) are given in Fig. 6, in which we notice, around the origin of the frequency plane, the presence of the interference terms coming from the interaction between positive and negative spatial frequencies. In either quadrant I or III, we remark two greater and brighter spots, between which are present some bright annulus representing the interference terms coming from the interaction between the two components. These two white Vol. 19, NO. 3, March 1990
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Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
(a)
(b)
(b)
Fig. 4. 2-D WD slices of the analytic image associated with the real image illustrated in Fig. 2a. (a) 2-D WD slice corresponding to the point (32, 32). (b) 2-D WD slice corresponding to the point (96, 96).
Fig. 5. Simulated test image with two components and its 2-D Hilbert transform. (a) Simulated test image. (b) Its 2-D Hilbert transform.
spots are in fact the two f r e q u e n c y c o m p o n e n t s r e p r e s e n t i n g r e s p e c t i v e l y the two i m a g e c o m p o n e n t s . W e also o b s e r v e , in the two local s p e c t r a from the W D , t h a t one o f the two f r e q u e n c y c o m p o n e n t s k e e p s a l w a y s the s a m e p o s i t i o n in the f r e q u e n c y p l a n e when p a s s i n g f r o m the local spec-
t r u m at p o i n t (33, 33) to that at (47, 47), this o n e r e p r e s e n t i n g the cosine i m a g e c o m p o n e n t , a n d that the other, in contrast, d i s p l a c e s t o w a r d h i g h e r frequenties, it r e p r e s e n t i n g the c h i r p i m a g e c o m p o n e n t . This is precisely in a g r e e m e n t with the t h e o r e t i c a l p r e d i c t i o n , a n d shows again the ability
Signal Processing
Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
(a)
(b)
213
(a)
(b)
Fig. 6. 2-D WD slices of the real image illustrated in Fig. 5a. (a) 2-D WD slice corresponding to the point (33, 33). (b) 2-D WD slice corresponding to the point (47, 47).
Fig. 7. Smoothed versions of the same 2-D WD slices illustrated in Fig. 6. (a) Smoothed 2-D WD slice corresponding to the point (33, 33). (b) Smoothed 2-D WD slice corresponding to the point (47, 47).
of the WD to track in space the spectral variations in an image. However, it can be seen that the above spectral analysis by means of the 2-D WD slices is not easily achieved because of the presence of the interference terms. To cope with this problem, we can seek to process, as usually done until now by
most of the researchers, some smoothing operations to attenuate the interference terms. But, as we shall show in the following, smoothing operations alone would not be sufficient to minimize the influence of the interference terms and to eliminate redundant information. For illustrating this, we give in Fig. 7 the smoothed version of the VoL 19, No. 3. March 1990
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Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
same 2-D WD slices as illustrated in Fig. 6. The smoothing procedure was the following: first choose a 7 x 7 smoothing neighbourhood centered at the considered point, then compute the 49 2-D WD slices within the zone, finally averaging all the obtained 2-D WD slices produces the smoothed local spectrum corresponding to the central point. C o m p a r e d with Fig. 6, interference terms in Fig. 7 have been largely reduced. This is particularly evident in Fig. 7b. However, the interference terms around the origin of the frequency plane have not been completely eliminated, and the symmetry redundancy remains still. Now, if we use both the 2-D analytic image and the smoothing procedure, the legibility of the local spectra can be considerably improved. In Fig. 5b, we have represented the 2-D Hilbert transform of the original test image. Figure 8 shows the 2-D slices of the 2-D analytic image WD, in which we observe, compared with Fig. 6, the total disappearance of the interference terms around the origin of the frequency plane as well as the redundant symmetry structure. A more apparent result is again found in Fig. 9 where we show the smoothed versions of the same 2-D WD slices as illustrated in Fig. 8. This example shows that the 2-D analytic image allows to efficiently eliminate the interference terms due to the interaction between positive and negative spatial frequencies, and that after that, further reduction of residual interference terms can be accomplished by using other smoothing techniques.
im
(a)
(b) EXAMPLE
C. As a last example, we take a phy-
sical image of size 512x512 shown in Fig. 10. Generally, such kind of images cannot be considered as m o n o c o m p o n e n t images. This means that its WD can present interference terms which may both come from the interaction between positive and negative spatial frequencies, and from the interaction between the various components in the image. In this case, as shown in the precedent example, the use of the 2-D analytic image should be the first thing to do in all tentatives of reducing them. Signal Processing
Fig. 8. 2-D WD slices of the analytic image associated with the real image illustrated in Fig. 5a. (a) 2-D WD slice corresponding to the point (33, 33). (b) 2-D WD slice corresponding to the point (47, 47).
We have, in this example, chosen two regions (128 x 128 pixels) of interest over which the WD was applied. The first one concerns a sub-image around the point (134, 168), and the second a sub-image around the point (262, 135) (illustrated respectively on the upper and lower left corner of
Y.M. Zhu et at / Hilbert transform for Wigner analysis of 2-D signals
(a)
(b) Fig. 9. Smoothed versions of the same 2-D slices illustrated in Fig. 8. (a) Smoothed 2-D WD slice corresponding to the point (33, 33). (b) Smoothed 2-D WD slice corresponding to the point (47, 47). Fig. 10b). A l m o s t all the p o i n t s o f significant a m p l i t u d e in the o b t a i n e d 128 x 128 2-D W D slices b e i n g c o n c e n t r a t e d in a s m a l l e r r e g i o n a r o u n d the origin, we r e p r e s e n t , in all the f o l l o w i n g figures, o n l y a 64 x 64 v e r s i o n a r o u n d the origin o f the original results (at the s a m e r e s o l u t i o n ) in o r d e r to a c h i e v e a b e t t e r visual a s s e s s m e n t o f the difference b e t w e e n the different local s p e c t r a f r o m the W D .
215
(a)
(b) Fig. 10. Original women image, its sub-images and the 2-D Hilbert transforms of the sub-images. (a) Women image. (b) Left half plane: sub-images; right half plane: 2-D Hilbert transforms of the sub-images.
F i g u r e 11 shows the 2-D W D slices (64 × 64) o f the w o m e n s u b - i m a g e s c a l c u l a t e d at the two a b o v e points. As b e f o r e , we first give in Fig. 12 the s m o o t h i n g results o f the 2-D W D slices i l l u s t r a t e d in Fig. 11. It is seen that there is little difference b e t w e e n the 2-D W D slices a n d their s m o o t h e d versions, e x c e p t that the latters have b e e n slightly Vol. 19, NO. 3, March 1990
Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
216
(a)
(a)
(b)
(b)
Fig. 11. 2-D W D slices of the women sub-images. (a) 2-D W D slice corresponding to the point (134, 168). (b) 2-D W D slice
Fig. 12. Smoothed versions of the same 2-D WD slices illustrated in Fig. 11. (a) Smoothed 2-D WD slice corresponding
corresponding to the point (262, 135).
to the point (134, 168). (b) Smoothed 2-D WD slice corresponding to the point (262, 135).
attenuated at high frequencies. Coarsely, as expected, the first local spectrum (Fig. 1 la) corresponding to the image point (134, 168) is characterized by a set of dominant spatial frequencies located almost along the v axis of the frequency plane. This is due to the fact that the first sub-image (illustrated on the u p p e r left corner of Fig. 10b)
presents more abrupt grey level variations in the y direction than in the x direction. In contrast, the second local spectrum (Fig. 1 lb) corresponding to the image point (262, 135) differs from the above one. The distribution of its dominant spatial frequencies is around the origin of the frequency plane. Sometimes, these frequencies are more
Signal Processing
Y.M. Zhu et aL / Hilbert transform for Wigner analysis of 2-D signals
(a)
(a)
(b)
(b)
217
Fig. 13. 2-D WD slices of the analytic images associated with the women sub-images in Fig. 10b. (a) 2-D WD slice corresponding to the point (134, 168). (b) 2-D WD slice corresponding to the point (262, 135).
Fig. 14. Smoothed versions of the same 2-D WD slices illustrated in Fig. 13. (a) Smoothed 2-D WD slice corresponding to the point (134, 168). (b) Smoothed 2-D WD slice corresponding to the point (262,135).
n u m e r o u s in q u a d r a n t s I a n d I I I t h a n in q u a d r a n t s II a n d IV. This set o f c h a r a c t e r i s t i c f r e q u e n c i e s is r e s p o n s i b l e for the r e l a t i v e l y h o m o g e n e o u s varia t i o n s t r u c t u r e o f the grey levels o f the s u b - i m a g e ( i l l u s t r a t e d o n the l o w e r left c o r n e r o f Fig. 10b).
have s h o w n in p r e c e d e n t s i m u l a t e d e x a m p l e s , the i n t e r f e r e n c e terms d u e to the i n t e r a c t i o n b e t w e e n positive a n d negative s p a t i a l frequencies can g r e a t l y m a s k useful i n f o r m a t i o n . In Fig. 13, we have r e p r e s e n t e d the 2-D W D slices o f the a n a l y t i c i m a g e s a s s o c i a t e d with the two w o m e n s u b - i m a g e s , the i n v o l v e d 2-D H i l b e r t t r a n s f o r m s ( i l l u s t r a t e d on
H o w e v e r , the f u r t h e r p r e c i s e a n d exact a n a l y s i s o f the a b o v e results m a y b e difficult, since, as we
Vol. 19. No. 3. M a r c h 1990
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Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
the right half plane of Fig. 10b) of the sub-images having been calculated by using Read and Treitel's formula. Comparing Fig. 13 with Fig. 11, we can observe the significant difference between the WDs of the real image and its associated analytic image. Indeed, careful examinations show that some dominant components, which did not appear in the local spectra from the WDs of the real images, are now revealed in those from the WDs of the analytic image. We can also remark the changement of the relative brightness of the dominant components in Fig. 13. In short, characteristic frequencies seem to be better represented in Fig. 13 than in Fig. 11. Sometimes, further improvement in local spectrum feature visibility can be achieved by employing the same smoothing procedure used above. The thus obtained results are shown in Fig. 14. Notice here that the combined use of the 2-D analytic signal and the smoothing procedure provides valuable results of primary interest. Of course, further work is needed before one can realize, for example, image feature extraction as well as pattern recognition using the WD based spectral analysis.
Appendix A. Read and Treitel 2-dimensional discrete Hilbert transform The definition of the 2-D discrete Hilbert transform (HT) proposed by Read and Treitel was introduced in the following manner [14]. Let f ( n , , n2) be a finite and causal discrete pulse verifying f ( n , , n2) = 0
for n, >! N,/2 or n2 >! N2/2
where n, varies over the discrete set {0, 1. . . . , N, 1} and n2 varies over the discrete set {0, 1 , . . . , N 2 - 1}. The discrete Fourier transform (DFT) of this pulse can be expressed as
F(k,, k2)= PR(k,, k2)+jPI(k,, k2),
Signal Processing
(A.2)
where PR(k,, k2) and PI(k,, k2) are respectively the real part and imaginary part of the complex spectrum F( k,, kz). Read and Treitel have defined a 2-D discrete Hilbert transform (DHT) by considering that PR(k,, k2) and PI(kl, k2) form a 2-D HT pair. By inverse DFT and because f ( n , , n2) is a real causal discrete pulse, they obtained DFT
6. Conclusion The WD offers a new approach for analysis of the local spectral characteristics of 2-D signals (or images). In practice however, the WD can produce interference terms which are often troublesome for interpretation of the results. In particular, in its applications to 2-D real signals, the influence of the interference terms coming from the interaction between positive and negative spatial frequencies may be not negligible. For reducing it, we have, in this paper, proposed a practical method by using a 2-D analytic signal based on a known 2-D discrete Hilbert transform. The proposed 2-D analytic signal WD exhibits an important reduction of interference terms, and has been shown to be more easily exploitable than the 2-D real signal WD.
(A.1)
'
PI(n,, n2) = - j [ s g n ( n , , n2)
+ b d y ( n , , n2)] D F T - ' PR(n,, n2) ,
(A.3)
with sgn(n,, n2) 0 < n, <½N,, 0 < n2<½N2,
f 1, 0,
½Nl
~l N 2 < 1"12< N 2 ,
elsewhere, (A.4)
and
{"
bdy(n,, n2)
n,=0, 0< n2<½N2, n,=0, ½N2
-1, 0,
n2=0, 0 < n, <½N1, nz = 0, 1N, < nl < N I , elsewhere.
(a.5)
Y.M. Zhu et al. / Hilbert transformfor Wigneranalysis of 2-D signals Now, if we call k(n~, n2) the kernel of the 2-D D H T according to the notation in Section 2, (A.3) can be written as
219
tigating the four following integrals: fR 2 sign(u + r/) sign(u - r/)
D F T 1 PI(nl, n2)
x exp{4j~r(xr/+y¢)} d r / d sc
= D F T l k(n~, n2) D F T -~ PR(nl, n2), 6(Y) sin(4~rxIu I) - J6(x, y), 2"rrx
-
(A.6) where
L
2 sign(u + r/) sign(v - ~:)
D F T 1 k(n~, n2) = - j ( s g n ( n , , n2) + b d y ( n ~ , n2)).
(A.7)
x exp{4j~r(xr/+ys¢)} d r / d sc 1
A p p e n d i x B. D e r i v a t i o n o f
=
Wa+jk
The Wigner distribution (WD) W~+jkof the 2-D distribution function (~ + j k ) is derived as follows: According to the bilinearity property of the WD, it can be written
47r2xy exp(--4j~r(xu --yv)),
~2 sign(v + ~:) sign(u -- rl) x exp{4jw(xr/+ys~)} d r / d sc 1
4~r2xy exp(4jrr(xu - yv)),
Wa+jk(X, y, u, v) = Wa+ W i t + 2 Re[ W~jk].
(B.1)
x exp{4j~r(xrl +ys¢)} d r / d s¢
It can easily be shown that
W~(x, y, u, v) = 6(x, y).
fu2 sign(v + sc) sign(v - ~:)
(B.2)
To calculate Wjk, we use the frequential definition (24). In this case, replacing the integrand function in (24) by ~ k defined in (19) yields
8(x) sin(4~y]v]) l a ( x , y ) . 2wy 4 Thus, we get
Wjk(X, y, U, V)
Wjk(X, y, U, V) = IR 2 [sign(u + rl) + s i g n ( v + ~)] • [sign(u - r/) + sign(v - ~:)] exp{4j'rr(xr/+ y~:)} d rI d~
sin(4~ylv]) = 8(y) sin(4~rxlu[) - - g(x) 27rx 2~ry
-~
cos 4,rr(xu-yv) 2~r2xy
1 ~6(x, y).
(B.4)
We now evaluate the term W~ak : = f~2 [sign(u + rl) sign(u - r/) W~ak(X, y, U, V) + sign(u + r/) sign(v - ~) = JR 2 sign(u -- r/) sign(v --~:) + sign(v + ~:) sign(u - 7/) x exp{4j~r(xr/+ y~:)} d r / d sc
+ sign(v + ~) sign(v - ~)] x exp{4j~r(xr I + ys¢)} d r / d s e.
(B.3)
The integrals in (B.3) can be evaluated by inves-
.{8(y) . 8(x) . = -J ~2-rrx exp(4j-rrxu) + 2-~-y exp(4j~ryv)).
(s.5) Vol. 19, No. 3, March 1990
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Y.M. Zhu et al. / Hilbert transform for Wigner analysis of 2-D signals
Consequently, 2 R e [ Wa,jk] = 8 ( y ) sin(4~rxu) + ~ ( x ) s i n ( 4 r r y v ) . ~rx ~ry
(B.6)
It results f r o m (B.2), (B.4) a n d (B.6) t h a t
Wa+jk(x, y, u, v) = I s ( x , 2 y ) q cos 4rr(xu - y u ) 2~rZxy + a ( y ) sin(4crxu)(½sign(u) + 1) ~x +
~(x) ~ry
s i n ( 4 ~ y v ) ( ½ s i g n ( v ) + 1).
(B.7)
References [1] R. Bamler and H. Gliinder, "The Wigner distribution function of two-dimensional signals. Coherent-optical generation and display", Optica Acta, Vol. 30, 1983, pp. 1789-1803. [2] G.F. Boudreaux-Bartels and T.W. Parks, "Time-varying filtering and signal estimation using Wigner distribution synthesis techniques", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-34, June 1986, pp. 442-451. [3] T.A.C.M. Claasen and W.F.G. Mecklenbraiiker, "The Wigner distribution--A tool for time-frequency signal analysis", Philips J. Res., Vol. 35, 1980, pp. 217-250, 276300, 372-383. [4] G. Cristobal, J. Bescos, J. Santamaria and J. Montes, "Wigner distribution representation of digital images", Pattern Recognition Letters, Vol. 5, 1987, pp. 215-221. [5] B. Escudi6, "'Repr6sentation en temps et fr6quence des signaux d'6nergie finie: analyse et observation des signaux', Annales des T~l~communications, Vol. 34, No. 3-4, 1979, pp. 101-111.
Signal Processing
[6] P. Flandrin, "Some feature of time-frequency representations of multicomponent signals", Proc. Internat. Conf. on Acoust. Speech Signal Process., 1984, pp. 41B4.1-B4.4. [7] P. Flandrin and B. Escudi6, "Time and frequency representation of finite'energy signals: A physical property as a result of an Hilbertian condition", Signal Process., Vol. 2, No. 2, 1980, pp. 93-100. [8] L. Jacobson and H. Wechsler, "A theory for invariant object recognition in the frontparallel plane", IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. PAM1-6, No. 3, May 1984, pp. 325-331. [9] L. Jacobson and H. Wechsler, "The Wigner distribution and its usefulness for 2D image processing", Proc. 6th Internat. Joint Conf. on Patter Recognition, Munich, October 1982, pp. 19-22. [10] C.P. Janse and A.J.M. Kaizer, "Time-frequency distributions of loudspeakers: the application of the Wigner distribution", J. Audio. Engrg. Soc., Vol. 31, No. 4, 1983, pp. 198-223. [11] W. Martin and P. Flandrin, "Wigner-Ville spectral analysis of nonstationary processes", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-33, December 1985, pp. 1461-1470. [12] F. Peyrin and R. Prost, "A untied definition for the discrete-time, discrete frequency and discrete time-frequency Wigner distribution", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-34, August 1986, pp. 858-867. [13] F. Peyrin, Y.M. Zhu and R. Goutte, "Extension of the notion of analytic signal for multidimensional signals. Applications to images", in: I.T. Young et al., eds., Signal Processing 111: Theories and Applications, North-Holland, Amsterdam, 1986, pp. 677-680. [14] P.R. Read and S. Treitel, "The stabilization of twodimensional recursive filters via the discrete Hilbert transform", IEEE Trans. Geosci. Electron., 1973, Vol. GE-I1, pp. 153-207. [15] T. Reed and H. Wechsler, "Tracking of nonstationaries for texture fields", Signal Process., Vol. 14, 1988, pp. 95-102. [16] Y.M. Zhu, F. Peyrin and R. Goutte, "Transformation de Wigner-Ville: description d'un nouvel outil de traitement du signal et des images", Ann. Tdl~communications, Vol. 42, No. 3-4, 1987, pp. 105-118. [17] Y.M. Zhu, F. Peyrin and R. Goutte, "Utilisation de la transformation de Wigner-Ville pour le traitement num6rique des signaux et des images: mise en oeuvre et interpretation", llbme CoIL GRETSI, Nice, 1987, pp. 21-24.