1018 OPTICS LETTERS / Vol. 22, No. 13 / July 1, 1997

Optoelectronic implementation of the triple correlation

David Mendlovic, Adolf W. Lohmann,* David Mas,† and Gal Shabtay

Department of Physical Electronics, Faculty of Engineering, Tel-Aviv University, 69978 Tel-Aviv, Israel

Received January 31, 1997

The auto triple correlation has several fundamental advantages over the ordinary autocorrelation of secondorder. We present an optoelectronic processor for the computation of the auto triple correlation. 1997Optical Society of America

The second-order autocorrelation I2sxd is a well-knowntool for signal processing:

I2sxd ­Z

I1sx0 dI1psx0 2 xddx0. (1)

The Fourier transform of I2sxd is the power spectrum ofthe signal I1sxd:

I2sud ­Z

I2sxdexps22piuxddx ­ jI1sudj2, (2)

where I1sud is defined as

I1sud ­Z

I1sxdexps22piuxddx . (3)

Unfortunately the Fourier phase of I1sud is apparentlylost. The auto triple correlation (ATC), defined as

I3sx, yd ­Z

I1sx0 dI1sx0 1 xdI1sx0 1 yddx0, (4)

does not suffer from this loss. It is possible for oneto recover the Fourier phase by manipulating theso-called bispectrum I3su, vd, which is the Fouriertransform of the ATC1,2:

I3su, vd ­ZZ

I3sx, ydexpf22pisux 1 vydgdxdy

­ I1sudI1svdI1s2u 2 vd . (5)

This triple correlation analysis has improved the an-gular resolution in optical astronomy by a factor of30 or more.2,3 Similar success was also achieved inother applications of the ATC, including noise sup-pression.1 Actually, the Fourier phase cannot be com-pletely recovered from the ATC. The linear term islacking. Hence the absolute location of the input re-mains unknown. So far, optical implementations ofthe ATC have been based on time integration of pro-cessors and thus has resulted in dynamic optical sys-tems.4 This type of system cannot provide real-timeperformance and require high technical complexity.

The proposed optoelectronic implementation is basedon the fact that one can substitute the ATC com-putation for a one-dimensional signal I1sxd with acomputation of the cross correlation of two relatedtwo-dimensional signals. This substitution is valu-able because second-order cross correlations can be per-formed easily with optoelectronic hardware.

In our system the generation of the two related sig-nals is done by a digital computer. The cross corre-

0146-9592/97/131018-03$10.00/0

lation is performed optically. Before actually describ-ing our system, we present the underlying theoreticalconcept.

We split the bispectrum into two factors

I3su, vd ­ TAsu, vdTBpsu, vd , (6)

where

TAsu, vd ­ I1sudI1svd ,

TBpsu, vd ­ I1s2u 2 vd . (7)

Note that if I1sxd is real the second part of Eqs. (7)becomes

TB su, vd ­ I1su 1 vd . (8)

The inverse Fourier transform of Eq. (6) yields the ATCof the one-dimensional signal I1sxd as a second-ordercross correlation of the two substitute signals TAsx, ydand TB sx, yd:

I3sx, yd ­ZZ

I3su, vdexpf2pisux 1 vydgdudv

­ZZ

TAsu, vdTBpsu, vdexpf2pisux 1 vydgdudv

­ZZ

TAsx0 1 x, y 0 1 ydTBpsx0, y 0 ddx0dy 0. (9)

We compute this two-dimensional cross correlation ac-cording to the concept of the joint transform corre-lator (JTC).5,6 We prefer the JTC approach over theequivalent spatial-filtering approach for practical rea-sons: The signal enters the system directly, not indi-rectly as with a spatial f ilter, and the alignment of theoptical setup is easier for the JTC. The input for theJTC is

Jsx, yd ­ TAsx 2 x0, yd 1 TB sx 1 x0, yd . (10)

The two shifted signals TAsx, yd and TB sx, yd can bededuced from Eqs. (7):

TAsx, yd ­ I1sxdI2syd ,

TB sx, yd ­ I1psxddsx 2 yd . (11)

Equations (11), together with Eq. (6), describe theessence of our method. They show how to convertthe relatively complex ATC operation into a con-ventional two-dimensional cross-correlation operation.

1997 Optical Society of America

July 1, 1997 / Vol. 22, No. 13 / OPTICS LETTERS 1019

The first step of the JTC processor consists of a Fouriertransform followed by a modulus square operation:

jJsu, vdj2 ­ jTAsu, vdj2 1 jTBsu, vdj2

1 TAsu, vdTBpsu, vdexps4piux0d

1 TBsu, vdTApsu, vdexps24piux0d . (12)

The second step is another Fourier transform. Ourprime interest lies within the third term of Eq. (12).This term yields the ATC after the second step accord-ing to Eq. (9). The other terms in Eq. (12) can be sup-pressed if the shift parameter x0 in Eq. (10) is largeenough.

Our experimental setup, depicted in Fig. 1, consistedof a spatial light modulator (SLM) as an input device,a 2-F Fourier transformer, and a CCD camera as anoutput device. Both steps were performed with thesame hardware. The output of the f irst step [Eq. (12)]was fed back to the SLM at the input station.

The signal I1sxd was chosen to be a windowed, biasedsinusoid function [Fig. 2(a)]. The corresponding jointsignal Jsx, yd is shown in Fig. 2(b). Figure 3 presentsthe intermediate signal jJsu, vdj2. The ATC is dis-played as a simulation in Fig. 4(a) and as an experi-mental result in Fig. 4(b). The proposed system wasalso tested for the one-dimensional rect function thatis depicted in Fig. 5(a). The input that was used asamplitude transmission of the SLM device for the f irstcycle is presented in Fig. 5(b). The computed ATC forthis object is shown in Fig. 6(a), and the output ob-tained is presented in Fig. 6(b). The experiment couldbe even faster if two SLM’s were used in cascade, in-stead of using a single SLM in two cycles.

Fig. 1. Optoelectronic system for obtaining the ATC.F. T., Fourier transform; J. T., joint transform.

Fig. 2. Windowed, biased sine function. (b) Input for thefirst cycle.

Fig. 3. Output of the f irst cycle ( joint spectrum).

Fig. 4. (a) Computed ATC for the sine case. (b) Experi-mentally obtained ATC (upper right) and the zero order(lower left).

Fig. 5. (a) Rect function. (b) Input for the f irst cycle.

Fig. 6. (a) Computed ATC for the rect case. (b) Experi-mentally obtained ATC (upper right) and the zero order(lower left).

In summary, we have presented a processor for com-puting the auto triple correlation of a one-dimensionalsignal. We used free-space optics, optoelectronic inter-faces (SLM, CCD), and a digital computer, each tech-nology for what it can do best.

1020 OPTICS LETTERS / Vol. 22, No. 13 / July 1, 1997

The authors acknowledge a Diffractive OpticsInfrastructure Fund grant provided by the IsraeliMinistry of Science and Technology. G. Shabtaythanks the Eschol Foundation for its f inancial support.A. Lohmann acknowledges support from DeutscheForschungsemeinschaft. D. Mas acknowledges a grantfrom the Generalitat Valenciana, University of Valen-cia, Spain.

*Permanent address, Physikalisches Institut, Er-langen-Nurnberg University, Erwin Rommelstrasse 1,Erlangen, Germany.

†Permanent address, Department Interuniversitarid’Optica, University of Valencia, Dr. Moliner 50, 46100,Burjassot, Spain.

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