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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 13 — Jun. 20, 2011
  • pp: 12594–12604
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Parallel lensless optical correlator based on two phase-only spatial light modulators

Xu Zeng, Takashi Inoue, Norihiro Fukuchi, and Jian Bai  »View Author Affiliations


Optics Express, Vol. 19, Issue 13, pp. 12594-12604 (2011)
http://dx.doi.org/10.1364/OE.19.012594


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Abstract

In this paper, we proposed a parallel phase-only lensless optical correlator based on two pieces of Liquid Crystal on Silicon Spatial Light Modulators. Phase Fresnel Lens Array and specialized grating are implemented to realize multi-channel and multiplexed LOC. Experimental results of Chinese characters’ recognitions are given as demonstration of proposed technique. High uniformity of processing channels has been verified by autocorrelation process of four same Chinese characters. The technique is programmable and adjustment of optical path could be realized without changing of optical setup. The implementations could be performed on the same configuration as single channel optical correlator without mechanical alternation.

© 2011 OSA

1. Introduction

2. Principle and configuration

The architecture of parallel LOC for optical processing is illustrated in Fig. 1
Fig. 1 (color online) Basic configuration of multichannel (a) and multiplexed (b) LOC. SLM1 and SLM2 are parallel positioned. Collimated beam propagates between SLM1 and SLM2 along a zig-zag path and the correlation results are observed by a CCD camera.
which is same as that in reference 11

11. N. Fukuchi, T. Inoue, H. Toyoda, and T. Hara, “Lensless Vanderlugt optical correlator using two phase-only spatial light modulators,” Chin. Opt. Lett. 7(12), 1131–1133 (2009). [CrossRef]

. Two reflective type SLMs are parallel arranged. A collimated beam is incident obliquely on SLM1 and reflected to SLM2 along a zigzag path. The input and filter patterns are displayed on SLM1 and SLM2, correspondingly. A camera is placed at the focal plane of the system to record the correlation results. Figure 1(a) illustrates the optical path of a multichannel LOC. The processing channels are spatial arranged on the different positions of SLM. The input beam is divided into several small beams after reflecting from SLM1 and incident on the SLM2 individually. While for a multiplexed LOC, each channel is frequency multiplexed, the incident beam is split after reflecting from SLM1, as shown in Fig. 1(b).

To realize spatial multiplexing in multichannel LOC, a FLA pattern, which are consisted of Fresnel Lens Patterns (FLPs) located on different positions of SLM, are employed to realize parallel correlation calculations between input and reference targets. Figure 2
Fig. 2 Schematic diagram of constructing input pattern of multichannel LOC. The input targets are N × N targets array represented as T(i,j) (i, j = 0, 1, 2 … N). The FLA is consisted of N × N FLPs. The input pattern is created by adding input targets and FLA.
shows the procedures to create the input pattern of multichannel LOC. Detailed descriptions are as follows.

  • 1. Input targets to be identified are arranged into a N×N array pattern, and the elementary unit located are denoted as T(i,j) (i, j = 0, 1, 2 … N). Each T(i,j) is phase encoded, and the total size of T(i,j) should be smaller than liquid crystal size of SLM.
  • 2. A phase type FLA composed of N×N FLPs is programmed to realize multichannel Fourier Transform. The focal length of each FLP is identical to each other and designed by the strategy proposed previously [12

    12. X. Zeng, J. Bai, C. Hou, and G. Yang, “Compact optical correlator based on one phase-only spatial light modulator,” Opt. Lett. 36(8), 1383–1385 (2011). [CrossRef] [PubMed]

    ]. Moreover, the size and center position of each FLP should be same as that of input targets, correspondingly.
  • 3. Add the FLA with the input targets and wrap the phase within 2π.

Filter pattern generation procedures of multichannel LOC are analogous to that of constructing input pattern, and the concept is illustrated in Fig. 3
Fig. 3 Principle of constructing the filter pattern in parallel LOC. The N × N reference target array denoted as R(i,j) (i, j = 0, 1, 2 … N) is phase-encode. Phase-only filter array F(i,j) is created by Fourier transforming of individual reference targets. The filter pattern is attained by adding a FLA with the phase-only filter array.
. The reference targets, represented as R(i,j) (i, j = 0, 1, 2, … N), are constructed and arranged as input targets. Fourier transform is performed on each reference target to form phase-only filter array denoted by F(i,j) (i, j = 0, 1, 2, … N), respectively. After that, another FLA is superimposed on filter array to create filter pattern of multichannel LOC.

In multichannel LOC, the input channel number is limited by the liquid crystal chip size of SLM. In order to increase the input channels, multiplexed LOC is proposed. Compared with multichannel LOC, the input targets are loaded on different spatial frequency carriers to achieve parallel processing. The procedure of generating filter pattern of multiplexed LOC is the same as that of multichannel LOC. Yet the approaches utilized of yielding input pattern are different. Two methods were explored in multiplexed LOC, denoted as Type I and Type II. Type I multiplexed LOC is constructed as follows, which is illustrated in Fig. 4
Fig. 4 Basic procedure to create the Type I input pattern of multiplexed LOC. By adding the input targets Tk with frequency carriers Gk in complex value, the spectrum of input targets could be split onto corresponding channel on SLM2, as illustrated in Fig. 1(b).
.

  • 1. Input targets are phase-encoded and denoted as Tk (k = 0, 1, 2, … N). Also, the size of each Tk should be smaller than that of liquid crystal on SLM.
  • 2. Different frequency carriers Gk (k = 0, 1, 2, … N) are applied to shift the spectrum of input targets to different channels on SLM2. In order to reduce high frequency components, Tk is added with Gk in complex value and the phase image of the superposition results is exported and denoted as Ik (k = 0, 1, 2, … N).
  • 3. Make a summation of the phase image Ik (k = 0, 1, 2, … N) and denoted the phase pattern as It, additionally, a FLP is added with It to obtain the input pattern of multiplexed LOC.

Through this technique, the Fourier Transform of different input targets are loaded on different spatial frequencies and split to different positions of SLM2. The multiplexed LOC could corporate many input targets into one hologram regardless of the liquid crystal size of SLM. But this approach is not suitable to process complex phase targets with several features to be classified. The reason is that the above multiplexed technique needs to add the same input targets with different frequency carriers, which will generate high frequency components in input pattern. Hence, Type II multiplexed LOC technique is introduced. As shown in Fig. 5
Fig. 5 Principle to construct the input pattern of Type II multiplexed LOC, where the input target T is superimposed with a specific grating G in complex value. The phase part I of superimposing image is overlapped with a FLP to form the input pattern.
, the phase target to be recognized is represented as T. Similarly in this case, a specific designed high frequency carrier G, which duplicates the input targets into two dimension array, is added with T in complex value. Likewise, the phase image of the summation results is exported as I and overlapped with a FLP. Compared with Type I multiplexed LOC, this method is more suitable and easier for the input target with sophisticated features to be classified.

For a multichannel LOC, the correlation signal is stronger than multiplexed LOC. On the contrary, the multiplexed LOC could carry more information without the limitation of liquid crystal size of SLM.

The capability decrease on single channel when extending to parallel processing can be theoretically indicated by the Space-Bandwidth Product (SBP). Furthermore, the SBP affects the optical processing performance of LOC. For a SLM, the SBP is expressed as
SBP=HxHyfxfy
(1)
where Hxand Hyrepresent the width and length of liquid crystal of SLM, fxand fy denote the maximum spatial frequencies along width and length directions, respectively. For simplicity, we assumed that Hx = Hy = H.

Based on the Nyquist sampling criteria, the maximum spatial frequencies are determined by the pixel pitch and is given as
fx=1/(2dx)
(2)
fy=1/(2dy)
(3)
where dx and dy are the width and length of one pixel, for our case, we set dx = dy = d and concluded from Eq. (2) and Eq. (3) following relationship is apparent
fx=fy
(4)
for a SLM consisted of M×Mpixels and individual pixel size is d×d, the width H of liquid crystal chip is
H=Md
(5)
substituting Eqs. (2), (3), (4), and (5) into Eq. (1), the SBP of SLM can be rewritten as
SBP=M2/4
(6)
Equation (6) directly shows that in a LOC, the SBP is proportionate to the pixel number, which means, for a specified SLM, when the channel number of a multichannel LOC is increased, the SBP of each channel is decreased and the performance of each processing channel is degraded correspondingly.

In a N×N channel LOC implementation, FLA restricts the performance of SLM as it taks the high frequency band of the system, and the smaller the focal length of FLP, the higher the frequency band is required. The focal length of FLP on the input targets is determined by the pixel pitch of SLM. The Fourier Transform image width Di of the input targets can be expressed as
Di=λf/d
(7)
where λ is the wavelength of incident light, f is the focal length of the FLP. While the filter width Df of each channel on SLM2 is
Df=Md/N
(8)
Assuming the scaling factor of Di over Df is K:1, where K is an integer, and combined with Eq. (7) and Eq. (8), the focal length of FLP on the input targets can be deduced and given as
f=KMd2/(λN)
(9)
From Eq. (9), it can be concluded that the focal length of FLP on each channel of LOC is inversely proportional to N.

While, for multiplexed implementation, the input targets are multiplexed in frequency domain and the SBP is shared among each channel. Analysis for multiplexed SBP is just like the situation in multichannel implementation, with the difference that the interested plane is the frequency plane (SLM2). As the filter patterns of each channel are arranged independently on SLM2, the input channel of multiplexed LOC is limited by the filter number displayed on SLM2. Furthermore, multiplexed LOC suffers from the low diffraction efficiency of the grating structures. With the increment of channel number, the input pattern would be more complex and the diffraction efficiency will be reduced.

From the above discussion, with the increment of channel number in parallel LOC, the performance of each channel is reduced due to changes of channel’s SBP, yet the bandwidth required by FLP of each channel is decreased accordingly. The maximum channel number is expected to be demonstrated in the future experiment.

3. Experimental results and discussions

In this section, results of experimental applications of this parallel LOC are given to verify the proposed techniques. Optical Characters Recognition (OCR) is the translation of images of characters into machine-editable text, which plays an important part in pattern recognition, computer vision and artificial intelligence. Thus, verification of parallel LOC is performed by Chinese characters recognition.

The SLM used in parallel LOC is phase-only reflective type Liquid Crystal On Silicon Spatial Light Modulator (LCOS-SLM, Hamamatsu Photonics K.K. X10468 series), which is characterized by pure, linear and precise phase control. The liquid crystal chip of LCOS-SLM is consisted of 792 × 600 square pixels, where the pixel size is 20μm. The light utilization efficiency and the fill factor is approximately 90% and 95%, respectively. The SLM is addressed with 256 gray-scale levels, and the phase modulation capability at the wavelength of 633 nm is 0 to 2π corresponding to a gray range from 0 to 255. The input and filter pattern are displayed on SLM1 and SLM2 with a size of 512 × 512 pixels, 10.24 × 10.24 mm on SLMs. A He-Ne laser with the wavelength of 633 nm is utilized as the light source. The beam is collimated and obliquely incident on SLM1 with an angle less than 10°. In addition, the size of input beam is shaped to approximately 10.24 × 10.24 mm to keep consistent with input pattern.

As ordinary Chinese characters are composed by elementary parts, therefore, it’s reasonable to recognize characters’ radicals to accomplish Chinese characters identification. The experiment is designed under the principle that if reference radicals are parts of input characters, strong correlation peak will appear. During this process, the feasibility, and even efficiency can be verified and evaluated. The input target is composed of four same Chinese characters patterns, as shown in Fig. 6(a)
Fig. 6 Input targets (a) are composed of four same Chinese characters, reference targets (b) is consist of four different Chinese characters’ parts represented as 1, 2, 3, 4, where reference target 1, 2, and 3 are intercepted from the input character and reference target 4 is selected randomly. The white cross lines on input and reference targets are drew for isolating the different channels, which is not displayed on SLM. The correlation signals between (a) and (b) is obtained by a CCD camera shown in (c).
. Each pattern is consist of 256 × 256 pixels, with the center located at (127, 127), (127, 383), (383, 127), (383, 383), respectively, where we define the top left corner of the input target pattern as (0, 0) and the right bottom corner as (511, 511). Each pattern has a Chinese character with gray value of 127 and the other part is 0. A 2 × 2 FLA with focal length of 648 mm is superimposed with input targets. The size and center of each FLP is coincident with the individual character pattern. Figure 6(b) gives the reference pattern that is consisted of four different Chinese character parts of 256 × 256 pixels denoted as 1, 2, 3, 4. The part 1, 2, 3 are intercepted from the input target, while the 4th reference target is not. The filter pattern is programmed from the reference pattern and overlapped with a FLA following the procedures presented in section 2. The FLA written onto the phase-only filter possesses a focal length of 324 mm. The input and filter patterns are displayed on the center of SLM1 and SLM2, respectively. At the focal plane of the system, approximately 648 mm from SLM2, a CCD camera is placed to capture the correlation signal. Figure 6(c) provides the correlation signals intensity between input and reference targets. The results shows that at the top-left, bottom-left and top-right of correlation plane, three correlation peaks are observed that correspond to the reference pattern 1, 2 and 3. However, at the position of the bottom-right that is represented as 4, no obvious correlation peak is detected, which means that there has no relationship between the input targets and reference target in the 4th correlation channel. This proves that four-channel LOC is feasible for parallel Chinese characters parts recognition.

Demonstration for multiplexed LOC is performed in a similar manner. As Fig. 7(a)
Fig. 7 Experimental results and patterns of 4-channel multiplexed LOC. (a) shows the input targets in the left column, the corresponding high frequency carriers in the middle column, which are retrieved from the spot patterns in the right column. The reference targets (b) are four same Chinese characters, which are identical to the input target at row 2. On the correlation plane (c), a strong correlation peak can be observed at the right-up.
shows, the left column represents four different Chinese characters are chosen as input targets, each has a size of 512 × 512 pixels. Then, the input targets will be superimposed with appropriate frequency carriers for the purpose of shifting them into separate channels, as shown in the center column of Fig. 7(a). Precise frequency carrier patterns are constructed by Iterative Fourier Transform Algorithm (IFTA) such as G-S algorithm [13

13. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 227–246 (1972).

] from object intensity patterns. In this implementation, spot patterns, as the object intensity patterns, adopted for G-S algorithm are shown in the right column of Fig. 7(a). Correspondingly generated phase patterns are shown in the middle column of Fig. 7(a). Each spot pattern is consisted of 512 × 512 pixels with only one pixel has gray value 255. The bright spots locate sequentially at (127, 127), (383, 127), (127, 383) and (383, 383) in the row 1, 2, 3 and 4, which are identical to channel center on the filter pattern. After the procedure in Fig. 7(a), a FLP with a focal length at 648mm is overlapped on SLM1 to Fourier transform the input targets. Four identical Chinese characters, which are same as the input characters at line 2 in Fig. 7(a), are utilized as reference targets (as shown in Fig. 7(b)). The reference pattern is composed of 512 × 512 pixels and the filter pattern is constructed from this reference pattern. Likewise, a FLP with a focal length of 324mm is used in filter pattern. And CCD camera was placed at the system focal plane for correlation signal as shown in Fig. 7(c). An obvious correlation peak is appeared on the top-right of correlation plane, while no apparent signal is seen on the other area. This result indicates that the input target at row 2 in Fig. 7(a) is successfully identified and the Type I multiplexed correlation technique is feasible based on two SLMs.

Verification for type II multiplexed LOC shares the same configuration as multichannel LOC. Input target is one Chinese character that is identical to the character in Fig. 6(a) yet the image size is 512 × 512 pixels. While in this application, the frequency carrier (the middle column in Fig. 8(a)
Fig. 8 Experimental results and input target to demonstrate Type II multiplexed LOC. Input target is one Chinese character (left image in (a)). The frequency carrier (middle image in (a)) is retrieved from the four spots pattern (right image in (a)). On the correlation plane (c), three correlation peaks can be observed at the position denoted as 1, 2 and 3. No obvious signal is appeared on part 4.
) is retrieved from the 4-spots pattern (the right image in Fig. 8(a)). To be specific, the spot pattern is composed of 512 × 512 pixels and at the position of (127, 127), (383, 127), (127, 383) and (383, 383), the gray value is 255. To realize Fourier Transform, FLPs are implemented on SLM1 and SLM2 just as Type I multiplexed LOC. Resulting correlation signal is given in Fig. 8(b). The correlation plane is divided into four parts which are represented as 1, 2, 3, 4 correspond to four processing channels. At the part 1, 2 and 3 of correlation plane, three correlation peaks are emerged from background. In contrast, at the 4th part, no sharp correlation signal appears. From the results, it can be demonstrated that the technique in Type II multiplexed LOC is suitable for character recognition. Moreover, by comparing the correlation results between Fig. 6(c) and Fig. 8(b), it also can be observed that the correlation peaks of multiplexed LOC are weaker than that of multichannel LOC. The reason is that the input pattern of multiplexed LOC contains more high frequency components than that of multichannel LOC, which will influence the performance of SLM.

Channel uniformity is a key factor in parallel optical processing which affects the discriminability of each channel. To evaluate the degree of uniformity of parallel LOC, identical Chinese characters are utilized as input and reference targets, resulting in autocorrelation in all the channels. The intensity of correlation signal is further measured by a photodiode, and uniformity can be evaluated by ratio between minimum intensity and maximum intensity. Uniformity evaluation has been done both on multichannel configuration and Type II multiplexed configuration. We use the input targets as in Fig. 6(a). Figure 9(a)
Fig. 9 Autocorrelation results of the Chinese characters in Fig. 6(a). (a) multichannel LOC (b) multiplexed LOC.
shows the experimental results of four-channel LOC. On the correlation plane, four apparent correlation peaks denoted as 1, 2, 3 and 4 are observed at corresponding positions. Table 1

Table 1. Normalized Correlation Signal Intensity of Four-Channel LOC in Fig. 9(a)

table-icon
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shows the results of normalized correlation signal intensity of each processing channel. From the results we can conclude that the difference among four-channel LOC is less than 10%. While, autocorrelation results for Type II multiplexed LOC is illustrated in Fig. 9(b).

Similarly, normalized intensity for 1 to 4 channels are measured and reported in Table 2

Table 2. Normalized Correlation Signal Intensity of Four-Channel Multiplexed LOC in Fig. 9(b)

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. Under this circumstance, we can obtain that the minimum intensity versus maximum intensity is 0.92, which implies that the uniformity of multiplexed and divergence LOC is acceptable.

4. Conclusion

In this paper, we presented a theoretical and experimental study of the parallel LOC concept based on two phase-only LCOS-SLMs. Employing the approach of adding FLA with input targets and filter pattern, multi-channel LOC for Chinese characters’ parts recognition is realized. Through multiplexing algorithm, multiplexed LOC is described and demonstrated for characters recognition applications. Uniformity of four-channel parallel LOC is verified and performance differences within 10% are achieved through experimental results.

In summary, one of the major advantages of this new approach is that various LOCs are able to be performed in the same optical setups without hardware alignment or alternation. This technique will lead to developing a compact, multifunctional, universal system, which could be used in digital image processing, optical neural networks, and furthermore, optical computer. Currently, the processing channel numbers of parallel LOC is mainly limited by the liquid crystal area of LCOS-SLM. The development of SLM technique will expand the processing capability of parallel LOC. Further research through associating methods from traditional optimization algorithms could be applied to improve the performances (SDF, SNR etc.) of our system.

Acknowledgments

This work was supported by Hamamatsu Photonics K. K., State Key Laboratory of Mordern Optical Instrumentation, Zhejiang University, and the National Natural Science Foundation of China (NSFC) 60877008. The authors are grateful to T. Hiruma, Y. Suzuki and T. Hara for their encouragement and H. Toyoda and N. Matsumoto for their useful discussions.

References and links

1.

F. T. S. Yu and X. J. Lu, “Real-time optical scanning correlator,” Appl. Opt. 23(18), 3109–3116 (1984). [CrossRef] [PubMed]

2.

F. T. S. Yu and M. S. Dymek, “Optical information parallel processing: a technique,” Appl. Opt. 20(8), 1450–1453 (1981). [CrossRef] [PubMed]

3.

D. A. Gregory and H. K. Liu, “Large-memory real-time multichannel multiplexed pattern recognition,” Appl. Opt. 23(24), 4560–4570 (1984). [CrossRef] [PubMed]

4.

H. K. Liu and J. G. Duthie, “Real-time screen-aided multiple-image optical holographic matched-filter correlator,” Appl. Opt. 21(18), 3278–3286 (1982). [CrossRef] [PubMed]

5.

T. H. Chao and H. K. Liu, “Real-time optical holographic tracking of multiple objects,” Appl. Opt. 28(2), 226–231 (1989). [CrossRef] [PubMed]

6.

S. K. Case, “Pattern recognition with wavelength-multiplexed filters,” Appl. Opt. 18(12), 1890–1894 (1979). [CrossRef] [PubMed]

7.

T. H. Chao and M. Chen, “Pattern recognition with a wavelength-angle multiplexed optical scanning correlator,” Opt. Eng. 25, 828–833 (1986).

8.

C. F. Hester and D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19(11), 1758–1761 (1980). [CrossRef] [PubMed]

9.

G. F. Schils and D. W. Sweeney, “Rotationally invariant correlation filtering for multiple images,” J. Opt. Soc. Am. A 3(7), 902–908 (1986). [CrossRef]

10.

M. Villarreal, C. Iemmi, and J. Campos, “Parallel classification of multiple objects using a phase-only multichannel optical correlator,” Opt. Eng. 42(8), 2354–2361 (2003). [CrossRef]

11.

N. Fukuchi, T. Inoue, H. Toyoda, and T. Hara, “Lensless Vanderlugt optical correlator using two phase-only spatial light modulators,” Chin. Opt. Lett. 7(12), 1131–1133 (2009). [CrossRef]

12.

X. Zeng, J. Bai, C. Hou, and G. Yang, “Compact optical correlator based on one phase-only spatial light modulator,” Opt. Lett. 36(8), 1383–1385 (2011). [CrossRef] [PubMed]

13.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 227–246 (1972).

OCIS Codes
(070.4550) Fourier optics and signal processing : Correlators
(070.5010) Fourier optics and signal processing : Pattern recognition
(200.4960) Optics in computing : Parallel processing
(230.6120) Optical devices : Spatial light modulators

ToC Category:
Fourier Optics and Signal Processing

History
Original Manuscript: April 20, 2011
Revised Manuscript: May 25, 2011
Manuscript Accepted: June 2, 2011
Published: June 14, 2011

Citation
Xu Zeng, Takashi Inoue, Norihiro Fukuchi, and Jian Bai, "Parallel lensless optical correlator based on two phase-only spatial light modulators," Opt. Express 19, 12594-12604 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-13-12594


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References

  1. F. T. S. Yu and X. J. Lu, “Real-time optical scanning correlator,” Appl. Opt. 23(18), 3109–3116 (1984). [CrossRef] [PubMed]
  2. F. T. S. Yu and M. S. Dymek, “Optical information parallel processing: a technique,” Appl. Opt. 20(8), 1450–1453 (1981). [CrossRef] [PubMed]
  3. D. A. Gregory and H. K. Liu, “Large-memory real-time multichannel multiplexed pattern recognition,” Appl. Opt. 23(24), 4560–4570 (1984). [CrossRef] [PubMed]
  4. H. K. Liu and J. G. Duthie, “Real-time screen-aided multiple-image optical holographic matched-filter correlator,” Appl. Opt. 21(18), 3278–3286 (1982). [CrossRef] [PubMed]
  5. T. H. Chao and H. K. Liu, “Real-time optical holographic tracking of multiple objects,” Appl. Opt. 28(2), 226–231 (1989). [CrossRef] [PubMed]
  6. S. K. Case, “Pattern recognition with wavelength-multiplexed filters,” Appl. Opt. 18(12), 1890–1894 (1979). [CrossRef] [PubMed]
  7. T. H. Chao and M. Chen, “Pattern recognition with a wavelength-angle multiplexed optical scanning correlator,” Opt. Eng. 25, 828–833 (1986).
  8. C. F. Hester and D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19(11), 1758–1761 (1980). [CrossRef] [PubMed]
  9. G. F. Schils and D. W. Sweeney, “Rotationally invariant correlation filtering for multiple images,” J. Opt. Soc. Am. A 3(7), 902–908 (1986). [CrossRef]
  10. M. Villarreal, C. Iemmi, and J. Campos, “Parallel classification of multiple objects using a phase-only multichannel optical correlator,” Opt. Eng. 42(8), 2354–2361 (2003). [CrossRef]
  11. N. Fukuchi, T. Inoue, H. Toyoda, and T. Hara, “Lensless Vanderlugt optical correlator using two phase-only spatial light modulators,” Chin. Opt. Lett. 7(12), 1131–1133 (2009). [CrossRef]
  12. X. Zeng, J. Bai, C. Hou, and G. Yang, “Compact optical correlator based on one phase-only spatial light modulator,” Opt. Lett. 36(8), 1383–1385 (2011). [CrossRef] [PubMed]
  13. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 227–246 (1972).

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