- Email: [email protected]
Detection of the difference between two images: An improved method
September 1972
OPTICS COMMUNICATIONS
Volume 6, number 1
DETECTION
OF THE DIFFERENCE AN IMPROVED
BETWEEN TWO IMAGES: METHOD
S.DEBRUS, M.FRANCON and C.P.GROVER* Institut d’optique,
Universitide
Paris VI, 75005 Paris, France
Received 27 June 1972
The method for the detection of the difference between two images, proposed by the authors, can be modified to improve the signal/noise ratio. An amplitude distribution proportional to COS~LQ is obtained in the Fourier plane using three exposures in the ratio 1 : 2 : 1. In the neighbourhood of q = n/2 the gradient of this distribution is one order of magnitude inferior to that of cosq obtained with two equal exposures. Consequently a wider slit can be used as a filter. A further improvement can be obtained if the number of exposures is increased.
In a recent communication [l] we have described a new method for the detection of the difference between two images using a random diffuser. The problem consists of comparing two photographs A and B(=A+b) in order to determine the difference b = B -A. Both A and B are modulated by the same diffuser D in such a way that the diffuser occupies the same position with respect to identical regions in A and B. For modulation, the image of A (or B) is formed on the diffuser. A photographic plate, kept immediately behind the diffuser, is exposed successively with A and B; the diffuser remains undisturbed during the exposures. On the photographic plate an exact superposition of the identical regions in A and B is absolutely necessary. A, B and b represent the irradiances and D is the transmittance of the diffuser. The total irradiance recorded in the two exposures is: NP(5)
+&S-C&S-Co)
(1)
XD@VMO),
because a translation
is equivalent
A X D@ P(0
+W--&,)I + b XD@W-fo),
(2)
which can be rewritten as
+ b X D@'s({-&,)
.
(3)
After development under the usual conditions of linearity, the negative H is illuminated by a parallel beam of light (fig. 1). The observation is made either in the focal plane of the objective 0, or in the plane H’ conjugate to H. The amplitude transmittance of the negative H is proportional to the irradiance recorded. In the focal plane the spectrum can be writ-
_zjlr+$qF.-
to a convolution
* On leave from the National Physical Laboratory of India, New Delhi-l 2, India.
(1)
.
This can be rewritten as A X DCS(S)+B
by a delta function. The direction of translation has been chosen parallel to the f axis. We have from eq.
Fig. 1. Optical arrangement for the reconstruction ference between two photographs.
of the dif-
15
Volume
6, number
1
OPTICS
COMMUNICATIONS
The irradiance,
ten as
September
1972
now, recorded is
(A+b) X DC3 6(C)
(5)
where symbol 2 denotes the Fourier transform of X, cp= w{,,/X and v is the angle which characterises the position of an arbitrary point M in the focal plane of 01. A slit is placed in the focal plane of 0, in such a manner that it coincides with a zero of cos cp (Young’s fringes). The first term of eq. (4) is stopped and in the image H’ we see only b X D, i.e., the difference between A and B. The undesired modulation by the diffuser D can be tolerated if the diffuser contains high spatial frequencies. As the slope of cos q for cp= 7r/2 is non-zero (curve 1, fig. 2), the slit used as filter has to be fine. This limitation on the slit can be waved by using, instead of two, three successive exposures as shown in table 1 . Amphtudc
t 1
-9
0
+bXD@W’),
(6)
and the spectrum in the focal plane of 01 is (7)
2‘@Ocos2cp+b@E.
(8)
In the amplitude the modulating function cos2cp is shown by curve (2) in fig. 2. The improvement achieved is apparent. In the neighbourhood of cp= rr/2, the variation of cos2 cpis one order smaller than that of coscp. Thus it is possible to use a comparatively wide slit as filter. Consequently, there is an improvement in the signal/noise ratio. A further improvement can be obtained if the number of exposures is increased. The exposure time and the amount of translation are calculated to obtain a factor cf modulation proportional to even powers of cos cp.If (Ntl) is the number of exposures (N is an even number), e.q. (8) becomes 2Nii@ihos’\iq+C$‘2b@D,
Fig. 2. Amplitude
Number of exposure
variation
Photograph
in the focal plane of the objective 01.
where C{l’ are the binomial coefficients. As an example, for 5 and 7 exposures eq. (9) is given respectively by
Table 1
16A”@i? cos4q + 62;@5
Translation
0
Time of exposure
1
B=A+b
2
A
+50
1
3
A
-to
1
16
(9)
and
2
The corresponding translations times are given in table 2.
and the exposure
Volume 6, number 1
OPTICS COMMUNICATIONS Table 2
Total number of exposures
Photograph
5
B=A+b
7
Translation
Time of exposure
0
6
A
f Po
4
A
* 2co
1
B=A+b
A A
0
20
September 1972
Curve (3) of fig. 2 shows the case with N = 6. The slit can be replaced by a binary grid of suitable frequency to give better resolution. A similar method of unequal exposures with equal displacements was suggested by Burch and Tokarski (2) to suppress the secondary maxima in the production of multiple beam fringes from a photographic scatterer.
References
[l] S.Debrus, M.Frangon and C.P.Grover, Opt. Commun. 4 (1971) 172.
[ 21 J.M.Burch and J.M.J.Tokarski, Opt. Acta 15 (1968) 101.
17
OPTICS COMMUNICATIONS
Volume 6, number 1
DETECTION
OF THE DIFFERENCE AN IMPROVED
BETWEEN TWO IMAGES: METHOD
S.DEBRUS, M.FRANCON and C.P.GROVER* Institut d’optique,
Universitide
Paris VI, 75005 Paris, France
Received 27 June 1972
The method for the detection of the difference between two images, proposed by the authors, can be modified to improve the signal/noise ratio. An amplitude distribution proportional to COS~LQ is obtained in the Fourier plane using three exposures in the ratio 1 : 2 : 1. In the neighbourhood of q = n/2 the gradient of this distribution is one order of magnitude inferior to that of cosq obtained with two equal exposures. Consequently a wider slit can be used as a filter. A further improvement can be obtained if the number of exposures is increased.
In a recent communication [l] we have described a new method for the detection of the difference between two images using a random diffuser. The problem consists of comparing two photographs A and B(=A+b) in order to determine the difference b = B -A. Both A and B are modulated by the same diffuser D in such a way that the diffuser occupies the same position with respect to identical regions in A and B. For modulation, the image of A (or B) is formed on the diffuser. A photographic plate, kept immediately behind the diffuser, is exposed successively with A and B; the diffuser remains undisturbed during the exposures. On the photographic plate an exact superposition of the identical regions in A and B is absolutely necessary. A, B and b represent the irradiances and D is the transmittance of the diffuser. The total irradiance recorded in the two exposures is: NP(5)
+&S-C&S-Co)
(1)
XD@VMO),
because a translation
is equivalent
A X D@ P(0
+W--&,)I + b XD@W-fo),
(2)
which can be rewritten as
+ b X D@'s({-&,)
.
(3)
After development under the usual conditions of linearity, the negative H is illuminated by a parallel beam of light (fig. 1). The observation is made either in the focal plane of the objective 0, or in the plane H’ conjugate to H. The amplitude transmittance of the negative H is proportional to the irradiance recorded. In the focal plane the spectrum can be writ-
_zjlr+$qF.-
to a convolution
* On leave from the National Physical Laboratory of India, New Delhi-l 2, India.
(1)
.
This can be rewritten as A X DCS(S)+B
by a delta function. The direction of translation has been chosen parallel to the f axis. We have from eq.
Fig. 1. Optical arrangement for the reconstruction ference between two photographs.
of the dif-
15
Volume
6, number
1
OPTICS
COMMUNICATIONS
The irradiance,
ten as
September
1972
now, recorded is
(A+b) X DC3 6(C)
(5)
where symbol 2 denotes the Fourier transform of X, cp= w{,,/X and v is the angle which characterises the position of an arbitrary point M in the focal plane of 01. A slit is placed in the focal plane of 0, in such a manner that it coincides with a zero of cos cp (Young’s fringes). The first term of eq. (4) is stopped and in the image H’ we see only b X D, i.e., the difference between A and B. The undesired modulation by the diffuser D can be tolerated if the diffuser contains high spatial frequencies. As the slope of cos q for cp= 7r/2 is non-zero (curve 1, fig. 2), the slit used as filter has to be fine. This limitation on the slit can be waved by using, instead of two, three successive exposures as shown in table 1 . Amphtudc
t 1
-9
0
+bXD@W’),
(6)
and the spectrum in the focal plane of 01 is (7)
2‘@Ocos2cp+b@E.
(8)
In the amplitude the modulating function cos2cp is shown by curve (2) in fig. 2. The improvement achieved is apparent. In the neighbourhood of cp= rr/2, the variation of cos2 cpis one order smaller than that of coscp. Thus it is possible to use a comparatively wide slit as filter. Consequently, there is an improvement in the signal/noise ratio. A further improvement can be obtained if the number of exposures is increased. The exposure time and the amount of translation are calculated to obtain a factor cf modulation proportional to even powers of cos cp.If (Ntl) is the number of exposures (N is an even number), e.q. (8) becomes 2Nii@ihos’\iq+C$‘2b@D,
Fig. 2. Amplitude
Number of exposure
variation
Photograph
in the focal plane of the objective 01.
where C{l’ are the binomial coefficients. As an example, for 5 and 7 exposures eq. (9) is given respectively by
Table 1
16A”@i? cos4q + 62;@5
Translation
0
Time of exposure
1
B=A+b
2
A
+50
1
3
A
-to
1
16
(9)
and
2
The corresponding translations times are given in table 2.
and the exposure
Volume 6, number 1
OPTICS COMMUNICATIONS Table 2
Total number of exposures
Photograph
5
B=A+b
7
Translation
Time of exposure
0
6
A
f Po
4
A
* 2co
1
B=A+b
A A
0
20
September 1972
Curve (3) of fig. 2 shows the case with N = 6. The slit can be replaced by a binary grid of suitable frequency to give better resolution. A similar method of unequal exposures with equal displacements was suggested by Burch and Tokarski (2) to suppress the secondary maxima in the production of multiple beam fringes from a photographic scatterer.
References
[l] S.Debrus, M.Frangon and C.P.Grover, Opt. Commun. 4 (1971) 172.
[ 21 J.M.Burch and J.M.J.Tokarski, Opt. Acta 15 (1968) 101.
17