Moiré properties of Walsh functions

Optics & Laser Technology,

Vol. 29, No. 5, pp. 261-265, 8

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1997 Elsevier

in Great Britain.

1997

Science Ltd

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ELSEVIER

Moirb properties functions C. COLAUTTI,

0. TRABOCCHI,

of Walsh

E. E. SICRE

A new type of moire experiment

is presented. Each grating is synthesized by Walsh functions. These functions have the property of being cyclically periodic, and the resulting moire is also cyclically periodic. Gratings synthesized by different Walsh functions were employed and a contouring measurement is described. 0 1997 Elsevier Science Ltd. KEYWORDS:

moirb, Walsh functions,

gratings

Introduction

I.4 d x,/2

In a typical moire experiment, a distorted image of a grating produced by the object under study, is superimposed upon a reference grating of the same period, from which a beat pattern results. In this way, information about the surface state of the object can be obtained’92. In this paper, a new type of moire experiment is presented, in which each grating is synthesized by a Walsh function3. As is well known, Walsh functions form an orthogonal and complete set of bivalued f 1 functions, and they have the property of being cychcally periodic; that is, they can be considered as square wave functions with step-like amplitude changes occurring at periodic intervals. These functions have been employed for a wide variety of applications in optical and digital image processing4-9. In our case, we have employed Walsh functions in a moire arrangement where sharply peaked moire fringes are found. These fringes are closely related to the moire fringes generated when the gratings are coded by pseudo-random sequenceslo. The narrowness of the moire fringes is useful for those contouring applications in which large displacements as well as small detail near the fringes, are to be detected.

k=O

= [sign(-sin(n

functions:

moire

t(x) = W,(x) = k[l + Wal,(x)]rect

5 0 x0

(2)

When two of these gratings are placed coplanar and each one rotated by an angle e/2, the resulting transmittance is given by T(X?Y) = 0, =

W2)

t(x

cos

812 + y sin 1!?/2)t(xcos 812 - y sin 812)

(3) We assume that the resulting moire pattern is viewed through an optical system with the following line-spread function

properties

qx, y) = y

(4) .x

where L, is a normalizing length. This line-spread function can be accomplished by a suitable filter and it can be also considered as an analytical approximation to the average in the x-direction, achieved when the high frequency details along the moire fringes are ignored.

The authors are in the Centro de lnvestigaciones Opticas (ClOp) C. C. 124, (1900) La Plata, Argentina. OT is also at the Universidad National de La Plata. Received 29 July 1996. Accepted 27 March 1997. 19:5-c

(1)

In (l), {&(x)} denotes the orthogonal but incomplete set of the Rademacher functions, which are periodic square waves of amplitude + 1 having a period dk = 2’ -k~o. The integer m is the rank of the binary expansion of n, in which the gk are the corresponding bits, 0 or 1; that is, 12= 2”‘g, + . . . + 2’go. In order to describe the transmittance function of a certain grating, it is more convenient to rewrite the Walsh functions as defined in the whole space and restricted to take only values 0 or 1; that is

Inside a certain finite domain 1x1< x0/2, the orthogonal and complete set of Walsh functions { W&(x); n = 1, 2, .} can be written as

Jar

E))r

x [sign(cos(2k-‘rc))]gk

First, we analyse the image irradiance of the resulting moire pattern. Some computer simulated moire patterns are shown to illustrate this result. Then, a contouring experiment is performed as a possible application of the present approach. Walsh

$))rfi[sign(cos(2ln

261

262

Moire’ properties

of Walsh functions:

C. Colautti

et al.

Fig. 1 Computer generated moire pattern, produced by two coplanar gratings rotated by an engle 0=20”. t(x) = w,,,(x)

(a) t(x) = k&(x);

and (b)

MoirC properties The intensity

distribution +r

of Walsh functions:

C. Colautti

263

et al.

is given by

+-* (5)

S(u - X, v - y)l T(x, ,~)l* d.u dq’

Z(U, v) = ss-X

--II

Introducing (3) and (4) in the last expression, and taking into account that 1T(x, y)12 = T(x, y), the image irradiance results 1 I(% r) = I(v) = L,

s

f% __% T(s)T(s

(6)

- 2vsin O/2) ds

where the change of variables: s = x cos e/2 + _rsin 012 was made. When (2) is introduced in this expression, it becomes +r.

I I@)=L,

s _-r

ff’,,W W,,b - VI ds

(7)

with q = 2v sin e/2. Equation (7) represents the autocorrelation of the Walsh function W,(x). Introducing the

Fig. 2 pattern

Experimental

set-up for generating

and analysing

the moirB

(a)

(b)

1.1

0.9

2 2

0.8

X 0.7

0.6

I

I

I

I

I

I

I

I

22

44

66

88

110

132

154

176

I

I

198 220

I

I

I

242

264

286

N (noof pixels) Fig. 3 (a) Result of a contouring experiment. The object is a tilted plane with a slight irregularity t(x) = W,,,(x). (b) Intensity profile of the fringes shown in Fig. 3(a)

in depth.

The grating

transmittance

is

264

Moire properties

of Walsh functions:

C. Colautti

J

I

I

I

I

I

150

175

200

225

250

275

et al.

1.1

0.9

0.8

0.6

0.3

N (no of pixels) intensity profiles of the moir6 fringes. Fig. 4 to the intensity profile to the right of it

Fourier expansions we obtain

x rect

of the Rademacher

The intensity

functions

to the left of the step is shown

in (2)

(8)

z 0 X0

where the coefficients ck, are non-zero only for odd values of lzi. In order to simplify the mathematical treatment, the finite size of the grating will not be taken into account. Introducing the last expression in (7) the image irradiance results

x exp(-2ni,i

C

9)

by the dotted

line and the dashed

line corresponds

patterns have been generated by computer simulation, they have been obtained by the superposition of two rotated gratings synthesized by Wxgd(x) = R7(x)Rg(x) and W&x) = R~(x)&(.Y), respectively. By comparing Figs l(a) and l(b), it can be observed that the former has a cyclical period lower than that of the latter. Since the cyclical period of the Walsh functions is determined by the lowest-order Rademacher component, the resulting moire from two Walsh functions will also have a cyclical period related to this Rademacher component.

Experimental

results

and conclusions

A contouring experiment was performed employing a grating synthesized by the Walsh function W,,,(x). A shadow moire technique was employed and the optical set-up is shown in Fig. 2. The object under study was a

(9)

where the indexes k, , . . . , k ,,,, k’, , . . . , ki,, must satisfy the following condition

c

g,(k, - k;) =’ d / I

(10)

As can be derived from (9) the moire pattern is cyclically periodic in a direction determined by ye= 2v sin O/2. The fringe pattern described in this equation resembles the original Walsh function, but with those coefficients ck, whose indices satisfy the condition given in (10). The beat patterns produced by two coplanar rotated gratings are shown in Figs l(a) and l(b). These moire

Fig. 5 Moire fringes produced by a Ronchi grating: the object under study is the same as has been employed to obtain the results shown in Fig. 3(a)

Moire properties

of Walsh functions:

tilted plane with a slight irregularity in depth. As a consequence of the sloping position the moire pattern modifies its cyclical period, as can clearly be seen in Figs 3(a) and (b), which correspond to the recorded moire pattern and the intensity profiles of the fringes, respectively. On the other hand, as a consequence of the small step, the moire fringes are discontinued. The thickness of this step was measured, and was found to be 0.47 + 0.005 mm. The shift of the fringes has also been measured and it corresponds to 0.20 + 0.005 mm, which is about one half of the line width. In Fig. 4 the intensity profiles to the left and right of the step are recorded and the shift in the maxima could be appreciated. In order to analyse the performance of the Walsh functions in the moire techniques, we compared the obtained moire patterns with the moire pattern generated under the same conditions but with a Ronchi grating. This pattern is shown in Fig. 5. As can be derived, by comparing Fig. 3(a) and Fig. 5, the Walsh moire pattern gives rise to an improved resolution since a shift in the fringe positions can be identified more clearly than in the case of the Ronchi moire. As has been previously mentioned, this technique should be useful in applications where large displacements are to be observed, while small displacements in the vicinity of the fringe line are also present.

C. Colautti

et al.

265

Acknowledgements This work was partially supported by the Consejo National de Investigaciones Cientificas y Tecnicas (CONICET, Argentina). References 1 2

3 4

5

6

Meadows, D.M., Johnson, W.M., Allen, J.B. Generation of surface contours by moire patterns, Appl Opr, 9 (1970) 942-947 Ikeda, T., Terada, H. Development of the moire method with special reference to its application to biostereometrics, Opt Laser Technol, 13 (198 I) 302-306 Beauchamp, K.G. Walsh Functions and their Applications Academic Press, New York (1975) Colautti, C., Ruiz, B., Sicre, E.E., Garavaglia, M. Walsh functions: analysis of their properties under Fresnel diffraction, J Mod Optics. 34 (1987) 1385-1391 Trabocchi, O., Colautti, C., Sicre, E.E. Diffraction properties of a periodic multiple-aperature system: an approach based on the Walsh functions, Opt Eng, 35 (1996) 94-101 Trabocchi, O., Colautti, C., Saavedra, G., Sicre, E.E. Spatial coherence properties of a multiple-aperture system: an approach based on the Walsh functions. Submitted for publication in J Mod Optics, 1996 Andrews, H.C. Computer Techniques in Image Processing, Academic Press, New York (1970) 75 Hazra, L.N. A new class of optimum amplitude filters, Optics Commun, 21 (1977) 232-236 Hazra, L.N., Guha, A. Far-field diffraction properties of radial Walsh filters, J Opt Sot Am A, 3 (1986) 843-846 Katyl, R.H. Moire screens coded with pseudo-random sequences, Appl Opt, I I (1972) 2278-2285

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