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Broadband image subtraction by spectral dependent sampling
OPTICS COMMUNICATIONS
Volume 59, number 5,6
BROADBAND G. GHEEN Electrmd
and
IMAGE SUBTRACTION
BY SPECTRAL
1 October 1986
DEPENDENT
SAMPLING
F.T.S. YU
Engineermg Department,
Pennsylvanra State University,
Umversity
Park, PA 16802, USA
Received 2 December 1985; revised manuscript received 28 May 1986
A white-light optical processor which can perform broad spectral band image subtraction is presented. This system spatially samples a wavelength smeared image of a broad-band source to establish the necessary coherence requirements for image subtraction. An analysis and experimental results of the color subtraction system are presented.
1. Introduction The source-encoded image subtraction system proposed by Wu and Yu [ 1,2] is an interesting example of how the coherence of light may be manipulated to extend the processing domain of a partially coherent optical processor. In that system the basic architecture of a coherent optical subtraction system [3] is used; however, the coherent light source is replaced with a spatially extended, monochromatic light source that is encoded by an optical mask. The encoding procedure establishes a point-pair spatial coherence which is sufficient for the subtraction operation. Here we shall extend the source-encoding procedure so that we can perform color image subtraction under white-light illumination. To do this it will be necessary to make some modifications to the monochromatic subtraction system since both the spatial filter and source-encoding mask are wavelength dependent. By
using a dispersion grating, we can separate the whitelight into its spectral components in the planes of the sourceencoding mask and spatial filter. We may then compensate for the wavelength dependence of the optical masks used in those two planes.
2. Description
and analysis
The proposed broadband subtraction system is shown in fig. 1, where S is an extended white-light source, DC is a dispersion grating, M is the sourceencoding mask, 0, and 0, are the two input objects that are to be subtracted, G is a spatial filter, and all lenses have the same focal lengthf. The front, off-axis portion of the system provides the proper spectral illumination for the sourceencoding mask M. With the source-encoding mask properly illuminated, a pointpair coherence will be established for points in the in-
DG
Fig. 1. Broadband image subtraction system. S is an extended broadband source, DG; dispersion grating, M; source encoding mask. 0 1 and O2 ; two input objects, G; fan-shaped sine-grating.
0 030-4018/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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put plane separated by 2H in the y-direction. This point-pair coherence is established for each wavelength component of the white-light. The remaining optical system acts to superimpose the images of the two input objects in the output plane with a X/2 phase shift between the two wavefronts. This condition is established for each spectral component of the white-light so each wavelength component will destructively interfere to subtract. The resulting subtracted color image consists of a superposition of the subtracted intensities of each wavelength. We will analyze the broadband subtraction system in a fashion which parallels the description given in the preceding paragraph. As explained by Cartwright [4] , the output of a broadband optical system may be determined by analyzing the system for a general wavelength X. The result for broadband illumination can then be obtained by integrating the output intensity over all wavelength components of the white-light. 2.1. Illumination of the source-encoding mask We begin our analysis of the broadband subtraction system by determining the mutual intensity of the light illuminating the sourceencoding mask. This may be accomplished by using the theory for the propagation of the mutual intensity [5]. IfJ~(x2.y2,x~.~~;) is the mutual intensity in plane P2 caused by the mutual intensityJL(xl ,yl ,x; ,v;) in plane Pl, then these two quantities are related by
X
K*(x;,y;,x;,y;)
dx, dy, dx; d$).
(1)
where K is the amplitude impulse response between the two planes, Note the subscript of J denotes that we are considering a single wavelength component X. To determine the mutual intensity from eq. (I), we tnust know the mutual intensity of the light source and the impulse response between the planes of the light source and the source-encoding mask. For ease of analysis we assume that the white-light source has a rectangular shape with equal intensity over its entire area. In this case, the mutual intensity at wavelength h of the light source is 336
J^b. P. cy’.P’, = C’(h) rect(a/a)
rect(fl/b) 6(cu--(Y’, /L $).
(2)
where C(h) is the intensity of the wavelength component h. the rectangular functions define the spatial limits of the source, and the Dirac delta function is a convenient way to express the incoherence of a ther-ma1 light source. For convenience. we assume C(h) is constant for all X and ignore it and all multiplicative constants in our analysis. The front. off-axis portion of system acts to illuminate the source-encoding mask with a wavelength smeared image of the light source. Assuming lenses ot infinite extent and I OO%,diffraction into the 1 order of the dispersion gr-ating. we may write the amplitude impulse response of this system as K(cu. p,x._v, = 6(x + o( +fiv(,._v
+ P).
(: 1
where f‘is the focal length of the lense. h is the wavelength of light, and vu is the spatial frequency of the dispersion grating. The factor fhvO is the translation due to the dispersion grating. Substituting eq. (2) and eq. (3) into eq. (1) and per-forming the integration, we obtain the mutual intensity illuminating the sourceencoding mask as J,(x.y,x’,y’) X rect(y/b)
= rect[(x +fxvo)/al 6(x ~ x’._v ~- .v’).
(4)
For large values off and vu, the wavelengths of light will be dispersed over a large distance. If the spatial extent of the light source is small in the xdirection, we obtain a high spectral purity and may effectively operate upon the different wavelength components with the source-encoding mask. Before proceeding to the next part of the analysis. we need to transform eq. (4) into the coordinate system of the sourceencoding mask. The optical axis ot the system following the sourceencoding mask lies along the principle ray of the mean wavelength x of the white-light. This means that the coordinate is translated by five in the xdirection. This translation can be taken into account by making the coordinate transform xO
=x+f~v(),
yo=y.
The mutual intensity
in the new coordinate
system is
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OPTICS COMMUNICATIONS
Jk(xo,yo,x~,y~~l = rectKxo + 0 X rect(yolb) “(x0 -
Qf~,)/al
xb,Y, - Yb>.
(5)
1 October
1986
where there are 2N t 1 slits, T is the distance between the slits at x,, = 0, and S is the width of the slits. The mutual intensity after the source encoding mask is
2.2. Determination of the mutual intensity in the input plane
J&Jo&$
To determine the mutual intensity in the input plane, we will again use eq. (1). The transfer function needed in this case is the Fourier kernal because of the Fourier transform relation between the source encoding and input planes. The mutual intensity of the light immediately after the source-encoding mask needs to be obtained. To determine the mutual intensity of the light immediately after the source-encoding mask, we need the transmittance function of the source-encoding mask. The source-encoding mask consists of a series of narrow slits arranged in a fan-shaped fashion as shown in fig. 2. We may express the transmittance of the source-encoding mask as
where the plus sign indicates the mutual intensity after passing through the optical mask and * denotes the complex conjugate. Substituting eq. (5) and eq. (6) into eq. (7), we arrive at
N
M(xO,y,)=
n=-lY C
-
rect
nT(1 + xO/vO~f) s
17(6)
=J,(xOYO’xb,yb)M(xo,vo)M*(xb,rb),
J’,(xO,~o,x~,~~)
(7)
= rectKxO + (A - QfvJal
X 6(x() - xb,yo - yb) x
,$ -N
+x,Jifv,,)
- nT(1 ret
(8)
s
where we have assumed that the spatial extent of the light source in they-direction (i.e. b) is sufficient to span the 2N t 1 slits. We are now in a position to solve for the mutual intensity in the input plane. Substituting K(xo ,yo, x 1, VI> = ew[-(ik/f)(xgl +yoy1)1 and eq. (8) into eq. (1) and solving, we obtain the mutual intensity in the input plane as JA(x1,y1,x;,~;)=exp[i2nvo(l
X sinc[ka(xl -x;)/2f]
-VW,
sinc[kS(yl
-+>I -y;)/2f]
N X _C, ev-inWyl
t
-
v;Yfl
,
(9)
where sine(x) = sin(x)/x and k = 27r/j;. Because the width of the slits is very small, the sine term containing the S parameter is approximately 1. We may also rewrite the sum of exponential terms in eq. (9) using the relation
5 exp(ix)=Sin[(2N+ 1)X/21 -N
Fig. 2. The source encoding mask of the color subtraction system. S, width of the slits; T, period of the slits for the mean wavelength.
sin(x/2)
Thus the mutual intensity expressed as
. in the input plane may be
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5.6
COMMUNICATIONS
I octobcl 1986
X/h)xl]
E,(xI,y,)=exp[i27m0(l
‘:‘*(X*,Y*) = F[(k/f)(x*
-- (h
X,fv,).
jk/j’)_Y*J , (15)
J~WJ’,Y’) .(ll) Y(X,Y,X’,Y’) = [J(x,Y,X,Y)J(X’,Y’,X’:Y’)l l’* Using eqs. (10) and (1 l), we see that the light in the input plane is coherent when x1 - xi = 0 and nrr for n = 0, 1, X.... To establish a WY1 ~ ri)/2f= coherence of 1 between the points on the two input objects that overlap in the input plane, we let Yr - Y; = 2H, where 2H is the separation between the two input objects in the Ydirection. This leads to the condition for the spacing of the slits on the sourceencoding mask as
2.3. Determination
(14)
where the linear phase term is taken from eq. (10) to express the position of the light source of wavelength h in the source encoding plane. The corresponding complex amplitude in the spatial filtering plane is
where M = 2N + 1. The coherence is related to the mutual intensity through the relationship [5]
T = fj;/2H.
f(xl..l’l),
where t; denotes the Fourier transform off’(x.4’). The spatial filter used in the Fourier plane is a fan-shaped sine grating whose transmittance is given by
G(x*,-‘.?)=+1+ _ [ sm(,X2:~:,.J
(10)
Multiplying eq. (16) by eq. (15) and performing an inverse Fourier transform, we arrive at an expression fol the complex amplitude in the output plane as
(12) of the output intensity
The final task in our analysis is to determine the intensity in the output plane of the broadband subtraction system. To do this, we use eq. (1); with the two points in the output plane (i.e. (x3,Y3) and (xi, y;)) set equal. The mutual intensity in the input plane after the input transparencies can be written as
X exp[(Wf)(x,x3 I . +y2y3)1 dx, Q7. Lettingp = (k/f)[x2 eq. (17) reduces to
~~(A
(17)
x)f’uo] andq = (k/j‘)~)~.
J~(xl,Yl.x;,Y;)=J,(xl,Yl,x;,Y;) X [O,(X,>Y,
-H)
+ O,(X,.Y,
X [j + sin (I +$,rV,,)I
+H)l
expIi(px3
+qY3 )I d/l dy. (18)
X Q(x;,Y;
+H)+O,(xr,yr
+H)l*>
(13)
where Jh is given in eq. (10) and 0, and 0, are the transmittances of the two input objects. The amplitude impulse response between the input and output planes needs to be derived. Because it is a difficult task to determine the impulse response between the input and output planes, we determine the transfer function for low spatial frequency input signals. This will adequately serve our purpose if 0, and 0, are low spatial frequency signals, Given a low spatial frequency signal f(x, y), we may write the complex amplitude in the input plane as 338
We may expand the sine term in eq. ( 18) in a Taylors expansion to obtain sin(l
+$nVO)=sin(H~)+~cos(H~),
(19)
where only the first two terms were retained since f(x, y) is a low frequency signal sop & 2nv0. Substituting eq. (19) into eq. (18) and solving the equation. we obtain
OPTICS COMMUNICATIONS
Volume 59, number 5,6
as the spatial filter. These distortions are an inate problem of this color subtraction system and is the price we pay to extend the source encoded subtraction system to operate under broadband illumination. To determine the effect of the distortions on the output subtracted image, we pick the two input images to be identical. Substituting 0, (x, y) = 02(x,y) into eq. (22) we obtain
X)x31
E,(x3,Y3) = expW@-
x U(x, r>* P&3 +Y,) + (l/W(x3,y3
+W - (l/2i)W3,Y3 - H)
x P(XpY3 +w+wgJJ3
-w111>
(20)
where the low spatial frequency consideration is taken into account by f(x, y) and the derivative arises from the pq term which multiplies the cosine term in eq. (19). Because we are concerned with the output intensity, we may ignore the linear phase term in eq. (20) and write the transfer function between the input and output plane as W,>.Y,,xSLPg)=W~
-x321
1 October 1986
10
= (H12nv,)2i2(a2/axay)01(x,y)i2 + [W2 - 1)112$
i2wx)01kY)i2.
(23)
Ideally the output should be zero; however, from eq. (23) we see that there is significant intensities where the transmittance changes rapidly. This is indicated by the derivatives. This distortion may be minimized by choosing vu as large as possible and by limiting the number of source-encoding slits (i.e. N).
-v,)
3. Experimental results -
U/2i)W1
- x3,y1
- y3 +W
+ vwq(a2iax3a~,)
x
W-X,J,
-Y3-H)+G(xl-X3’Y1-Y3+H)I (21)
We may now solve for the output intensity of the subtraction system. Substituting eq. (13) and eq. (21) into eq. (1) with x3 = xi and y3 we obtain the output intensity for wavelength h as
=y;,
I,(x3,.Y3)= + (Hln~)
IO, - 0212 Im[(OT - o”,)(a2/a~,ay3)(o,
+ [(N* - iw$l(apx,w,
+ o,)l
+02)i2,(22)
where the (x3,y3) parameter was dropped to avoid cluttering the equation, Im stands for the imaginary part, and only the terms which correspond to the subtracted result were retained. From eq. (22) we can see that the output result is independent of wavelength. The first term in eq. (22) corresponds to the desired subtracted result. The other terms in the equation are distortions caused by the fan-shaped sine-grating used
An experimental demonstration of the color subtraction system was performed to verify its operation and investigate the distortions indicated in eqs. (22) and (23). In our experiment a xenon-arc-lamp was used as the white-light source. All the lenses used in the system where the same with focal length of 400 nm. The spatial frequency of the dispersion grating was vu = 120 lines/mm. The source-encoding mask was designed to pass wavelength ranging from 4500 A to 6500 A. The average period of the slits on the encoding mask was T = 22 /.un, the width of the slits was s = 4 pm, and the number of slits was 2N + 1 = 19. The separation of the two input objects from the optical axis in the input plane was H = 5 mm. In one experiment, a pair of black-and-white binary transparencies was used as the input objects. The two input objects and the subtracted result are shown in fig. 3. The diameter of the circle is approximately 1 mm. As predicted by eq. (23) the edges of the circle did not subtract well. The broad lines at the top and bottom of the subtracted circle are due to the second term in eq. (23). In another experiment, color transparencies were used for input objects. The two input objects and the subtracted result are given in fig. 4. In the subtracted result we see the F-l 6 as expected. Also present in the 339
I5.g. 3. Binary
input
I&. 4. Color input
objects
objects
(a. b) and the subtracted
(a, b) and the subtracted
result
(c,.
result (c).
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subtracted image are bright areas along the mountain tops, the edges of the sand dune and along the shore line. The angular dependence of the first term in eq. (23) corresponds well with the effect seen along the shore line. That is, the first term in eq. (23) is most significant for edges oriented at an angle of 545’ from the x-axis.
4. Conclusion This paper introduced a sourceencoded subtraction system which operates under broadband illumination. Our analysis indicated that spectral dependent sampling can establish the point-pair coherence needed for the subtraction operation. Our analysis also indicated an inate distortion in the output of the subtraction system. This distortion is the price that is paid to extend the sourceencoded subtraction system to operate under white-light. This distortion can be minimized by limiting the number of sourceencoding slits and by using a dispersion grating with a large spatial
1 October
1986
frequency. Experimental results were presented which correspond well with the results of our analysis and demonstrate the viability of the system.
Acknowledgement We acknowledge the support of U.S. Air Force Rome Air Development Center, Hanscom Air Force Base, under contract no. F19628-84-K-0031.
References [l] ST. Wu and F.T.S. Yu, Optics Lett. 6 (1981) 452. [2] S.T. Wu and F.T.S. Yu, Appl. Optics 20 (1981) 4082. [3] S.H. Lee, S.K. Yao and A.G. Milnes, J. Opt. Sot. Am. 60 (1970) 1037. [4] S. Cartwright, Appl. Optics 23 (1984) 318. [5] M. Born and E. Wolf, Principles of optics (6th Ed., Pergamon Press, New York, 19’80). [6] F.T.S. Yu, Optical information processing (Wiley, New York, 1983).
341
Volume 59, number 5,6
BROADBAND G. GHEEN Electrmd
and
IMAGE SUBTRACTION
BY SPECTRAL
1 October 1986
DEPENDENT
SAMPLING
F.T.S. YU
Engineermg Department,
Pennsylvanra State University,
Umversity
Park, PA 16802, USA
Received 2 December 1985; revised manuscript received 28 May 1986
A white-light optical processor which can perform broad spectral band image subtraction is presented. This system spatially samples a wavelength smeared image of a broad-band source to establish the necessary coherence requirements for image subtraction. An analysis and experimental results of the color subtraction system are presented.
1. Introduction The source-encoded image subtraction system proposed by Wu and Yu [ 1,2] is an interesting example of how the coherence of light may be manipulated to extend the processing domain of a partially coherent optical processor. In that system the basic architecture of a coherent optical subtraction system [3] is used; however, the coherent light source is replaced with a spatially extended, monochromatic light source that is encoded by an optical mask. The encoding procedure establishes a point-pair spatial coherence which is sufficient for the subtraction operation. Here we shall extend the source-encoding procedure so that we can perform color image subtraction under white-light illumination. To do this it will be necessary to make some modifications to the monochromatic subtraction system since both the spatial filter and source-encoding mask are wavelength dependent. By
using a dispersion grating, we can separate the whitelight into its spectral components in the planes of the sourceencoding mask and spatial filter. We may then compensate for the wavelength dependence of the optical masks used in those two planes.
2. Description
and analysis
The proposed broadband subtraction system is shown in fig. 1, where S is an extended white-light source, DC is a dispersion grating, M is the sourceencoding mask, 0, and 0, are the two input objects that are to be subtracted, G is a spatial filter, and all lenses have the same focal lengthf. The front, off-axis portion of the system provides the proper spectral illumination for the sourceencoding mask M. With the source-encoding mask properly illuminated, a pointpair coherence will be established for points in the in-
DG
Fig. 1. Broadband image subtraction system. S is an extended broadband source, DG; dispersion grating, M; source encoding mask. 0 1 and O2 ; two input objects, G; fan-shaped sine-grating.
0 030-4018/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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5.6
put plane separated by 2H in the y-direction. This point-pair coherence is established for each wavelength component of the white-light. The remaining optical system acts to superimpose the images of the two input objects in the output plane with a X/2 phase shift between the two wavefronts. This condition is established for each spectral component of the white-light so each wavelength component will destructively interfere to subtract. The resulting subtracted color image consists of a superposition of the subtracted intensities of each wavelength. We will analyze the broadband subtraction system in a fashion which parallels the description given in the preceding paragraph. As explained by Cartwright [4] , the output of a broadband optical system may be determined by analyzing the system for a general wavelength X. The result for broadband illumination can then be obtained by integrating the output intensity over all wavelength components of the white-light. 2.1. Illumination of the source-encoding mask We begin our analysis of the broadband subtraction system by determining the mutual intensity of the light illuminating the sourceencoding mask. This may be accomplished by using the theory for the propagation of the mutual intensity [5]. IfJ~(x2.y2,x~.~~;) is the mutual intensity in plane P2 caused by the mutual intensityJL(xl ,yl ,x; ,v;) in plane Pl, then these two quantities are related by
X
K*(x;,y;,x;,y;)
dx, dy, dx; d$).
(1)
where K is the amplitude impulse response between the two planes, Note the subscript of J denotes that we are considering a single wavelength component X. To determine the mutual intensity from eq. (I), we tnust know the mutual intensity of the light source and the impulse response between the planes of the light source and the source-encoding mask. For ease of analysis we assume that the white-light source has a rectangular shape with equal intensity over its entire area. In this case, the mutual intensity at wavelength h of the light source is 336
J^b. P. cy’.P’, = C’(h) rect(a/a)
rect(fl/b) 6(cu--(Y’, /L $).
(2)
where C(h) is the intensity of the wavelength component h. the rectangular functions define the spatial limits of the source, and the Dirac delta function is a convenient way to express the incoherence of a ther-ma1 light source. For convenience. we assume C(h) is constant for all X and ignore it and all multiplicative constants in our analysis. The front. off-axis portion of system acts to illuminate the source-encoding mask with a wavelength smeared image of the light source. Assuming lenses ot infinite extent and I OO%,diffraction into the 1 order of the dispersion gr-ating. we may write the amplitude impulse response of this system as K(cu. p,x._v, = 6(x + o( +fiv(,._v
+ P).
(: 1
where f‘is the focal length of the lense. h is the wavelength of light, and vu is the spatial frequency of the dispersion grating. The factor fhvO is the translation due to the dispersion grating. Substituting eq. (2) and eq. (3) into eq. (1) and per-forming the integration, we obtain the mutual intensity illuminating the sourceencoding mask as J,(x.y,x’,y’) X rect(y/b)
= rect[(x +fxvo)/al 6(x ~ x’._v ~- .v’).
(4)
For large values off and vu, the wavelengths of light will be dispersed over a large distance. If the spatial extent of the light source is small in the xdirection, we obtain a high spectral purity and may effectively operate upon the different wavelength components with the source-encoding mask. Before proceeding to the next part of the analysis. we need to transform eq. (4) into the coordinate system of the sourceencoding mask. The optical axis ot the system following the sourceencoding mask lies along the principle ray of the mean wavelength x of the white-light. This means that the coordinate is translated by five in the xdirection. This translation can be taken into account by making the coordinate transform xO
=x+f~v(),
yo=y.
The mutual intensity
in the new coordinate
system is
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Jk(xo,yo,x~,y~~l = rectKxo + 0 X rect(yolb) “(x0 -
Qf~,)/al
xb,Y, - Yb>.
(5)
1 October
1986
where there are 2N t 1 slits, T is the distance between the slits at x,, = 0, and S is the width of the slits. The mutual intensity after the source encoding mask is
2.2. Determination of the mutual intensity in the input plane
J&Jo&$
To determine the mutual intensity in the input plane, we will again use eq. (1). The transfer function needed in this case is the Fourier kernal because of the Fourier transform relation between the source encoding and input planes. The mutual intensity of the light immediately after the source-encoding mask needs to be obtained. To determine the mutual intensity of the light immediately after the source-encoding mask, we need the transmittance function of the source-encoding mask. The source-encoding mask consists of a series of narrow slits arranged in a fan-shaped fashion as shown in fig. 2. We may express the transmittance of the source-encoding mask as
where the plus sign indicates the mutual intensity after passing through the optical mask and * denotes the complex conjugate. Substituting eq. (5) and eq. (6) into eq. (7), we arrive at
N
M(xO,y,)=
n=-lY C
-
rect
nT(1 + xO/vO~f) s
17(6)
=J,(xOYO’xb,yb)M(xo,vo)M*(xb,rb),
J’,(xO,~o,x~,~~)
(7)
= rectKxO + (A - QfvJal
X 6(x() - xb,yo - yb) x
,$ -N
+x,Jifv,,)
- nT(1 ret
(8)
s
where we have assumed that the spatial extent of the light source in they-direction (i.e. b) is sufficient to span the 2N t 1 slits. We are now in a position to solve for the mutual intensity in the input plane. Substituting K(xo ,yo, x 1, VI> = ew[-(ik/f)(xgl +yoy1)1 and eq. (8) into eq. (1) and solving, we obtain the mutual intensity in the input plane as JA(x1,y1,x;,~;)=exp[i2nvo(l
X sinc[ka(xl -x;)/2f]
-VW,
sinc[kS(yl
-+>I -y;)/2f]
N X _C, ev-inWyl
t
-
v;Yfl
,
(9)
where sine(x) = sin(x)/x and k = 27r/j;. Because the width of the slits is very small, the sine term containing the S parameter is approximately 1. We may also rewrite the sum of exponential terms in eq. (9) using the relation
5 exp(ix)=Sin[(2N+ 1)X/21 -N
Fig. 2. The source encoding mask of the color subtraction system. S, width of the slits; T, period of the slits for the mean wavelength.
sin(x/2)
Thus the mutual intensity expressed as
. in the input plane may be
337
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COMMUNICATIONS
I octobcl 1986
X/h)xl]
E,(xI,y,)=exp[i27m0(l
‘:‘*(X*,Y*) = F[(k/f)(x*
-- (h
X,fv,).
jk/j’)_Y*J , (15)
J~WJ’,Y’) .(ll) Y(X,Y,X’,Y’) = [J(x,Y,X,Y)J(X’,Y’,X’:Y’)l l’* Using eqs. (10) and (1 l), we see that the light in the input plane is coherent when x1 - xi = 0 and nrr for n = 0, 1, X.... To establish a WY1 ~ ri)/2f= coherence of 1 between the points on the two input objects that overlap in the input plane, we let Yr - Y; = 2H, where 2H is the separation between the two input objects in the Ydirection. This leads to the condition for the spacing of the slits on the sourceencoding mask as
2.3. Determination
(14)
where the linear phase term is taken from eq. (10) to express the position of the light source of wavelength h in the source encoding plane. The corresponding complex amplitude in the spatial filtering plane is
where M = 2N + 1. The coherence is related to the mutual intensity through the relationship [5]
T = fj;/2H.
f(xl..l’l),
where t; denotes the Fourier transform off’(x.4’). The spatial filter used in the Fourier plane is a fan-shaped sine grating whose transmittance is given by
G(x*,-‘.?)=+1+ _ [ sm(,X2:~:,.J
(10)
Multiplying eq. (16) by eq. (15) and performing an inverse Fourier transform, we arrive at an expression fol the complex amplitude in the output plane as
(12) of the output intensity
The final task in our analysis is to determine the intensity in the output plane of the broadband subtraction system. To do this, we use eq. (1); with the two points in the output plane (i.e. (x3,Y3) and (xi, y;)) set equal. The mutual intensity in the input plane after the input transparencies can be written as
X exp[(Wf)(x,x3 I . +y2y3)1 dx, Q7. Lettingp = (k/f)[x2 eq. (17) reduces to
~~(A
(17)
x)f’uo] andq = (k/j‘)~)~.
J~(xl,Yl.x;,Y;)=J,(xl,Yl,x;,Y;) X [O,(X,>Y,
-H)
+ O,(X,.Y,
X [j + sin (I +$,rV,,)I
+H)l
expIi(px3
+qY3 )I d/l dy. (18)
X Q(x;,Y;
+H)+O,(xr,yr
+H)l*>
(13)
where Jh is given in eq. (10) and 0, and 0, are the transmittances of the two input objects. The amplitude impulse response between the input and output planes needs to be derived. Because it is a difficult task to determine the impulse response between the input and output planes, we determine the transfer function for low spatial frequency input signals. This will adequately serve our purpose if 0, and 0, are low spatial frequency signals, Given a low spatial frequency signal f(x, y), we may write the complex amplitude in the input plane as 338
We may expand the sine term in eq. ( 18) in a Taylors expansion to obtain sin(l
+$nVO)=sin(H~)+~cos(H~),
(19)
where only the first two terms were retained since f(x, y) is a low frequency signal sop & 2nv0. Substituting eq. (19) into eq. (18) and solving the equation. we obtain
OPTICS COMMUNICATIONS
Volume 59, number 5,6
as the spatial filter. These distortions are an inate problem of this color subtraction system and is the price we pay to extend the source encoded subtraction system to operate under broadband illumination. To determine the effect of the distortions on the output subtracted image, we pick the two input images to be identical. Substituting 0, (x, y) = 02(x,y) into eq. (22) we obtain
X)x31
E,(x3,Y3) = expW@-
x U(x, r>* P&3 +Y,) + (l/W(x3,y3
+W - (l/2i)W3,Y3 - H)
x P(XpY3 +w+wgJJ3
-w111>
(20)
where the low spatial frequency consideration is taken into account by f(x, y) and the derivative arises from the pq term which multiplies the cosine term in eq. (19). Because we are concerned with the output intensity, we may ignore the linear phase term in eq. (20) and write the transfer function between the input and output plane as W,>.Y,,xSLPg)=W~
-x321
1 October 1986
10
= (H12nv,)2i2(a2/axay)01(x,y)i2 + [W2 - 1)112$
i2wx)01kY)i2.
(23)
Ideally the output should be zero; however, from eq. (23) we see that there is significant intensities where the transmittance changes rapidly. This is indicated by the derivatives. This distortion may be minimized by choosing vu as large as possible and by limiting the number of source-encoding slits (i.e. N).
-v,)
3. Experimental results -
U/2i)W1
- x3,y1
- y3 +W
+ vwq(a2iax3a~,)
x
W-X,J,
-Y3-H)+G(xl-X3’Y1-Y3+H)I (21)
We may now solve for the output intensity of the subtraction system. Substituting eq. (13) and eq. (21) into eq. (1) with x3 = xi and y3 we obtain the output intensity for wavelength h as
=y;,
I,(x3,.Y3)= + (Hln~)
IO, - 0212 Im[(OT - o”,)(a2/a~,ay3)(o,
+ [(N* - iw$l(apx,w,
+ o,)l
+02)i2,(22)
where the (x3,y3) parameter was dropped to avoid cluttering the equation, Im stands for the imaginary part, and only the terms which correspond to the subtracted result were retained. From eq. (22) we can see that the output result is independent of wavelength. The first term in eq. (22) corresponds to the desired subtracted result. The other terms in the equation are distortions caused by the fan-shaped sine-grating used
An experimental demonstration of the color subtraction system was performed to verify its operation and investigate the distortions indicated in eqs. (22) and (23). In our experiment a xenon-arc-lamp was used as the white-light source. All the lenses used in the system where the same with focal length of 400 nm. The spatial frequency of the dispersion grating was vu = 120 lines/mm. The source-encoding mask was designed to pass wavelength ranging from 4500 A to 6500 A. The average period of the slits on the encoding mask was T = 22 /.un, the width of the slits was s = 4 pm, and the number of slits was 2N + 1 = 19. The separation of the two input objects from the optical axis in the input plane was H = 5 mm. In one experiment, a pair of black-and-white binary transparencies was used as the input objects. The two input objects and the subtracted result are shown in fig. 3. The diameter of the circle is approximately 1 mm. As predicted by eq. (23) the edges of the circle did not subtract well. The broad lines at the top and bottom of the subtracted circle are due to the second term in eq. (23). In another experiment, color transparencies were used for input objects. The two input objects and the subtracted result are given in fig. 4. In the subtracted result we see the F-l 6 as expected. Also present in the 339
I5.g. 3. Binary
input
I&. 4. Color input
objects
objects
(a. b) and the subtracted
(a, b) and the subtracted
result
(c,.
result (c).
Volume
59, number
5,6
OPTICS COMMUNICATIONS
subtracted image are bright areas along the mountain tops, the edges of the sand dune and along the shore line. The angular dependence of the first term in eq. (23) corresponds well with the effect seen along the shore line. That is, the first term in eq. (23) is most significant for edges oriented at an angle of 545’ from the x-axis.
4. Conclusion This paper introduced a sourceencoded subtraction system which operates under broadband illumination. Our analysis indicated that spectral dependent sampling can establish the point-pair coherence needed for the subtraction operation. Our analysis also indicated an inate distortion in the output of the subtraction system. This distortion is the price that is paid to extend the sourceencoded subtraction system to operate under white-light. This distortion can be minimized by limiting the number of sourceencoding slits and by using a dispersion grating with a large spatial
1 October
1986
frequency. Experimental results were presented which correspond well with the results of our analysis and demonstrate the viability of the system.
Acknowledgement We acknowledge the support of U.S. Air Force Rome Air Development Center, Hanscom Air Force Base, under contract no. F19628-84-K-0031.
References [l] ST. Wu and F.T.S. Yu, Optics Lett. 6 (1981) 452. [2] S.T. Wu and F.T.S. Yu, Appl. Optics 20 (1981) 4082. [3] S.H. Lee, S.K. Yao and A.G. Milnes, J. Opt. Sot. Am. 60 (1970) 1037. [4] S. Cartwright, Appl. Optics 23 (1984) 318. [5] M. Born and E. Wolf, Principles of optics (6th Ed., Pergamon Press, New York, 19’80). [6] F.T.S. Yu, Optical information processing (Wiley, New York, 1983).
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