Changing the pseudocolor of a grating encoded phase picture using a grating

Changing the pseudocolor of a grating encoded phase picture using a grating

Optics Communications90 ( 1992) 27-31 North-Holland OPTICS COMMUNICATIONS Changing the pseudocolor of a grating encoded phase picture using a gratin...

322KB Sizes 0 Downloads 20 Views

Optics Communications90 ( 1992) 27-31 North-Holland

OPTICS COMMUNICATIONS

Changing the pseudocolor of a grating encoded phase picture using a grating Fanglin Peng Department of Physics, Beijing Normal University, Beijing, 100875 China

Received 17 December 1991

Analystic formulae, based on Fourier optics, are presented to explain the experiment of changing the pseudocolorof a grating encoded phase picture using a grating superposed on the phase picture. The superposed grating and the encoded picture are considered as a compositegratingtheoreticallyand its Fourier spectra are calculated.The Fourier spectra of the compositegrating are different from that of the encoded phase picture, and can be changedby translating the superposedgrating, so the pseudocolor image generatedby the compositegratingwill vary with the translation of the superposedgrating.

I. Introduction

The optical pseudocoloring of black-and-white images always deserves special attention because it offers some distinct advantages, such as that the system is simple and economical to operate and its pseudocolor images have continuous chromaticity change. The first density pseudocolor encoding by halftone screen implementation with a coherent processor was reported by Liu and Goodmen [ 1 ]. A similar method with a while-light processor, an incoherent processor which reduced coherent noise, was reported by Tai, Yu and Chen [2]. Although good results have been obtained, however there is a spatial resolution loss with the halftone technique and number of discrete lines due to sampling are generally present in the pseudocolor image. A method of addition of an incoherent image and a coherent contrast reversed image, each of them with a different primary color, was reported by Santamaria, Geo and Besc6s [ 3 ] for density pseudocolor. Another method based on the addition of a positive image and a negative image has been proposed by Chao, Zhuang and Yu [4], where the two images are color encoded through modulation by a Ronchi grating and color filtering. These two techniques give much better resolution than the halftone technique,

but have the limitation of moderate range of the chromaticities. Later a white-light density pseudocoloring, based on superposition of three different primary color encoded images (i.e., a positive image, a negative image and their product image, each of them with a different color) was proposed by M6ndez and Besc6s [ 5] for extending the moderate range of chromaticities, where the three images were obtained by a diffraction grating moderating and color filtering. A similar method with a different grating encoding technique was reported by Yu, Chert and Chao [6 ]. Both methods need three exposures with a grating and three-color filtering at the Fourier plane, which introduce the additional processing complexity. A convenient white-light density pseudocolor technique, called grating encoded phase picture pseudocoloring, which utilized the diffraction feature of the phase grating, was reported by Zhang et al. [ 7,8 ]. The simple technique, which only required one spatial encoding and one spatial filtering, offered several major advantages that previous techniques also have offered. Its pseudocolor images were free from coherent artifact noise and as colorful as that generated by superposition of three primary color coded images. When a prepared photographic plate with insinsitive grating code was used for encoding, the processing procedure was simplified and better results were obtained [ 9 ]. This technique also could

0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

27

Volume 90, number 1,2,3

OPTICS COMMUNICATIONS

generate reflective density pseudocolor images [ 10 ]. Very recently, Zhou and Chen [ 11 ] reported the experiment of changing the pseudocolor image with a grating superposed on the phase and presented approximate expression to explain their experiment. Although their experimental results were interesting, no more detailed theoretical discussion has been given so far, except this approximate expression. In this communication we shall analyse the interesting experiment theoretically according to Fourier optics and present analytic formulae which help in explaining the experiment and promoting the technique. We consider the superposed grating and the phase picture as a composite phase grating theoretically, and then determine its diffraction light field distribution using Fourier transforming. This method is similar to that used in refs. [ 12-15 ], where the versatile phase grating has been used for nonlinear optical processing [ 15 ], the color filter [ 12 ] and the beam spliter [ 13,14 ]. This analytic method not only direct leads to analytic formulae to explain the experiment, but also evidently reveals the inherent relation among the different usage of the phase grating. In the next section we review the grating encoded phase picture pseudocoloring technique briefly. The section is followed by the detailed discussion of the role of the grating in changing the pseudocolor image.

1 June 1992

_p••Y L

--

~Yl

Let us discribe the grating encoded phase picture pseudocoloring method briefly [ 9,10 ]. A black-andwhite image transparency in contact with a Ronchi grating is put on a fresh photographic plate and exposured them simultaneously. After developing and bleaching the exposured plate, we obtain a surface relief phase object, called grating encoded phase picture. When the phase picture is put on the input plane of a while-light processor, as shown in fig. 1, a density pseudocolor image at the output plane is generated by spatial filtering at the Fourier plane, which allows only a single diffraction order to pass through. The amplitude transmittance of the phase picture is [7,8] T' (x, y) =exp[iCt(x, y)B(x) ] , 28

(1)

I

Fig. 1. A White-light pseudocolor encoder. The coordinate systems (x, y), (x~, y~ ) and (Xo, Yo) are on input, filter and output planes, respectively. The Fourier transforming lenses are L, situated at a focal distance from each plane. The light source P is the point source of white-light.

where i = x / ~ , C is an appropriate proportionally constant for monochromatic light, (x, y) is the coordinate system of the input plane, t(x, y) is the intensity transmittance of the black-and-white transparency and B(x) is the transmittance of a Ronchi grating of period 2a, defined as

+~

. [ 2 x - a ( 2 k + 1 ).)

B ( x ) = k=E-~ rect k

~a

'

(2)

A simple special case is that t(x, y) is a constant, that is, the black-and-white image has equal density everywhere, then the phase picture becomes a binary-phase grating, which can be written as

T(x) =exp [iDB(x) ] , 2. Review of grating encoded phase picture pseudocoloring

L

Xl -- ~ - k

(3)

where D = Ct(x, y). It should be noted that D = 2hn/ 2 where h is the optical path difference of two parts in a phase grating period and 2 is the light wavelength. Generally, t(x, y) is an arbitrary density distribution. In this case we can divide the black-and-white image into many small regions, each of them has equal density. Following above argument, the phase picture shouldconsist of many different binary-phase gratings. For simplicity, we may analyse the diffraction intensity distribution of any binary-phase grating in the white-light processor instead of the encoded phase picture. According to Fourier optics, the amplitude transmittance, g(x), of a grating of period L can be written as +oo

G(x)=

~ n

=

--

Cnexp(i2nnx/L) , oc~

(4)

L

if

C. = ~

g(x) exp(-i2n~x/L).

(5)

o The amplitude distribution of nth diffraction order on the output plane was C~ exp(i2nrcx/L) [ 14], the corresponding intensity distribution is I, = (C,) 2 .

(6)

Substituting eq. (4) into eqs. (5) and (6), we have Io= (1 + c o s D ) / 2 , I,=(1-cosD)(l-cosn~)/nZg

1 June 1992

OPTICS COMMUNICATIONS

Volume90, number 1,2,3

(7) 2,

(n>0).

(8)

These results are the same as those obtained in refs. [7-9]. Obviously, In depends on ~. and n. For given ;t and n, In is a certain numerical value, which represents even monochromatic density distribution on the output plane. This is just the phenomenon observed in the experiment. For different 2 and same n, the values of In, i.e., the monochromatic densities are different. Since all these unequal monochromatic intensities are superposed incoherently, this superposition will generate even chromatic light other than white-light at the output plane. Of course the chromatic light can vary with n. In addition, the boundary function of the phase grating has already modulated all the Fourier spectra of the phase grating, so it remains after spatial filtering and determines the region where the chromatic light appears. Thus the generated pseudocolor image remains both information of the gray levels and its distributions of the original black-and-white image.

3. Changing the pseudocolor using a grating If we superpose a grating on the phase picture in the input plane, keep their lines parallel to each other and transversely translate the grating in the direction perpendicular to their lines, the pseudocolor of the image on the output plane can be changed in realtime. To discuss the experiment we first examine the amplitude transmittance resulting from superposition of the grating and phase picture. We assume that the phase picture is a phase grating and the superposed grating is a Ronchi grating. The amplitude transmittance of the phase grating is represented by

eq. (3). The amplitude transmittance of the Ronchi grating with translation distance, s, can be written as B(x-s)=

+~ .[2x-2s-a(2k+ k=X-o~rect~, 2aa

1 )'~ ).

(9)

The resulting amplitude transmittance, G(x, s), of the superposed two gratings is G(x, s) = B ( x - s ) T ( x ) ,

(10)

which is a periodic function of x with period 2a because it is the product of two periodic functions of x with the same period 2a. It also is a periodic function ofs with period 2a because eq. (lO) has the following features G(x, s + 2 a ) = B [ x - ( s + 2 a ) ] T(x) =B[ ( x - 2 a ) - s ] =B(x-s)

T(x)

T(x)=G(x,s).

So we will calculate In only for O<.s~2a. The resuiting transmittance G(x, s) is specified below according to the values of s: (i) When s=0, the Ronchi grating is at initial position, we have G(x, 0) = e x p ( i D ) , =0,

(O<~x<~a), (a
(11)

Fig. 2a illustrates this expression in graphical form. In fig. 2, the dotted lines represent the phase profile of phase grating, the oblique lines represent the opaque regions of the Ronchi grating, where the transmittance is zero, and the solid lines represent the phase profile of the resulting grating. Similarly, we have: (ii) When O<~s~a (see fig. 2b) G(x,s)=O ,

(O<~x~s) ,

=exp(iD) ,

(s
=1 ,

(a
=0,

(a+s
(12)

(iii) When a
(O
=0,

(s-a
= 1,

(s
(13) 29

Volume 90, number 1,2,3

OPTICS COMMUNICATIONS

a)

1

DB~x) m- t Ir-qI

Ir--1I

~

a

1 June 1992

C2m-I- 2 ~ ( 2 m - 1 ) { [ 1 - e x p ( i D ) ] _~

2a

- [l-exp(iD)] I2m--

D

exp[ - i ( 2 m - 1 ) n s / a ] ) ,

1 (1-cosD) (2m~)2

(

1-cos

2m0 -

-

(17) (18)

x 0

a 2m--I

b>

t ,-i

--

(2m_l)2zr2

1-sinDsin(2m-1)ma

(19)

D,B(x)

r-I

F-]

a

0

_

////~

"

~

2a

_-x

When a
y////

Co=(1/2a)[(2a-s)+(s-a) e x p ( i D ) ] ,

(20)

Io = ( 1/4a 2) [ ( s - a ) 2 + ( 2 a - s ) 2 /D

/ Z/

/

. . . . OS a a+S

~

X

+2(2a-s)

c) r-1

-

r-1 I I m I

I I 0

a

~X

2a

The phase profile of the composite grating when (b) O<~s<~a,(c) a
Fig. 2.

(a) s =

0,

To calculate the intensity distribution on the output plane we consider the function G(x, s) as the transmittance of a composite grating theoretically and utilize eqs. ( 5 ) and (6) again. Substituting G (x, s) into eqs. (5) and (6), we find: When O<~s
(14)

Io=(1/4a2)[(a-s)2+sZ+2s(a-s)cosD].

(15)

The values of C, (n > 0) could be divided into two groups according to n. Let m = 1, 2, ..., we have

30

(21)

At this time the expressions of C2m+l, 12,. and I2m+J are the same as that when O<~s<~a,but C2m has an additional negative sign. It can be seen that Io and I, in eqs. ( 14 ) - (21 ) are functions of 2 and n. So the pseudocolor image, colors of which vary with n, can be generated by spatial filtering according to the discussion similar to above section. A new conclusion from eqs. ( 1 4 ) - (21 ) is that lo and I, are functions of translation distance, s, between the Ronchi grating and the phase grating. Even if 2 and n are given, monochromatic intensities Io and I, will vary with the translation distance s. It has been mentioned that all the unequal monochromatic light constitute chromatic light, so any variation of monochromatic intensities will cause a color variation of the chromatic light. This is the reason why translating the Ronchi grating can change the pseudocolor image. A close examination reveals that, for given 2 and n, the plot of function of Io versus the translation distance, s, is parabolic. Theoretical plots of Io, Ii and 12 are drawn in fig. 3, where D=zr/4.

4. Conclusion

6'2,, = ( i / 4 n m ) { [exp(iO) - 11 - [ e x p ( i O ) - 1 ] X exp ( - i2mns/a) },

(s-a) cos D] .

( 16 )

We have analysed in detail the experiment of changing the pseudocolor of a grating encode phase picture using a grating, based on Fourier optics. We consider the superposed grating and the phase pic-

OPTICS COMMUNICATIONS

Volume 90, number 1,2,3

J

Io 2a'=S 0'15t

1 June 1992

a n d o r d e r n. Since the grating e n c o d e d phase picture p s e u d o c o l o r i n g t e c h n i q u e utilizes the diffraction F o u r i e r spectra o f the phase picture a n d t r a n s l a t i n g the s u p e r p o s e d grating can change these diffraction spectra, we can change the p s e u d o c o l o r image using a grating.

References

/ a

2a'~S

a

2a

0.151

I2

Fig. 3. The theoretical plots of/o, I~ and/2, where D= rt/4. ture as a c o m p o s i t e grating, specify its resulting a m plitude t r a n s m i t t a n c e a n d calculate the i n t e n s i t y o f each d i f f r a c t i o n order. A n a l y t i c f o r m u l a e are presented. T h e intensities, different f r o m that o f the phase picture, are a f u n c t i o n o f the t r a n s l a t i o n distance, s, b e t w e e n the s u p e r p o s e d grating a n d the phase picture, as well as a f u n c t i o n o f w a v e l e n g t h 2

[ 1] H.K. Liu and J.W. Goodmen, Nouv. Rev. Optics 7 (1976) 285. [ 2 ] A. Tai, F.T.S. Yu and H. Chen, Optics Lett. 3 ( 1978 ) 190. [3] J. Santamaria, M. Gea and J. Besc6s, J. Optics 10 (1979) 151. [ 4 ] T.H. Chao, S.L Zhuang and F.T.S. Yu, Optics Lett. 5 (1980) 230. [5] J.A. M6ndez and J. Besc6s, J. Optics 14 (1983) 69. [6] F.T.S. Yu, X.X. Chen and T.H. Chao, J. Optics 15 (1984) 55. [7] Z. Jingiiang, W. Shuying, L. Dehe and P. Fanglin, Acta Optica Sinica 5 (1985) 944. [8]Z. Jingiiang, W. Shuying and L. Dehe, Process. SPIE 523 (1985) 336. [ 9 ] S. Wang, J. Zhang and C. Lei, Optical Engineering 26 ( 1987 ) 60. [ 10 ] M. Guo-guang, D. Pei-jun and F. Zhi-liang, Optik 81 (1989) 89. [ 11 ] Z. Ying and C. Zhenpei, Proc. SPIE 974 ( 1988 ) 103. [ 12] K. Knop, Appl. Optics 17 (1978) 3598. [ 13 ] H. Damman and K. Gortler, Optics Comm. 3 ( 1971 ) 312. [ 14 ] J.N. Mait, J. Opt. Soc. Am. A 7 (1990) 1514. [ 15 ] S.R. Dashiell and A.A. Sawchuk, Appl. Optics 16 ( 1977 ) 1936.

31