1658 J. Opt. Soc. Am. A/Vol. 15, No. 6 /June 1998 Mendlovic et al.

Fractional triple correlation and its applications

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

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

Received August 26, 1997; revised manuscript received December 15, 1997; accepted January 26, 1998

Higher-order correlations are well known for their use in noise removal, image enhancement, and signal iden-tification. They are generalizations of the well-known second-order correlation. The fractionalization of thesecond-order correlation provides some interesting features that are related to the shift-variance property ofthe fractional-Fourier-transform operation. This project proposes the fractionalization of the triple-correlation operation (as well as other higher-order correlations). A suggested definition as well as some ap-plications are given. Computer simulations demonstrate some of the features this operation offers. © 1998Optical Society of America [S0740-3232(98)01106-5]

OCIS codes: 070.2580, 070.2590.

1. INTRODUCTIONThe triple correlation (TC) and its two-dimensional Fou-rier transform, the bispectrum, are well-known opera-tions, mainly in astronomical detection and noiseremoval.1 It is possible to recover a signal from its auto-triple correlation (ATC) if one is willing to sacrifice the ab-solute position of the image2,3; i.e., only the linear part ofthe Fourier phase of the signal is lost. Thus the ATCbears more information about the signal than does the au-tocorrelation, which contains no spectrum phase informa-tion. The TC can also be used to implement rotation orscale invariance.4 Moreover, when zero-mean Gaussiannoise is added to a signal, one can obtain the TC of thesignal itself, provided that ensemble averaging is per-formed on the TC of the distorted signal.5 Nevertheless,the conventional correlation performed on a one-dimensional signal results in a one-dimensional output,whereas the ATC of a signal has two dimensions.

Because of the above-mentioned properties of the TC,use of the TC has expanded to a large variety of fields,such as speckle masking in astronomy and high-resolution astronomical imaging,5–7 recovery of movingobjects from fixed backgrounds,8 optical characterrecognition,9 and invariant pattern recognition.4,10

Lately, an optoelectronic implementation for the TC,which provides the ability to perform the TC operation ata TV rate (up to 60 frames per second), was suggested.11

Recently, the fractional Fourier transform (FRT) wasintroduced to the optics community.12–16 This transfor-mation permits one to control the amount of shift vari-ance by changing the fractional order of the transform.If the fractional order p is chosen to be zero, then the re-sult is totally shift variant, since its output is the objectitself, while for p 5 1 the transform converges to the con-ventional Fourier transformation, which is totally shiftinvariant in its absolute value (shift of the input is ex-pressed as a linear phase factor in the transformationplane).

In this paper we suggest combining the TC and theFRT to define the fractional triple correlation (FTC) andthe fractional bispectrum. The advantages of the new

0740-3232/98/061658-04$15.00 ©

definition are discussed and demonstrated. Section 2provides a short review of the FRT and the TC. In Sec-tion 3 the FTC is defined and some of its applications areconsidered. An extension for higher-order fractional cor-relation is presented in Section 4. Computer simulationsare shown in Section 5.

2. BACKGROUNDThe FRT is a mathematical transformation that has asimple optical implementation.12–14 It is a simple gener-alization of the conventional Fourier transformation,which was found to be useful in pattern recognition andnoise removal.15 The FRT is defined as

lp~u ! 5 Fp@l~x !#

5 CE2`

`

l~x !expF ip~x2 1 u2!

tan~pp/2!2

2pixusin~pp/2!Gdx, (1)

where Fp denotes the FRT operator, p is the fractional or-der that controls the shift variance of the transform, and

C 5

expH 2iFp sgn~sin f!

42

f

2 G J~ usin fu!1/2 ,

where f 5 pp/2. Note that the role of the constant C isto preserve the energy from the object domain to the frac-tional Fourier domain. The ATC of the signal l(x) is de-fined as

l31~x1 , x2! 5 E

2`

`

l~x !l~x 1 x1!l~x 1 x2!dx. (2)

The Fourier transform of the ATC, also called the bispec-trum, can be expressed as

l31~u1 , u2! 5 E

2`

`

l31~x1 , x2!

3 exp@22pi~x1u1 1 x2u2!#dx1dx2

5 l1~u1! • l1~u2! • l1~2u1 2 u2!. (3)

1998 Optical Society of America

Mendlovic et al. Vol. 15, No. 6 /June 1998 /J. Opt. Soc. Am. A 1659

Here l1(u) is the Fourier transform (or the FRT of order1) of the signal l(x). As is shown below, the TC and thebispectrum as presented in Eqs. (2) and (3) are specialcases of the FTC and the fractional bispectrum.

3. FRACTIONAL TRIPLE CORRELATIONEvery definition of the FTC should be consistent with Eq.(2) for p 5 1. It is convenient to define the FTC indi-rectly. First we define the fractional bispectrum:

l3P~u1 , u2! 5 lp~u1! • lp~u2! • lp~2u1 2 u2!, (4)

and then we define the FTC as the p-order inverse FRT(or 2p-order FRT) of the fractional bispectrum:

l3P~x1 , x2! 5 F2p@ lp~u1! • lp~u2! • lp~2u1 2 u2!#.

(5)

The indirect approach leads immediately to another defi-nition for the fractional bispectrum:

l3P~u1 , u2! 5 Fp@l3~x1 , x2!#, (6)

where l3(x1 , x2) is the conventional TC. Note that thisdefinition is in general different from the previous one[Eq. (4)] and is useful mainly for fractional filtering of TCsignals. We wish to explore both avenues, although weshall concentrate on the first definition since fractionalfiltering of signals has been considered in previouspapers.15

The FTC, as defined in Eq. (5), shares some of the prop-erties of the TC; thus it might assist in applications simi-lar to those of the TC. For example, when zero-meanGaussian noise is added to the signal, the ensemble aver-age of the FTC of the distorted signal is equal to the FTCof the clean signal. This statement can be verified if onefollows the same steps in proving this property for the TCcase. This statement indicates that the FTC might be auseful tool for pattern recognition and noise removal pur-poses.

In many cases, aside from Gaussian noise, other noisesmay distort the signal. One example is what we call bor-der noise. In realistic problems, the signal we are inter-ested in is not isolated on its support vector but is sur-rounded by other undesired signals. In temporal signals,for instance, undesired information eventually may pre-ceed and follow the signal itself. Since the TC is shift in-variant, both the signal and the noise may overlap in theTC domain, resulting in a highly distorted TC chart. Un-fortunately, this noise does not admit a statistical treat-ment, so its removal must be done in some heuristic man-ner. One solution is windowing the signal, but this mayresult in a modulation or even a loss of part of the signalspectrum. With the definition given in Eq. (5), one cansee that, by replacing the Fourier transform with an FRTof order p, we give more weight to the central part than tothe other parts of the signal.16 The shift-variance prop-erty of the FRT, which is controlled by the parameter p,has been the main subject of many papers (see Ref. 17, forinstance) and may be used here to decrease the influenceof the border noise. The fractional order p is chosen ac-cording to the weight that one is willing to assign to thecentral part of the signal.

Thus these arguments indicate that, whenever bordernoise distorts a signal for which a reference is available, itmight be worthwhile to regard the FTC as a secondarysignal and compare the FTC of a distorted signal with theFTC of the reference instead of comparing the signalsthemselves.

Furthermore, based on the properties of the FRT,14,15

the FTC may have better properties than the TC for otherkinds of noise as well. For example, let us consider anonstationary chirp noise, which is expressed as n(x)5 A exp(2ibx2), where A is an unknown constant and bis a known constant (or may be easily measured). Theordinary TC of that expression is unbounded, and thus itwould not be effective for noise removal. On the otherhand, if we perform an FTC with fractional order p, de-fined by

tanS p

2p D 5

p

b, (7)

the fractional bispectrum of the noise will be a delta func-tion, and it may be easily filtered without distortion of thesignal’s information. Thus, for nonstationary noises, theFTC may exhibit better performance than the TC becauseof the acquired properties of the FRT, such as shift vari-ance and handling chirp signals.

The above two examples indicate that there are variouscircumstances in which, by replacing the TC with theFTC, better performance can be obtained.

4. HIGH-ORDER CORRELATIONOn the basis of the definition of the second-order correla-tion, the generalization toward higher-order correlationsfor real signals is made with the following definition:

ln1~x1 , x2 ,..., xn21! 5 E l~x !)

k51

n21

l~x 1 xk!dx, (8)

where l(x) is the signal and ln1(x1 , x2 ,..., xn21) is the

nth-order correlation function. Note that the nth orderresults in n 2 1 dimensions whenever the signal is onedimensional.

An equivalent definition can be derived in the spectraldomain. The spectrum of l(x) is

l1~u ! 5 E l~x !exp~2i2pxu !dx, (9)

and the spectrum of the nth order correlation is

ln1~u1 , u2 ,..., un21!

5 l1~2u1 2 u2 2 ¯ 2 un21!)k51

n21

l1~uk!. (10)

As with the FTC operation, several definitions for thehigher-order fractional correlation may be proposed. Themost simple definition is the symmetric one:

lnp~x1 , x2 ,..., xn21!

5 F2pF lp~2u1 2 u2 2 ¯ 2 un21!)k51

n21

lp~uk!G . (11)

1660 J. Opt. Soc. Am. A/Vol. 15, No. 6 /June 1998 Mendlovic et al.

5. COMPUTER SIMULATIONSComputer simulations were performed to demonstratethe advantages of the FTC over the conventional TC.The simulations are related to the case of a border noisethat surrounds the signal. These simulations rely on thefact that the fractional order may give more weight to thecentral part of the signal than to its edges, thus suppress-ing the effect of the border noise in the fractional tripledomain.

Figure 1 shows the signals that were used for the com-puter simulations. Figure 1(a) shows a rectangular sig-nal. Figure 1(b) shows the same signal surrounded by anoise signal that represents the border noise. Note thatthe edges of the border noise are considered to be un-known. Our assumption is that no a priori informationabout the noise exists.

Figures 2(a) and 2(b) show the ordinary TC of the sig-nals. One can see that the TC of the noisy signal is quite

Fig. 1. (a) Rectangular signal, (b) rectangular signal embeddedin border noise.

Fig. 2. (a) Triple correlation of a rectangular signal, (b) triplecorrelation of the signal depicted in Fig. 1(b), (c) fractional triplecorrelation of the order p 5 0.2 of a rectangular signal, (d) frac-tional triple correlation of the order p 5 0.2 of the signal de-picted in Fig. 1(b).

different from the TC of the original signal: It suffersfrom a high degree of nonlinear deformation. Figures2(c) and 2(d) show the FTC of order 0.2 for the clean andthe noisy signal, respectively. One can see that there is ahigh resemblance between the FTC of the corrupted sig-nal and that of the clean signal. This particular frac-tional order was chosen to show the improvement in usingthe FTC over the TC in this example. It is evident thatfor other examples, which involve other types of noise andobjects, different fractional orders should be considered.

Since the influence of the noise has been suppressed,we can easily identify the signal by correlating the FTCfunctions of the clean signal and of the corrupted signal.Further comparison of Fig. 2(a) with 2(c) and Fig. 2(b)with 2(d) shows that the dynamic range of the FTC can besmaller than that of the conventional triple correlation.

6. CONCLUSIONSNovel definitions for the fractional triple correlation andthe fractional bispectrum were proposed. The advan-tages and applications of the new definitions were dis-cussed. Several examples that demonstrate the capabili-ties of the FTC were given. Computer simulations wereundertaken to demonstrate the performances of the pro-posed FTC for the case of a border noise that surroundsthe signal. In addition, possible definitions have beensuggested for higher-order fractional correlations. Fur-ther investigation is needed to explore the advantages indefining the FTC and its higher- (or lower-)order gener-alizations with a different order assigned to each of theterms in Eq. (5) or Eq. (11). Another possible future di-rection would be to optimize the fractional order of theFTC for different noise models.

ACKNOWLEDGMENTSThis work was partially supported by Comision Intermin-isterial de Ciencia y Tecnologıa, Spain, project TAP96-1015-C03-03. David Mas acknowledges a grant from theGeneralitat Valenciana. The authors acknowledge theDiffractive Optics Infrastructure Fund provided by the Is-raeli Ministry of Science and Technology. Gal Shabtaythanks the Eshcol Foundation for its financial support.

*Permanent address, Departament d’Optica, Univer-sity of Valencia, Dr. Moliner 50, 46100 Burjassot, Spain.

†Permanent address, Erlangen-Nurnberg University,Physikalisches Institut, Erwin Rommelstrasse 1, 91058Erlangen, Germany.

REFERENCES1. A. W. Lohmann and B. Wirnitzer, ‘‘Triple correlation,’’ Proc.

IEEE 72, 889–901 (1984).2. A. W. Lohmann, ‘‘A new algorithm for image and signal re-

covery from triple correlation,’’ Optik 72, 71–76 (1986).3. H. Bartelt, A. W. Lohmann, and B. Wirnitzer, ‘‘Phase and

amplitude recovery from bispectra,’’ Appl. Opt. 23, 3121–3129 (1984).

4. A. W. Lohmann, ‘‘Pattern recognition based on the triplecorrelation,’’ Optik 78, 117–120 (1988).

Mendlovic et al. Vol. 15, No. 6 /June 1998 /J. Opt. Soc. Am. A 1661

5. A. W. Lohmann, G. Weigelt, and B. Wirnitzer, ‘‘Specklemasking in astronomy: triple correlation theory and appli-cations,’’ Appl. Opt. 22, 4028–4037 (1983).

6. B. Wirnitzer, ‘‘Bi-spectrum analysis at low light levels andastronomical speckle masking,’’ J. Opt. Soc. Am. A 2, 14–21(1985).

7. G. Weigelt, ‘‘Triple-correlation imaging in optical as-tronomy,’’ in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1991), Vol. 19, pp. 294–319.

8. A. W. Lohmann, ‘‘Recovery of an object, moving at randomin front of a fixed background,’’ Optik 73, 127–134 (1986).

9. B. Braunecker, R. Hauck, and A. W. Lohmann, ‘‘Opticalcharacter recognition based on nonredundant correlationmeasurements,’’ Appl. Opt. 18, 2746–2753 (1979).

10. H. Bartelt and B. Wirnitzer, ‘‘Shift invariant imaging ofphoton limited data using bispectral analysis,’’ Opt. Com-mun. 53, 13–22 (1985).

11. D. Mendlovic, A. W. Lohmann, D. Mas, and G. Shabtay,‘‘Optoelectronic implementation of the triple correlation,’’Opt. Lett. 22, 1018–1020 (1997).

12. D. Mendlovic and H. M. Ozaktas, ‘‘Fractional Fourier trans-formations and their optical implementation: I,’’ J. Opt.Soc. Am. A 10, 1875–1881 (1993).

13. D. Mendlovic and H. M. Ozaktas, ‘‘Fractional Fourier trans-formations and their optical implementation: II,’’ J. Opt.Soc. Am. A 10, 2522–2531 (1993).

14. A. W. Lohmann, ‘‘Image rotation, Wigner rotation, and thefractional Fourier transform,’’ J. Opt. Soc. Am. A 10, 2181–2186 (1993).

15. R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, andH. M. Ozaktas, ‘‘Chirp filtering in the fractional Fourier do-main,’’ Appl. Opt. 33, 7599–7602 (1994).

16. A. W. Lohmann, Z. Zalevsky, and D. Mendlovic, ‘‘Synthesisof pattern recognition filters for fractional Fourier process-ing,’’ Opt. Commun. 128, 199–204 (1996).

17. J. Garcia, D. Mendlovic, Z. Zalevsky, and A. W. Lohmann,‘‘Space variant simultaneous detection of several objects bythe use of multiple anamorphic fractional Fourier trans-form filters,’’ Appl. Opt. 35, 3945–3952 (1996).