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Interpolation method of computer-generated filters for large object formats
Volume 23, number 3
OPTICS COMMUNICATIONS
December 1977
INTERPOLATION METHOD OF COMPUTER-GENERATED FILTERS FOR LARGE OBJECT FORMATS Toyohiko Y A T A G A I The Institute of Physical and Chemical Research, 2-1, Hirosawa, Wako-Oty, Saitama 351, Japan Received 20 September 1977
An improved coding scheme of computer-generated spatial filters is described. A normal computer-generated hologram (CGH) is interpolated and new sample points are positioned between old ones, so that a hologram with a high carrier frequency is obtained. An effectual technique of interpolation is mentioned in terms of the discrete Fourier transform. When making a comparison between the present method and the Bartelt method, no false images in the Bartelt method can be observed. The experimental reconstruction of the CGH obtained by interpolating 3 X 3 times a hologram of 32 X 32 sample points verifies the theory.
1. Introduction One o f the most important applications o f computer-generated holograms (CGH) [1,2] is in optical data processing. They enable us to make a lot o f sophisticated filters, such as image enhancement filters [3], a matched filter [4], a Mellin transform filter [5] and so on. There exists, however, a serious problem that the input pattern processed b y the CGH filter is limited in size mainly because o f a low carrier frequency of the filter. As is well known, the desired output image is the cross-correlation o f an input pattern and the impulse response of a filter, and the extent o f the correlation is the sum o f the extents o f the input pattern and the impulse response o f the filter. If the carrier frequency is larger than the bandwidth of the correlation, the first diffraction order o f the desired correlation can be separated from the zero order corresponding to the undistorted image o f the input pattern and the other diffraction orders. This means that the carrier frequency o f the CGH filter is not decided only from the extent of the impulse response but also from that o f the input pattern. Thus, the large format o f the input pattern requires a large number o f sample points of the CGH filter to avoid undesired overlapping o f various diffraction orders. Lowenthal et al. have shown that the holographi-
cally copied spatial filter with a higher carrier frequency can be used for much larger patterns [6]. Recently Bartelt et al. have described a two-step method for producing CGH filters with high carrier frequencies [7]. In the first step a non-redundant hologram is plotted which reconstructs a desired image in the Mth order of diffraction. In the second step, M-time interlace is performed b y a step-and-repeat camera to suppress false images which appear between the zero and the Mth diffraction orders. The present paper describes an interpolation method of making CGH filters with high spatial carrier frequencies, which allows us to calculate a hologram having the effectively large number o f resolution cells b y using the data o f a hologram with a small number of sample points.
2. Background Before describing how we can synthesize a hologram with a high carrier frequency, we consider some properties of the discrete Fourier transform (DFT). For simplicity, calculations in this section are carried out in one dimension. The extension to the twodimensional situation is trivial because o f the separability of the DFT. We define here the DFT of a sequence o f N samples, f ( p A ) , 0 ~< p ~
Volume 23, number 3
OPTICS COMMUNICATIONS
N-1
F(n~2) = ~
f(pA) exp[--inp UZA]
(1)
p:0
where A denotes the sampling interval, and ~2 : 27r/(NA). With this specification of [2, there are only N distinct and independent values computable by eq. (1), namely, those for n = 0. t, ...,N - 1. If the sampling period ~ in the Fourier domain is divided into K subperiods, then the new period will be co = ~2/K. Since the sequence F(n[2) provides samples of the desired sequence only at intervals of K samples at the new sampling rate, the remaining samples must be filled in by interpolation [8]. For practical convenience of fabricating the CGH, we define here the kth subsequence made up of the kth sample in each period of [2. To see how this can be ,done, consider the discrete form of shifting property of the Fourier transform: N
1
F(n~2 + kco) = ~ ] f ( p A ) exp[--ikpcoA] exp[-inp~2A] p =0
( k : 0 , 1 ....
December 1977
quence ofN(K - 1) zero-valued samples are obtained. This procedure allows arbitrary high carrier frequency in synthesizing a CGH, as mentioned in the next section. It should be noted that the reversal of this procedure is well known as an alternative method of interpolation in the DFT [9]. On the other hand, Bartelt's hologram corresponds to making K-time interlace of N-sample sequence F(n~). The inverse DFT in this case is given by N-I K-1
1 ~ k:O ~ F(n~)exp[i(n~ + kco)rA] fB(rA)=N-Kn:o 1
N - I K - I N-1
= ~ ~ ~ f(pA) exp[--inp~2A] NKn=O k=0 p=0 X exp [i(n~ + kco)rA] :/(Jr] N A). w(r)
(5)
where [r]N =r modulo N, and
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f(PA)
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1
if@A) = - - ~ ~[] F(n[2+kco) exp [i(n~ + kco)rA] NK n=0 k=0 ( r = 0 , 1 .... , N K - 1). (3) Substituting eq. (2) into eq. (3), we have
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,0:
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(4)
Thus from the interpolated sequence F(ng + kco) the desired sequence f(rA), 0 K r K N - 1, and also a se348
O
o - ~ -15A 0 16A Fig.i.Interpolationmethod in one-dimensionalcase(N = 8,
× exp [--inp~A] exp [i(n~ + kco)rA] =
]
K = 4). (a) Original sequence f ( p A ) , 0 <~ p ~< 8, (b) its DFT F(n~2), (c) sequence in interpolation (o: first interpolated sequence, ~: second interpolated sequence), (d) interpolated sequence F(n~2 + kw), 0 ~< n ~< 8, 0 <~ k ~< 4, and (e) final sequence f ' ( r A ) , 0 <~ r <~ 32, which is given by the inverse DFT o f (d).
OPTICS COMMUNICATIONS
Volume 23, number 3 1 l -- exp(2zrir/N) w(r) = K 1 - exp(2zrir/NK)
(6)
The result is the product of the sequence f([r]NA ) which is the periodic extension o f f ( p A ) and the weighting factor w(r) which supresses the undesired sequences. To summarize the explanation done so far, a concrete numerical example (N = 8, K = 4) is given. The original sequence f(pA)is defined within the region: - N / 2 + 1 <~p <~N/2 as in fig. l(a). Its DFT and the interpolation process is shown in fig. l(b) and (c)-(d). It is seen from fig. l(e) that the desired sequence with a large number of zero-valued samples is obtained by the interpolation method. Fig. 2 shows the Bartelt method; (a): a sequence made by K-time interlace, (b): the periodic extension of the original sequence, (c): the weighting factor, and (d): the final sequence which is given by the product of (c) and (d). This figure clearly shows the effect of the weighting factor, but a large effect of supression is not expected for the sequence with a small zerovalued region. In the worst case when the boundary
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December 1977
sample f ( N / 2 • A) of the original sequence is not zerovalued, the weighting factor is Iw(N/2)l = [K sin Tr/2K] -1. For large K we have [w(N/2)l ~ 2/rr, and so the noise power of the Bartelt method is 4/7r2 0.41 in the worst case. For making the tail of the weighting factor w(r) even smaller a way has been proposed, which is carried out by doing more than K repetition steps [7].
3. Synthesis procedure and experimental verification Consider a Fourier transform CGH here. The hologram synthesized by the interpolation method consists o f N × N resolution cells. Each of resolution cells is divided into K X K subcells. We define the (k,/)th subhologram as a hologram made up of (k,/)th subcells. The first step in the interpolation method is to plot the (1,1)th subhologram which is calculated with the data of a normal CGH o f N X N sample points. Since we know the amplitude and the phase only at the interval of the resolution cells, the remaining subholograms must be calculated by the interpolation algorithm, such as eq. (2). The (k, l)th subhologram calculated is plotted in the place of the (k, l)th subcells. These procedures are repeated until all the subholograms are obtained. Figs. 3 and 4 show an experimental verification of the interpolation method. A letter "E" that contained 32 × 32 sample points served as a test object. All the holograms fabricated were the Lohmann type CGH. Fig. 3(a) is the CGH of 32 X 32 sample points without interpolation and fig. 3(b) the hologram made by 3 X 3 interlace of the hologram of (a). In fig. 4, we find reconstructed images (a) and (b) corresponding to the hologram of fig. 3(a) and (b), respectively. Comparison of the reconstructed image in fig. 4(a) with (b) shows the same image size, but the distance between the image and the zero order diffraction spot in fig. 4(b) is three times as long as (a).
If (rA) I 4. Concluding remarks
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Fig. 2. Sequence in the Barter method. (a) Interlaced sequence of fig. l(b), (b) periodic extension of fig. l(a), (c) weighting factor w(r), and (d) final sequencef'B(rA) which is given by the product of (b) and (c).
We have presented a new technique for the synthesis of a CGH with a high carrier frequency. Using interpolation which is easily performed with the DFT, a hologram made up of a large number of resolution 349
Volume 23, number 3
OPTICS COMMUNICATIONS
December 1977
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Fig. 3. Lohmann type CGHs. (a) Hologram of 32 X 32 sample points without interpolation, and (b) its interpolated hologram of 96 X 96 sample points.
3
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Fig. 4. (a) Reconstructed image of fig. 3(a), and (b) reconstructed image of fig. 3(b) 350
cells can be synthesized. The experimental reconstruction of the CGH obtained by interpolating 3 × 3 times a hologram of 32 × 32 sample points was successfully performed. The interpolation method mentioned here does not reduce the computing and plotting time in comparison with the Lowenthal and the Vartelt method. The main advantage of the interpolation method is that a sufficiently large size hologram of N K × ArK sample points is synthesized by K × K subholograms which are successively calculated from the data array o f N × N sample points and also successively plotted. This means that the required data size in a computer memory is only N × N to make a hologram o f N K × N K resolution cells. In a slightly different sense, we can make interpolation of the reconstructed image from the CGH. The outline of the method is that an original hologram o f N × N sample points is periodically extended K × K times and then the extended hologram is multiplyed by a weighting function such as eq. (6) to get a final hologram which reconstructs the image interpolated at K × K times as many points. The detailed discussions and experimental results will be given in a report which is in preparation. In the present paper we implicitly assumed that the number K of subperiods within the resolution cell is
Volume 23, number 3
OPTICS COMMUNICATIONS
integer, or that ~2 is divisible by w, but this assumption is not necessary [10]. Thus, the size of the hologram need not be integer multiples of the original data size of the object, but the carrier frequenc2¢ of the hologram can be controlled arbitrarily. The author would like to thank H. Saito for his encouragement of this work and for many helpful discussions.
References
December 1977
[2] [3] [4] [5] [6] [7]
A.W. Lohmann and D.P. Paris, Appl. Opt. 6 (1967) 1739. A.W. Lohmann and D.P. Paris, Appl. Opt. 7 (1968) 651. A. Kozma and D.L. Kelly, 4 (1965) 387. D. Casasent and D. Psalts, Opt. Commun. 19 (1976) 217. S. Lowenthal and P. Chavel, Appl. Opt. 13 (1974) 718. H. Bartelt, W.J. Dallas and A.W. Lohmann, Opt. Commun. 20 (1977) 50. [8] P. Chavel and J.-P. Hugonin, J. Opt. Soc. Amer. 66 (1976) 989. [9] L.R. Rabiner and B. Gold, Theory and application of digital signal processing (Prentice-Hall, Inc. Englewood Cliffs, 1975) p. 50. [10] R.W. Schafer and L.R. Rabiner, Proc. IEEE 61 (1973) 692.
[1] B.R. Brown and A.W. Lohmann, Appl. Opt. 5 (1966) 967.
351
OPTICS COMMUNICATIONS
December 1977
INTERPOLATION METHOD OF COMPUTER-GENERATED FILTERS FOR LARGE OBJECT FORMATS Toyohiko Y A T A G A I The Institute of Physical and Chemical Research, 2-1, Hirosawa, Wako-Oty, Saitama 351, Japan Received 20 September 1977
An improved coding scheme of computer-generated spatial filters is described. A normal computer-generated hologram (CGH) is interpolated and new sample points are positioned between old ones, so that a hologram with a high carrier frequency is obtained. An effectual technique of interpolation is mentioned in terms of the discrete Fourier transform. When making a comparison between the present method and the Bartelt method, no false images in the Bartelt method can be observed. The experimental reconstruction of the CGH obtained by interpolating 3 X 3 times a hologram of 32 X 32 sample points verifies the theory.
1. Introduction One o f the most important applications o f computer-generated holograms (CGH) [1,2] is in optical data processing. They enable us to make a lot o f sophisticated filters, such as image enhancement filters [3], a matched filter [4], a Mellin transform filter [5] and so on. There exists, however, a serious problem that the input pattern processed b y the CGH filter is limited in size mainly because o f a low carrier frequency of the filter. As is well known, the desired output image is the cross-correlation o f an input pattern and the impulse response of a filter, and the extent o f the correlation is the sum o f the extents o f the input pattern and the impulse response o f the filter. If the carrier frequency is larger than the bandwidth of the correlation, the first diffraction order o f the desired correlation can be separated from the zero order corresponding to the undistorted image o f the input pattern and the other diffraction orders. This means that the carrier frequency o f the CGH filter is not decided only from the extent of the impulse response but also from that o f the input pattern. Thus, the large format o f the input pattern requires a large number o f sample points of the CGH filter to avoid undesired overlapping o f various diffraction orders. Lowenthal et al. have shown that the holographi-
cally copied spatial filter with a higher carrier frequency can be used for much larger patterns [6]. Recently Bartelt et al. have described a two-step method for producing CGH filters with high carrier frequencies [7]. In the first step a non-redundant hologram is plotted which reconstructs a desired image in the Mth order of diffraction. In the second step, M-time interlace is performed b y a step-and-repeat camera to suppress false images which appear between the zero and the Mth diffraction orders. The present paper describes an interpolation method of making CGH filters with high spatial carrier frequencies, which allows us to calculate a hologram having the effectively large number o f resolution cells b y using the data o f a hologram with a small number of sample points.
2. Background Before describing how we can synthesize a hologram with a high carrier frequency, we consider some properties of the discrete Fourier transform (DFT). For simplicity, calculations in this section are carried out in one dimension. The extension to the twodimensional situation is trivial because o f the separability of the DFT. We define here the DFT of a sequence o f N samples, f ( p A ) , 0 ~< p ~
Volume 23, number 3
OPTICS COMMUNICATIONS
N-1
F(n~2) = ~
f(pA) exp[--inp UZA]
(1)
p:0
where A denotes the sampling interval, and ~2 : 27r/(NA). With this specification of [2, there are only N distinct and independent values computable by eq. (1), namely, those for n = 0. t, ...,N - 1. If the sampling period ~ in the Fourier domain is divided into K subperiods, then the new period will be co = ~2/K. Since the sequence F(n[2) provides samples of the desired sequence only at intervals of K samples at the new sampling rate, the remaining samples must be filled in by interpolation [8]. For practical convenience of fabricating the CGH, we define here the kth subsequence made up of the kth sample in each period of [2. To see how this can be ,done, consider the discrete form of shifting property of the Fourier transform: N
1
F(n~2 + kco) = ~ ] f ( p A ) exp[--ikpcoA] exp[-inp~2A] p =0
( k : 0 , 1 ....
December 1977
quence ofN(K - 1) zero-valued samples are obtained. This procedure allows arbitrary high carrier frequency in synthesizing a CGH, as mentioned in the next section. It should be noted that the reversal of this procedure is well known as an alternative method of interpolation in the DFT [9]. On the other hand, Bartelt's hologram corresponds to making K-time interlace of N-sample sequence F(n~). The inverse DFT in this case is given by N-I K-1
1 ~ k:O ~ F(n~)exp[i(n~ + kco)rA] fB(rA)=N-Kn:o 1
N - I K - I N-1
= ~ ~ ~ f(pA) exp[--inp~2A] NKn=O k=0 p=0 X exp [i(n~ + kco)rA] :/(Jr] N A). w(r)
(5)
where [r]N =r modulo N, and
(a)oOlOo
(2)
,K-l).
(b)
f(PA)
The summation in eq. (2) can be interpreted as the N-point DFT of the product of the sequence f ( p A ) and a proper phase factor exp[-ikpcoA]. This means that K subsequences can be calculated with the data f ( p A ) of N samples, so that the sequence of NK samples is obtained by successive interpolation procedure. The inverse DFT of the interpolated sequence f ( n ~ + kw) is considered here, which is given by N-1 K-1
1
if@A) = - - ~ ~[] F(n[2+kco) exp [i(n~ + kco)rA] NK n=0 k=0 ( r = 0 , 1 .... , N K - 1). (3) Substituting eq. (2) into eq. (3), we have
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O~r<~N-1
,0:
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(4)
Thus from the interpolated sequence F(ng + kco) the desired sequence f(rA), 0 K r K N - 1, and also a se348
O
o - ~ -15A 0 16A Fig.i.Interpolationmethod in one-dimensionalcase(N = 8,
× exp [--inp~A] exp [i(n~ + kco)rA] =
]
K = 4). (a) Original sequence f ( p A ) , 0 <~ p ~< 8, (b) its DFT F(n~2), (c) sequence in interpolation (o: first interpolated sequence, ~: second interpolated sequence), (d) interpolated sequence F(n~2 + kw), 0 ~< n ~< 8, 0 <~ k ~< 4, and (e) final sequence f ' ( r A ) , 0 <~ r <~ 32, which is given by the inverse DFT o f (d).
OPTICS COMMUNICATIONS
Volume 23, number 3 1 l -- exp(2zrir/N) w(r) = K 1 - exp(2zrir/NK)
(6)
The result is the product of the sequence f([r]NA ) which is the periodic extension o f f ( p A ) and the weighting factor w(r) which supresses the undesired sequences. To summarize the explanation done so far, a concrete numerical example (N = 8, K = 4) is given. The original sequence f(pA)is defined within the region: - N / 2 + 1 <~p <~N/2 as in fig. l(a). Its DFT and the interpolation process is shown in fig. l(b) and (c)-(d). It is seen from fig. l(e) that the desired sequence with a large number of zero-valued samples is obtained by the interpolation method. Fig. 2 shows the Bartelt method; (a): a sequence made by K-time interlace, (b): the periodic extension of the original sequence, (c): the weighting factor, and (d): the final sequence which is given by the product of (c) and (d). This figure clearly shows the effect of the weighting factor, but a large effect of supression is not expected for the sequence with a small zerovalued region. In the worst case when the boundary
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December 1977
sample f ( N / 2 • A) of the original sequence is not zerovalued, the weighting factor is Iw(N/2)l = [K sin Tr/2K] -1. For large K we have [w(N/2)l ~ 2/rr, and so the noise power of the Bartelt method is 4/7r2 0.41 in the worst case. For making the tail of the weighting factor w(r) even smaller a way has been proposed, which is carried out by doing more than K repetition steps [7].
3. Synthesis procedure and experimental verification Consider a Fourier transform CGH here. The hologram synthesized by the interpolation method consists o f N × N resolution cells. Each of resolution cells is divided into K X K subcells. We define the (k,/)th subhologram as a hologram made up of (k,/)th subcells. The first step in the interpolation method is to plot the (1,1)th subhologram which is calculated with the data of a normal CGH o f N X N sample points. Since we know the amplitude and the phase only at the interval of the resolution cells, the remaining subholograms must be calculated by the interpolation algorithm, such as eq. (2). The (k, l)th subhologram calculated is plotted in the place of the (k, l)th subcells. These procedures are repeated until all the subholograms are obtained. Figs. 3 and 4 show an experimental verification of the interpolation method. A letter "E" that contained 32 × 32 sample points served as a test object. All the holograms fabricated were the Lohmann type CGH. Fig. 3(a) is the CGH of 32 X 32 sample points without interpolation and fig. 3(b) the hologram made by 3 X 3 interlace of the hologram of (a). In fig. 4, we find reconstructed images (a) and (b) corresponding to the hologram of fig. 3(a) and (b), respectively. Comparison of the reconstructed image in fig. 4(a) with (b) shows the same image size, but the distance between the image and the zero order diffraction spot in fig. 4(b) is three times as long as (a).
If (rA) I 4. Concluding remarks
0.0__0__0__00 0 0 0 0 0.0__0__
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Fig. 2. Sequence in the Barter method. (a) Interlaced sequence of fig. l(b), (b) periodic extension of fig. l(a), (c) weighting factor w(r), and (d) final sequencef'B(rA) which is given by the product of (b) and (c).
We have presented a new technique for the synthesis of a CGH with a high carrier frequency. Using interpolation which is easily performed with the DFT, a hologram made up of a large number of resolution 349
Volume 23, number 3
OPTICS COMMUNICATIONS
December 1977
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Fig. 3. Lohmann type CGHs. (a) Hologram of 32 X 32 sample points without interpolation, and (b) its interpolated hologram of 96 X 96 sample points.
3
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Fig. 4. (a) Reconstructed image of fig. 3(a), and (b) reconstructed image of fig. 3(b) 350
cells can be synthesized. The experimental reconstruction of the CGH obtained by interpolating 3 × 3 times a hologram of 32 × 32 sample points was successfully performed. The interpolation method mentioned here does not reduce the computing and plotting time in comparison with the Lowenthal and the Vartelt method. The main advantage of the interpolation method is that a sufficiently large size hologram of N K × ArK sample points is synthesized by K × K subholograms which are successively calculated from the data array o f N × N sample points and also successively plotted. This means that the required data size in a computer memory is only N × N to make a hologram o f N K × N K resolution cells. In a slightly different sense, we can make interpolation of the reconstructed image from the CGH. The outline of the method is that an original hologram o f N × N sample points is periodically extended K × K times and then the extended hologram is multiplyed by a weighting function such as eq. (6) to get a final hologram which reconstructs the image interpolated at K × K times as many points. The detailed discussions and experimental results will be given in a report which is in preparation. In the present paper we implicitly assumed that the number K of subperiods within the resolution cell is
Volume 23, number 3
OPTICS COMMUNICATIONS
integer, or that ~2 is divisible by w, but this assumption is not necessary [10]. Thus, the size of the hologram need not be integer multiples of the original data size of the object, but the carrier frequenc2¢ of the hologram can be controlled arbitrarily. The author would like to thank H. Saito for his encouragement of this work and for many helpful discussions.
References
December 1977
[2] [3] [4] [5] [6] [7]
A.W. Lohmann and D.P. Paris, Appl. Opt. 6 (1967) 1739. A.W. Lohmann and D.P. Paris, Appl. Opt. 7 (1968) 651. A. Kozma and D.L. Kelly, 4 (1965) 387. D. Casasent and D. Psalts, Opt. Commun. 19 (1976) 217. S. Lowenthal and P. Chavel, Appl. Opt. 13 (1974) 718. H. Bartelt, W.J. Dallas and A.W. Lohmann, Opt. Commun. 20 (1977) 50. [8] P. Chavel and J.-P. Hugonin, J. Opt. Soc. Amer. 66 (1976) 989. [9] L.R. Rabiner and B. Gold, Theory and application of digital signal processing (Prentice-Hall, Inc. Englewood Cliffs, 1975) p. 50. [10] R.W. Schafer and L.R. Rabiner, Proc. IEEE 61 (1973) 692.
[1] B.R. Brown and A.W. Lohmann, Appl. Opt. 5 (1966) 967.
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