Localization of objects with circular symmetry in a noisy image using wavelet transforms and adapted correlation

Pattern Rrmwnirise . Vol, 27, No. 3, pp .3il 361,19% Elsevier Silence Lid Copyright © 1994 Pattern Recognition Society Printed a Great Britain. AU rights reserved 003132033/94 1601+ .110

® Pergamon

LOCALIZATION OF OBJECTS WITH CIRCULAR SYMMETRY IN A NOISY IMAGE USING WAVELET TRANSFORMS AND ADAPTED CORRELATION C. DUCOTTET, J . DANiERE, M . MoINE, J. P. ScHON and M . COURBON Laboratoire Traitement du Signal et Instrumentation, URA CNRS 842, Universite J . Monnet, 23, rue du Docteur P . Michelon, 42023 Saint-Etienne Cedes, France (Received 12 March 1993 ; received for publication

5

October 1993)

Abstract-Two methods of detection of the position of objects with circular symmetry and variable size are presented . The methods, based on wavelet transforms, allow the determination of the center of the objects in a noisy background due for example to speckle in holographic images . The first method uses the orthonormal wavelet transform of an image, while the second uses the correlation product of the image with an adapted representation of the object . The last method is valid even for high noise levels . Image processing Wavelet transform

Object recognition Holography Correlation Interference rings

l . INTRODUCTION The detection of objects or patterns with circular symmetry and variable size can be useful for several practical problems. In our case the problem was raised by automatic detection of the position of particles in a three-dimensional (3D) flow by microholography . This technique as described by Royer" 1 allows the recording of a 3D field of particles, the positions of which can be obtained by moving a video camera in this field until the camera is focused on the particle." ) For small particles (10 µm) however, the image of the particle can be confused with speckle noise and consequently, its detection by a direct technique is difficult . One of the methods to determine the position of the particle is to use the fact that images obtained by focusing the camera in the vicinity of a particle contain interference rings which are due to the interaction between the light diffused by the particle and the direct laser light . The detection of the particles is then equivalent to the detection of these rings. This problem was at the origin of the present work . From the image analysis point of view, the problem can be expressed as the detection of rings in a noisy image, these rings being, in some cases, strongly blurred by noise due to speckle (Fig . 1). Several methods may be used to detect these rings . Two approaches are proposed each of them having their advantages and their limits, The first of these approaches uses the wavelet transform to locate the rings . This method allows the detection of the rings independently of their diameter . However, the center is not determined with much accuracy in a noisy environment . The second method is derived from the former in the sense that a possible interpretation of the wavelet trans-

Circular objects localization

form of a signal is the correlation between this signal and a wavelet at a given scale . More precisely, a particular ring system is revealed in the signal by calculating its correlation product either with an orthonormal wavelet or with a theoretical representation of the system . So if we replace the primary wavelets of variable scale by a sum of theoretical models of the rings of different diameters, we will be able to get an accurate determination of the center of the observed rings . However, more information on the size and the form of the object is needed in this method, than in the first one . We finally discuss the influence of the signal to noise ratio on localization .

2.

WAVELET TRANSFORM THEORY APPROACH

2 .1 . Wavelet transform principle

The wavelet transform (WT) has been introduced in order to get some spatial information in the Fourier space. If a specific frequency is detected in a Fourier spectrum, it is difficult to determine its spatial origin in the signal. This is due to the basic functions used in this transform: they are very well localized in the frequency domain, and not localized in the spatial domain . In order to introduce a spatial localization, Gabor!s 1 proposed to multiply complex exponential functions by a window, function . This window function is assumed to be spatially well localized, as for example a gaussian function . The basic idea of WT is to use a family of functions localized both in space and in the frequency domain . These functions are related to each other by translations and by changes of scale . The frequency variable used with the Fourier transform is then replaced by the scale parameter. 351



3 52

C. Duc orT[T et at . 2 .2. Continuous transform Now, we consider the L2 ( M 2 ) space of squared integrable functions with two real variables x and y . Let w x and m ), be the corresponding Fourier spatial pulsations along the x and y axes . To define a wavelet transform,(° ) it is necessary to introduce a function Y'(x, y) which verifies the condition +f ~w(mwmr)h dm x dm,,<+zt l W W,, 0 0

(I)

where 'P is the Fourier transform of 4' . From this function, a family of functions can be derived according to the relation T, .b=,b,.(x, y) = sT (s(x - b.), sly - by) ) where s is a scale parameter and b 1 and by are translation parameters . Then, the wavelet transform Wf of a function feLr (3t') is a decomposition on the family

W1(s,brby)= J

f

f(x,y)st •(s(x-b..), s(y-b y)) dxdy

(2)

where 'P is the complex conjugate of Y' . Condition (1) makes the transformation reversible . However, it is generally redundant : indeed wavelet coefficients of a given function are not independent . Another characteristic induced by condition (1) is that the function ' presents local oscillations and has a zero mean value. 2 .3 . Discrete transform

2 .3 .1 . Continuous transform discretization . In practice, to compute the transform, s, b,. and by parameters must be restricted to a discrete sublattice . Taking into account the scaling properties, the translation step has to depend on this scale . With a dilatation step s o > 1, and a translation step b o # 0, the chosen sampling is s=sL bx =nbo sL by = mb0 so

where (n,m,j)EZ 3 .

The discrete wavelet transform (DWT) is then defined as the decomposition on the family of functions (' ) fs)d2 P(so(x - nboso'),

Fig. 1 . Interferences observed near a particle registered by microholography .

so(y - tnboso ) ))1,+ .m .nEz , .

Depending on the s o and b 0 values, the transform is more or less redundant . This redundancy can be eliminated in a particular case corresponding to multiresolution analysis(b) where s o =2 and b o =1 . This wavelet family can thus be chosen as an orthonormal basis . The corresponding transform is the orthonormal wavelet transform (OWT) . From the concept of multiresolution analysis, Mallat found a fast algorithm to compute the OWT. He showed



Localization of objects with circular symmetry in a noisy image

353

(b)

L first order . y filtering direction 2 : first order, x and y filtering directions 3 : first order. s : filtering direction 4 : second order, V filtering direction 5 : second order, x and y filtering directions 6 : second order, x filtering direction

Fig. 2 . (a) Example of a 256 x 256 pixel image made out of a horizontal grid with a 4 pixel period, a vertical grid with an 8 pixel period, a two-dimensional grid with a horizontal period of 4 and a vertical period of 8 and an oblique grid with a 16 pixel period . (b) Wavelet transform modulus of (a) computed up to the third order using a symmetrical wavelet (c) . The black parts correspond to high values .



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C.

DUCOTTET

that some orthonormal wavelet basis can be obtained using two quadratic mirror filters h and g which are respectively a low pass and a high pass filter . The computing process is made recursive by filtering the signal along both the x and y axes . The resulting filterings by h along both the x and y axes give an approximation of the signal on a scale of 2t - ' . Similarly, the filterings by h along the x axis and by g along the y axis, by g along the x axis and h along the y axis, and by g along both the x and y axes give the wavelet coefficients on a scale of 2i - ' . The wavelet transform can thus be interpreted as a signal decomposition into a set of independent, spatially oriented frequency channels . This leads to an anisotropic transformation and it can be shown that it is impossible to build an isotropic transformation using a scale factor of 2 .t" 2 .3 .2 . Example. Figure 2 presents an example of a wavelet decomposition using the Mallat algorithm . The initial 256 x 256 pixel image (Fig . 2(a)) is made out of four sinusoidal grids: a horizontal grid with a 4 pixel period, a vertical grid with an 8 pixel period, a twodimensional grid with a horizontal period of 4 and a vertical period of 8 and an oblique grid with a 16 pixel period. With an initial resolution set to 1 (which corresponds to the order j = 0), the wavelet transform has been computed up to the third order using the wavelet of Fig . 2(c). Three sub-images are associated to each order as shown in Fig . 2(b). For example, the double grid on Fig. 2(a) top left is enhanced in Fig. 2(b) (number 3 for the x period and number 4 for they period) . This example shows clearly that the wavelet transform can be used for several applications :

et at.

cross section is given by =Posin(k , r/z P.0

z

)sinl

k 2,r_~

k,r/z

z

where r= .J(x 2 +y2 ) is the radius of the ring system, z the distance from the particle plane and P 0 ,k„k2 some physical constants. This equation can be written as P,(r)=s2 A(s 2 r)S(sr)

(3)

where 1 I J

A(s 2r)

z

= Pv sin (k, S 2 r) k,s 2 r Sin (k2(Sr) ' )

S(Sr)

indicating that the cross section is the product of two terms linked to the scaling properties . The form of this cross section is very peculiar to holography . It does not have exactly the scaling property, as its shape changes with the scale . Therefore, the localization methods exposed will consider a more general but more simple definition for the cross section where objects with circular symmetry are assumed to be constructed using only one reference cross section p(r) and a scaling parameters (P.(r))t,E,,=(sp(sr))(SEm'-

(4)

The factor s is necessary to keep the signal energy constant +6 +'O

• localization of frequencies found in a signal ;t s ' • edge and discontinuity detection by zero crossing of coefficients" • recognition of a noisy pattern hidden in an image (provided the wavelet looks like the pattern) . The following section is devoted to an example of this last application .

p(sr) '=

f f S2

p'(sJ(x2

+y2 ))dxdy

11p(r)

On these bases, two methods of detecting the center of the objects are developed as previously announced . Both methods use transforms which create images with peak intensity at the center of the objects

3 . LOCALIZATION OF O&JECTS WITH CIRCULAR SYMMETRY

3 .2 . Localization using orthonormal wavelet transform 3 .1 . Introduction Figure I shows typical images to analyse . They come from a microholographic recording of particles in a three-dimensional flow . Images are taken in the vicinity of a particle at three different video camera-particle distances . The interference rings produced by a particle can be used to localize this particle . These rings have variable diameter and noise due to speckle is always present . For example, in the images of Fig . 1, the ratio between the maximum amplitude of the rings and the r .m.s . of the noise level is smaller than 3 . In fact, the analysis of the interference phenomenon which produces the rings shows that the intensity in a

3 .2.1, Determination of the wavelet basis . According to the continuous definition of the wavelet transform Wf (s, b„ b,,) of a function f (see equation (2)), values of coefficients on a scale of s represent the correlation product of f with the wavelet at the same scale . So, if the shape off is close to the shape of Y', a peak will appear for the best fitting position (b„ br ). This property is useful because the structures with circular symmetry have a cross section of variable size . A maximum value at a given scale indicates the presence of a ring system at the same scale. So if a wavelet with a shape close to the object is used, the localization of this object with circular symmetry is possible .



Localization of objects with circular symmetry in a noisy image

(a)

355

(b)

I

(c)

(d)

Fig . 3 . Different wavelets with compact support constructed using h filters : (a) 6 sample filter ; (b) 10 sample filter; (c) 20 sample filter; (d) 22 sample symmetrical filter . are not symmetrical whereas wavelet 3(d) is symmetrical but does not oscillate as much. Only some tests will determine the best wavelet (see Section 3 .2 .3) .

Fig. 4. Theoretical cross section of the interference rings to

detect . The Mallat algorithm being fast, it is interesting to use an orthonormal wavelet basis . As the construction of an adapted orthonormal wavelet basis is a difficult mathematical task, we have used the h and g filters computed by Daubechies .° With these filters, the effective shape of the associated wavelet is given by the inverse transform of a dirac function . Figure 3 presents some wavelets computed with different g and It filters. Wavelets 3(a), 3(b) and 3(c) have been constructed with g and h filter coefficients given by Daubechies"I whereas wavelet 3(d) has been obtained by making wavelet 3(c) symmetrical . For that, the h filter has been computed once again with its phase set to zero . The comparison of these wavelets with the theoretical cross section of an object (Fig . 4) shows that wavelets 3(b), 3(c) and 3(d) fit in quite well with this cross section ; wavelets 3(b) and 3(c) oscillate a lot but

3 .2 .2 . Practical realization with images . From a single source image, the orthonormal wavelet decomposition produces three detailed images for each scale corresponding to the horizontal, vertical and combined computation directions . In Fig. 5, the information for each scale is given in an image obtained by the sum of the absolute values of the three subimages corresponding to each order . Then one or more scales containing ring information are identified (here channels 3 and 4), and they are added after putting all of them at the same resolution by interpolation . The final image (Fig. 6(a)) shows a maximum value close to the ring system center . The object position has been approximately localized. As the coefficients are positive if the wavelet is fitted by a dark ring and negative if it is fitted by a clear ring, the absolute value is necessary . On the other hand, the fact that the transformation is not isotropic can introduce some inaccuracy on the peak position used to detect the object . Non-isotropic noise can produce similar effects. 3 .2.3 . Choice of the wavelet . In order to choose the most efficient wavelet for localization, the method is applied to the same image using each wavelet of Fig . 3 . The results are presented in Fig . 6 where it is clear that the symmetrical wavelet is more efficient . The contrast between the main peak and the others is greater in this case .

356

C . DUCOTTET

(a)

et at .

(b)

Fig . 5 . Successive scales of the wavelet transform of image 1(b) computed using the sum of the absolute values of the three subimages corresponding to each order : (a) first order ; (b) second order ; (c) third order ; (d) fourth order; (e) fifth order.

(0)

i

(d)

(6)

ig .6. Images and their corresponding vertical cross sections presenting the influence of the wavelet shape on localization computed with image 1(b) using scale numbers 3 and 4 ; (a) wavelet of Fig. 3(b) ; (b) wavelet of Fig 31c) : (c) wavelet of Fig. 3(d) .



J

4

(a)

4

(b)

(c)

Fig. 7. Images and their corresponding vertical cross sections . The localization process computed with images 1(a), 1(b) and I(c), uses the 22 samples symmetrical wavelet at scale 4 .





Localization of objects with circular symmetry in a noisy image The choice of scale numbers to use is linked to the size of the objects to detect .

359

and g(x,Y)eL'(i)

3.2 .4 . Localization test using symmetrical wavelet .

Images presented in Fig. 1 come from the same interference phenomenon observed in three different observation planes . In this case, the scale of the cross section in image 1(c) is about two times smaller than the scale of image 1(a) . As the size of the object varies by a factor of 2, only one scale is necessary for the localization process. Figure 7 shows the result using the scale number 4. The presence of the rings is clearly detected in spite of the noise present in the image. However, the center of these rings is not determined very accurately because of the difference of shape between the object and the wavelet . On the other hand, these results raise another drawback of the chosen discrete transform : the translation parameter sampling is not small enough to allow an accurate localization. This method is a fast way to detect the presence of circular objects with a cross section which looks like the used wavelet . As this condition is the only one necessary to keep the method efficient, it can be used as well to detect objects with various shapes (for example pieces of rings). In the particular case of holography, another method based on the same concepts, but able to reach an accuracy of about one pixel for the localization of the ring center, will be presented in the next section .

3 .3 . Localization by correlation with a sum of objects of different scales 3 .3 .1 . Introduction. In the previous method, the best results would be obtained when the wavelet fits perfectly the object (and particularly, with one having the circular symmetry). Generally, such a wavelet does not allow an orthonormal basis to be constructed, but fortunately, this property is not necessary to solve the localization problem . Moreover, the low accuracy of localization is partly due to spatial sampling. The use of the continuous transform should overcome this problem . The wavelet definition (see equation (2)) can be seen as a correlation between the signal and the wavelet . To get the object center it is interesting to compute a correlation product series with different scales of the theoretical model of this object. It is important to note that this operation is equivalent to a continuous wavelet transform where the wavelet is derived from the object by adjusting the mean value to zero (provided condition (1) is verified) . More precisely, let us consider the normalized correlation Cf,(u, v), defined for two functions f(x,y)eL2 (R) PR 27 : 3

8

f

f(x,y)g(x-u,y-u)dxdy

J

Cf,(u, u) = If [I

II till

If P, is the normalized function describing the object at scale s, the correlation of a given function f with this function is Cf,p,(u, u)=

f f f(x,Y)P,(J((x-u)2+(y-r)2))dxdy L

Ilf11

3 .3 .2. Correlation with a sum of scales . If the object size is not "a priori" known, the correlation product has to be computed for several scales . This can lead to long computing times and to a dilution of information . This drawback can be eliminated by using an appropriate new function . As the correlation product is linear, this new function can be constructed as a sum of functions of different scales representing objects . Let be a discrete sampling of the scale parameter, the sum associated with the cross section family based on P,(r) is N

£P(r)=

Y P,,(r) .

It is important to note that the correlation peak, given by the sum £p and an object at a scale s, is smoother than the one given by the correlation of the object by itself. Indeed, the correlation of an object at a given scale with a term of the sum at a different scale can have a negative contribution to the sum itself . The efficiency of the method is obviously linked with the choice of the scale parameter sampling and the scale parameter interval . 3 .3 .3 . Determination of the sum of scales.

• General properties Let us consider a sampling (s e), for the scale parameter s and the corresponding sum of scales £ p (r) . To evaluate the efficiency of the sum, let us compute the correlation function at 0 of this sum with an object at scale t C1(t)Cx .P,(0,0)

+s N 2m f Y P,,,(r)P,(r)rdr o . ,0 11 £p(r) 11

2x

N

11 £e(r) 11 .YO 1

f P,.(r)P,(r)rdr

N E C,(O II£r(r)II Jo

(5)



3 60

C . DuconET et at.

with

It can be shown that if C,(t)=2x

J

P,(r)P,(r)rdr

F(x)=27 f rp(xr)p(r)dr

0

0

= Cr, .r,(O, O).

then

Equation (5) shows that C5 (t) can be deduced from the sum of correlations between the different terms of L, with an object at scale t. The sampling of the scale parameter can be obtained from the study of the Qt) functions . In order to get an efficient detection, C,,(t) must have significantly high values . According to equation (5), for a Fixed sampling step, C,,(t) decreases when the interval Es,, s,1 is increasing (the numerator is a function of the sampling step whereas the denominator depends mainly on the number of samples) . Consequently, as the interval [s , s N] increases, the likeness between the sum and the object decreases .

Qt)= s F( s )

case of

cross

sections

Qt e'-'F(e ) e'-° e'-').

(a) the Qt) curves have the scaling property, (b) if t, and t 2 verify t, t 2 =s 2 then 1 )=C,(t 2) . Let us consider the new coordinate u =In (t) and the new parameter a = In (s). Equation (6) becomes °-' = F( C,(u) = Thus, on a logarithmic scale, the curves are symmetrical with respect to a and can be obtained from each other by translations . According to equaton (5), C5 (t) can be deduced from Qt) with se {s , s,, . . ., s,v } . Using the variables u and a, the C (u) curves are shifted and symmetrical . To allow an efficient localization of objects with a scale in the interval [s0 , s ,], C,,(t) must be as flat as possible in this range . This result can be obtained if a is sampled with a constant step which corresponds to an exponential sampling of s

with constant

° o

If the cross section family verifies equation (4) C,(t)=2a

f sp(sr)tp(tr)rdr . 0

s

a„=6,+n6

09 08 0 .7 -0 .6 N 05 U04 0 .3 0 .2 0 .1 950

(6)

Two important properties can be deduced from this equation :

o

• Particular shape

t F( t )

so s° = e ° " = e a° (e a 'r = dod,

1330

1710

2090

2470

3 .3 .4 . Application to holographic rings. The cross section of the interference rings has not exactly the scale property (see Section 3 .1) . Moreover, the study of the C,(t) curves cannot be done analytically because of the expression of P,(r) (see equation (3)) . Therefore, a numerical approach is necessary . More significant scale variables to use for holography are z = 1/s 2 and v= 1/t 2 . Figure 8 presents different plots of Qv) obtained with the theoretical cross section of the interference

2850

v

Fig . 8 . C,(v) curves of interference rings for z = 1330 ( •), z = 1900 (+), z =2470 (x ) .

(a)

M=100% MEN

m,m -

MM I_/a

f~

0 .4 0 .3 0 .2 0 .1 950

1330

1710

2090

2470

2850

V

Fig. 9. Cross section of the sum of theoretical objects (a), C5 (v) curve (b +) and C, ,,,(v) curve (b ∎ ) .

(b)

(c)

Fig. 10. Positive values of images and their corresponding horizontal cross sections . The correlations are computed with images I(a), I(b) and I(c) using a sum of theoretical objects with different scales .

(a)

I P

3 62

C . DUCOTTET et al.

Fig . 11 . Synthetic rings (a) and different types of noise used for the simulation : Gaussian white noise (b), white noise of Rayleigh (c), speckle noises (d) and (e) .



Localization of objects with circular symmetry in a n

tmag

63

4 . CONCLUSION

a

0 .5

1

1

Fig . 12 . Effect of noise on localization by a superposition of theoretical synthetic ring system with noise : x, Gaussian white noise ; O, white noise of Rayleigh ; ∎ and +, speckle noises .

rings . As they are nearly symmetrical and invariant by translation, a linear sampling of z is correct . The sampling step can be determined from the curves by choosing the value of v which produces a given attenuation of C,(v) Figure 9 shows the resulting cross section of the sum £p(r) and the corresponding CE (v) compared with C,"(v) where z o represents the mean value of z . In Fig . 10, correlations of images on Fig. 1 with the sum £ p (r) are presented . The localization is accurate (better than one pixel) for all the images . Moreover, the peak value is high compared to the noise level and its shape is circular . 3 .3 .5 . Analysis of the effect of noise . The aim of this section is to evaluate the noise sensitivity of the method with respect to detection and localization. A theoretical ring system (Fig. 11(a)) has been superposed with different types of noise using the ratio (a) between the maximum amplitude of the rings and the r .m.s . of the associated noise . The efficiency of the localization has been measured by the ratio (D) between the correlation peak produced by the rings and the highest peak produced by the noise . The following D =f(a) curves (Fig . 12) present the influence of two synthetic white noises, one having a Gaussian distribution (Fig . 11(b)) and the other a Rayleigh distribution (Fig. 11(c)) as well as two speckle noises (Figs I I(d) and (e)) coming from holographic recordings . One can remark that the two speckle noises have a stronger influence on detection than the synthetic ones. This fact seems due to the relatively long correlation length of this type of noise which contain much more low frequencies . If the detection threshold is D = 1 .2, localization is possible for a > 1 .5 with a speckle noise and a > 0 .2 with a white noise . The localization error for the center is about 2 pixels for D > 1 .2 during all the tests .

A wavelet transform interpreted as the correlation of the signal with a family of functions whose elements are deduced from each other by a dilatation, is a useful tool to localize objects with circular symmetry and variable size . The orthonormal transform leads to fast calculations using the Mallat algorith Moo but the primary wavelet choice and discretization technique limit the accuracy of localization . It is difficult to find a wavelet having exactly the same shape as the object with these particular bases . The first method can then be generalized to the localization of objects of cross section close to the wavelet shape . A precise model of these objects is not necessary . If the shape of the object is well known, it is possible to compute directly the correlations with a theoretical model of the object which is equivalent to a continuous wavelet transform . The computing time can be significantly reduced by the use of a sum of theoretical models at different scales . This last way to localize a system of rings in a noisy background is particularly efficient in microholography where the ratio between the amplitude of the interference rings and the r .m .s . of the noise is small . Acknowledgements--We would like to thank T . Fournel. J . Pigeon and D . Blanc for helpful discussions .

REFERENCES H . Royer, Holographic de particules micrometriques pour Is velocimetrie dans les ecoulements gazeux, 4"' cotloque national de visualisation et traitement d'images, Lille (1990) . 2 . C. Geiler, J . P. Sehon, M . Stanislas et H . Royer, Depouillement automatique d'hologrammes de microparticules . application a In granulometrie et a In velocimetrie, 4""` colloque national de visualisation et traitement d'images . Lille (1990). 3 . D . Gabor, Theory of communication, J . Insin Elect . Engrs (London) 93(111), 429-457 (1946) . 4 . A . Grossmann and J . Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J . Math . Analysis 15(4), 723-736 (1984) . 5 . 1. Daubechies, Orthonormal bases of compactly supported wavelets, Commas . Pure Appl. Math. 41(7),909-996 (1988) . AT&T Bell Laboratories, 600 Mountain Avenue .. Murray Hill, NJ 07974, U .S.A . 6. S . G . Mallat. A theory for multiresolution signal decomposition : the wavelet representation, GRASP tab, Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA 19104-6389, U .S .A. 7. J . C. Feauveau, Analyse multiresolution pour les images avec on facteur de resolution /2, Traitement do Signal 7(2), 117-128(1990). 8 . R . Kronland-Martinet, J . Morlet and A . Grossmann, Analysis of sound pattern through wavelet transform, Centre de Physique Theorique . CNRS, Luminy, Case 907, 13288 Marseille Cedex 9, France (1987) . 9. A . Grossmann, Wavelet transforms and edge detection, Stochastic Processes in Physics and Engineering, P . Blanchard, L . Strelt and M . Hazewinkel, eds . Reldel, Boston (1988) . Centre de Physique Thhorique, Section II, CNRS, Luminy, Case 907,13288 Marseille Cedex 9, France. 1.



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About the Author-CHBLSTOPHE DUCOTTET graduated from "Ecole Nationale Superieure de Physique de Marseille", France, in 1990. He received a D .E .A. ("DiplSme d'Etudes Approfondies') from the University of Marseille, France, in 1990. Since 1991, he has worked at the "Laboratoire de Traitement du Signal et Instrumentation" in Saint-Etienne, France, in order to present his thesis concerning wavelet transform applications in image analysis .

About the Author-JoANNns DAMIEEE received his Ph .D . from Lyon I University in 1977 and is presently professor at Saint Etienne University . He has taught control system since 1976 and digital signal processing since 1988 . From 1966 to 1978 he worked on nuclear spectroscopy in the Nuclear Physics Institute (Lyon) . In 1979, he joined TSI Laboratory to develop sensors in collaboration with several industries . Since 1990 his research interests have been in the field of image processing .

About the Author-MARTINE MotNE received her D .E.A . in image processing in 1990. Since 1990 she has been working on a thesis of image analysis applied to fluid mechanics at the "Laboratoire de Traitement du Signal et Instrumentation" in Saint-Etienne, France .

About the Author-JEAN-PAUL- SCHON graduated from the "Institut National Polytechnique de Grenoble" (hydraulics, 1967). His Ph .D. which was obtained at the "Ecole Centrale de Lyon" (1974), was linked to the simulation in a wind tunnel of diffusion in the atmospheric boundary layer. Since 1977, he has worked on quantitative flow visualization techniques : measurement of concentrations and velocities of particles by laser tomography and microholography .

About the Author-MICHEL COULtaoN received his "Doctoral es Sciences" in 1986 from the University Jean Monnet, France, where he has been researcher and "Maitre de conference" since 1987 . He has been engaged in studies on flow visualization with image processing .