Abstract

Joint-transform correlation architecture is employed for digital matrix multiplication. Real-valued matrix–vector, complex-valued matrix–vector, real-valued matrix–matrix, and complex-valued matrix–matrix multiplication operations can all be realized simply by programming of the data arrangement in the input plane of a multiple-input joint-transform correlator. The proposed method benefits from the advantages of speed because of the real-time processing capability of the joint-transform correlator and of high accuracy because of the digital representation of the multiplied numbers. Computer-simulation results are provided in which the negative binary encoding method is used to encode matrix elements.

© 1999 Optical Society of America

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References

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1998 (2)

S. P. Kozaitis, M. A. Getbehead, “Multiple-input joint transform correlator for wavelet feature extraction,” Opt. Eng. 37, 1325–1331 (1998).
[CrossRef]

C. Li, S. Yin, F. T. S. Yu, “Nonzero-order joint transform correlators,” Opt. Eng. 37, 58–65 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

G. Li, L. Liu, C. Zhou, “Simplified optical complex multiplication using quarter-imaginary number representation,” Opt. Commun. 122, 16–22 (1995).
[CrossRef]

1994 (5)

1993 (1)

F. Cheng, P. Andres, F. T. S. Yu, “Removal of the intra-class associations in a joint power transform spectrum,” Opt. Commun. 99, 7–12 (1993).
[CrossRef]

1992 (1)

1987 (1)

1986 (1)

1984 (3)

Y. Z. Liang, H.-K. Liu, “Optical matrix–matrix multiplication method demonstrated by the use of a multifocus hololens,” Opt. Lett. 9, 322–324 (1984).
[CrossRef] [PubMed]

F. T. S. Yu, X. J. Lu, “A programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).
[CrossRef]

W. T. Rhodes, P. S. Guilfoyle, “Acoustooptic algebraic processing architectures,” Proc. IEEE 72, 820–830 (1984).
[CrossRef]

1983 (1)

1970 (1)

1966 (1)

Alam, M. S.

Andres, P.

F. Cheng, P. Andres, F. T. S. Yu, “Removal of the intra-class associations in a joint power transform spectrum,” Opt. Commun. 99, 7–12 (1993).
[CrossRef]

Artman, J. O.

Awwal, A. A. S.

M. A. Karim, A. A. S. Awwal, Optical Computing: An Introduction (Wiley, New York, 1992).

Bocker, R. P.

Bromley, K.

Carlotto, M.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate numerical computation by optical convolution,” in International Optical Computing Conference II, W. T. Rhodes, ed., Proc. SPIE232, 151–156 (1980).

Casasent, D.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate numerical computation by optical convolution,” in International Optical Computing Conference II, W. T. Rhodes, ed., Proc. SPIE232, 151–156 (1980).

Casasent, D. P.

Cheng, F.

F. Cheng, P. Andres, F. T. S. Yu, “Removal of the intra-class associations in a joint power transform spectrum,” Opt. Commun. 99, 7–12 (1993).
[CrossRef]

Cheng, L.

Clayton, S. R.

Deutsch, M.

Garcia, J.

Getbehead, M. A.

S. P. Kozaitis, M. A. Getbehead, “Multiple-input joint transform correlator for wavelet feature extraction,” Opt. Eng. 37, 1325–1331 (1998).
[CrossRef]

Goodman, J. W.

Guilfoyle, P. S.

W. T. Rhodes, P. S. Guilfoyle, “Acoustooptic algebraic processing architectures,” Proc. IEEE 72, 820–830 (1984).
[CrossRef]

Hemz, R. A.

Hotate, K.

Javidi, B.

J. Wang, B. Javidi, “Multiobject detection using the binary joint transform correlator with different types of thresholding methods,” Opt. Eng. 33, 1793–1804 (1994).
[CrossRef]

Kalivas, D. S.

D. S. Kalivas, “Real-time optical multiplication with accuracy,” Opt. Eng. 33, 3427–3430 (1994).
[CrossRef]

Karim, M. A.

Kozaitis, S. P.

S. P. Kozaitis, M. A. Getbehead, “Multiple-input joint transform correlator for wavelet feature extraction,” Opt. Eng. 37, 1325–1331 (1998).
[CrossRef]

Lee, M. C.

Lee, S. H.

Li, C.

C. Li, S. Yin, F. T. S. Yu, “Nonzero-order joint transform correlators,” Opt. Eng. 37, 58–65 (1998).
[CrossRef]

Li, G.

Liang, Y. Z.

Liu, H.-K.

Liu, L.

Lu, G.

Lu, X. J.

F. T. S. Yu, X. J. Lu, “A programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).
[CrossRef]

Mendlovic, D.

Nakano, H.

Neft, D.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate numerical computation by optical convolution,” in International Optical Computing Conference II, W. T. Rhodes, ed., Proc. SPIE232, 151–156 (1980).

Perlee, C.

Psaltis, D.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate numerical computation by optical convolution,” in International Optical Computing Conference II, W. T. Rhodes, ed., Proc. SPIE232, 151–156 (1980).

Rhodes, W. T.

W. T. Rhodes, P. S. Guilfoyle, “Acoustooptic algebraic processing architectures,” Proc. IEEE 72, 820–830 (1984).
[CrossRef]

Shao, L.

Song, Q. W.

Talbot, P.

Wang, J.

J. Wang, B. Javidi, “Multiobject detection using the binary joint transform correlator with different types of thresholding methods,” Opt. Eng. 33, 1793–1804 (1994).
[CrossRef]

Wang, Z.

Weaver, C. S.

Wu, S.

Yin, S.

C. Li, S. Yin, F. T. S. Yu, “Nonzero-order joint transform correlators,” Opt. Eng. 37, 58–65 (1998).
[CrossRef]

Yin, Y.

Yu, F. T. S.

C. Li, S. Yin, F. T. S. Yu, “Nonzero-order joint transform correlators,” Opt. Eng. 37, 58–65 (1998).
[CrossRef]

G. Lu, Z. Zhang, S. Wu, F. T. S. Yu, “Implementation of dc spectra-free joint transform correlator using phase-shifting techniques,” Appl. Opt. 36, 470–483 (1997).
[CrossRef] [PubMed]

F. Cheng, P. Andres, F. T. S. Yu, “Removal of the intra-class associations in a joint power transform spectrum,” Opt. Commun. 99, 7–12 (1993).
[CrossRef]

F. T. S. Yu, X. J. Lu, “A programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).
[CrossRef]

Zhang, Z.

Zhou, C.

G. Li, L. Liu, C. Zhou, “Simplified optical complex multiplication using quarter-imaginary number representation,” Opt. Commun. 122, 16–22 (1995).
[CrossRef]

Appl. Opt. (9)

Opt. Commun. (3)

F. T. S. Yu, X. J. Lu, “A programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).
[CrossRef]

G. Li, L. Liu, C. Zhou, “Simplified optical complex multiplication using quarter-imaginary number representation,” Opt. Commun. 122, 16–22 (1995).
[CrossRef]

F. Cheng, P. Andres, F. T. S. Yu, “Removal of the intra-class associations in a joint power transform spectrum,” Opt. Commun. 99, 7–12 (1993).
[CrossRef]

Opt. Eng. (4)

C. Li, S. Yin, F. T. S. Yu, “Nonzero-order joint transform correlators,” Opt. Eng. 37, 58–65 (1998).
[CrossRef]

J. Wang, B. Javidi, “Multiobject detection using the binary joint transform correlator with different types of thresholding methods,” Opt. Eng. 33, 1793–1804 (1994).
[CrossRef]

S. P. Kozaitis, M. A. Getbehead, “Multiple-input joint transform correlator for wavelet feature extraction,” Opt. Eng. 37, 1325–1331 (1998).
[CrossRef]

D. S. Kalivas, “Real-time optical multiplication with accuracy,” Opt. Eng. 33, 3427–3430 (1994).
[CrossRef]

Opt. Lett. (3)

Proc. IEEE (1)

W. T. Rhodes, P. S. Guilfoyle, “Acoustooptic algebraic processing architectures,” Proc. IEEE 72, 820–830 (1984).
[CrossRef]

Other (2)

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate numerical computation by optical convolution,” in International Optical Computing Conference II, W. T. Rhodes, ed., Proc. SPIE232, 151–156 (1980).

M. A. Karim, A. A. S. Awwal, Optical Computing: An Introduction (Wiley, New York, 1992).

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Figures (9)

Fig. 1
Fig. 1

(a) Data arrangement in the input plane for real MVM and (b) the corresponding correlation output.

Fig. 2
Fig. 2

(a) Data arrangement in the input plane for complex MVM and (b) the corresponding correlation output.

Fig. 3
Fig. 3

(a) Data arrangement in the input plane for real MMM and (b) the corresponding correlation output.

Fig. 4
Fig. 4

(a) Data arrangement in the input plane for multiplying two 2 × 2 real matrices according to Fig. 3(a) and (b) the corresponding correlation output.

Fig. 5
Fig. 5

(a) Compact data arrangement for multiplying two 2 × 2 real matrices and (b) the corresponding correlation output.

Fig. 6
Fig. 6

(a) Compact data arrangement for multiplying two 3 × 3 real matrices and (b) the corresponding correlation output.

Fig. 7
Fig. 7

(a) Data arrangement in the input plane for multiplying two 2 × 2 complex matrices and (b) the corresponding correlation output.

Fig. 8
Fig. 8

(a) Joint input image for complex MVM, (b) the corresponding correlation output, (c) the resultant vector elements after those undesired correlations are blocked.

Fig. 9
Fig. 9

(a) Joint input image for real MMM, (b) the corresponding correlation output, (c) the resultant matrix elements after those undesired correlations are blocked.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

fx, y=k=1K fkx-ak, y-bk,
Fu, v=k=1K Fku, vexp-i2πuak+vbk,
|Fu, v|2=k=1K Fku, vexp-i2πuak+vbk2=k=1K |Fku, v|2+j,kjkK Fju, vFk*u, vexp-i2π×uaj-ak+vbj-bk+j,kjkK Fj*u, vFku, vexpi2π×uaj-ak+vbj-bk,
fx, y=k=1K fkx, yfkx, y+j,kjkK fjx, yfk*x, y  δx-aj-ak, y-bj-bk+j,kjkK fj*x, yfkx, y  δ×x+aj-ak, y+bj-bk,
Dx2H,  Dy2V.
a11a12a1Na21a22a2NaN1aN2aNNb1b2bN=c1c2cN.
cp=q=1N apqbq,  p=1,, N.
h2,  w2L,
cp=q=1N apqbq,  p=1,, N,
apq=apqR+iapqI,  bq=bqR+ibqI, cp=cpR+icpI,
cpR=q=1N apqRbqR+q=1N apqI-bqI,
cpI=q=1N apqRbqI+q=1N apqIbqR.
cR=ARbR+AI-bI,
cI=ARbI+AIbR.
cpq=k=1N apkbkq,  p, q=1,, N.
ck=Abk,  k=1,, N,
bk=b1kb2kbNk,  ck=c1kc2kcNk.
C=k=1N Ck,
Ck=a1ka2kaNkbk1 bk2  bkN.
cpqk=apkbkq.
C=AB=a11b11a11b12a21b11a21b12+a12b21a12b22a22b21a22b22=C1+C2.
CR=ARBR+AI-BI,
CI=ARBI+AIBR.
AR+iAIBR+iBI=a11R+ia11Ia12R+ia12Ia21R+ia21Ia22R+ia22Ib11R+ib11Ib12R+ib12Ib21R+ib21Ib22R+ib22I=a11Rb11R+a11I-b11I+a12Rb21R+a12I-b21Ia11Rb12R+a11I-b12I+a12Rb22R+a12I-b22Ia21Rb11R+a21I-b11I+a22Rb21R+a22I-b21Ia21Rb12R+a21I-b12I+a22Rb22R+a22I-b22I+ia11Rb11I+a11Ib11R+a12Rb21I+a12Ib21Ra11Rb12I+a11Ib12R+a12Rb22I+a12Ib22Ra21Rb11I+a21Ib11R+a22Rb21I+a22Ib21Ra21Rb12I+a21Ib12R+a22Rb22I+a22Ib22R=CR+iCI.
12-1217i14-i5-2i-2+i1420-i4212+i183+i7-3+i41=89+i36-118+i55-320+i252.
011100+i000000110100+i000000010001+i000000000000+i010010000000+i001111000000+i000010000010+i010010010100+i101010011100+i010110000111+i011011001101+i000100000001+i000000=00122462201+i0013323210000010131110+i0000233433100031253320+i00113453340.
1301820-411-524-714-210-5-5038=-91236118-55048817355-6832
011101000000010110010100101011000001001111000110000100001001010010000010011110001111001111000000000111011000=000111121010011224322000111321010011323423100021332433200112335221000023342110001125442000001233320.

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