JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Quadratic Filter Theory and Partially Coherent Optical Systems*

R. KEITH RANEY

Radar and Optics Laboratory, Institute of Science and Technology, The University of Michigan,Ann Arbor, Michigan 48107

(Received 11 March 1969)

Quadratically nonlinear systems may be analyzed and synthesized by linear methods by exchanging anN-dimensional nonlinear problem for a 2N-dimensional linear formulation. This paper describes the basis forsuch linearization of a quadratic functional, and applies the method to partially coherent transilluminatedoptical systems.INDEX HEADINGS: Coherence; Filter; Optical systems; Image formation; Fourier transforms; Modulation

transfer.

The techniques of linear, time-invariant filter theoryhave become important tools for the analysis of prob-lems of optical data processing and image assessment.However, linear methods to date have been confinedto systems that are strictly coherent or noncoherent,since such systems are linear in amplitude or irradi-ance, respectively. Partially coherent imaging systemsare not linear, although such systems have a veryspecial type of nonlinearity. In an optical system, theimage irradiance (under quite broad restrictions) isrelated to the object transmittance by a quadraticform. If a quadratic filter theory could be devisedanalogous to that which exists for linear filters, thenpartially coherent imaging systems could be morereadily and consistently investigated.

The first purpose of this paper is to describe a practicaltheory for quadratically nonlinear systems.1 The mainresult of the general discussion is an exact linearizationof a quadratic filter. This result is applied to thepartially coherent imaging configuration, showing thatthe transfer function for the linearization is the trans-mission cross-coefficient, which is the Fourier transformof the expression representing the combined effect ofthe illumination and the optics.

The second purpose of this paper is to set the stagefor the logical synthesis of partially coherent optical-imaging and data-processing systems. In several appli-cations, use of partially coherent processing allowsimproved performance when compared to purely co-herent or noncoherent methods. This topic will beconsidered in future papers.

A complete description of detector nonlinearity in anoptical system would include consideration of effectsthat are not quadratic, such as detector saturation.However, departures from quadratic detection are notconsidered here, for several reasons. First, the imagingmodel considered below is a very reasonable represen-tation for many situations. Second, the nonlinearities

* This work was supported by the Air Force Avionics Labora-tory, Air Force Systems Command, United States Air Force.Paper presented at San Diego meeting of Optical Society, March1969 [J. Opt. Soc. Am. 59, 488A (1969)].

I Quadratic-filter theory is appropriate whenever a system tobe studied includes an energy-flux measurement, or has availableas the input signal only a second moment of the observed process,a class of problems in which optical systems include perhaps themost important examples.

that might be included have received considerabletheoretical and experimental attention, in contrast tothe general quadratic nonlinearity which is the subjectof this work. Finally, our objective is to provide asimple and unified framework for the analysis andsynthesis of quadratic systems in general and opticalsystems in particular, so that consideration of othernonlinearities would lead us astray.

BACKGROUND AND RELATED WORK

At present, there exists no unified subject honoredby the term "quadratic filter theory," although certainobvious parallels to linear filter theory would seem tojustify it. There are several diverse disciplines in whichresults are available which bear directly on the problem,including the theory of correlation functions2 3' andlinear filter theory.3

As is well known, the observable output of a partiallycoherent optical system may be expressed in terms of aquadratic functional. Two countervailing commentsmay be directed to this fact. In the first place, althoughthe quadratic form is well recognized and physicallymeaningful in optics, little progress has been reportedthat goes beyond the apparent difficulty representedby the nonlinear formulation. A particularly concisestatement of these facts is given by Becherer andParrent,4 who build on the mutual coherence functiondeveloped by Wolf.- In the second place, the impactof communication theory on optics has been confinedalmost exclusively to linear models. Early Europeanwork on optical-filter theory was publicized and ex-panded in this country by O'Neill,5 followed by aclassic summary article by Cutrona, Leith, Palermo,and Porcello.7 Other contributions of filter theory inoptics are represented by a minimum-mean-square error

2 J. L. Dobb, Stochastic Processes (John Wiley & Sons, Inc.,New York, 1953).

'W. B. Davenport and W. L. Root, An Introduction to theTheory of Random Signals and Noise (McGraw-Hill Book Co.,New York, 1958).

4 R. J. Becherer and G. B. Parrent, Jr., J. Opt. Soc. Am. 57,1479 (1967).

5 M. Born and E. Wolf, Principles of Optics (Pergamon Press,Inc., New York, 1959).

6 E. L. O'Neill, IRE Trans. IT-2, 56 (1956).7 L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello,

IRE Trans. IT-6, 386 (1960).

1149

VOLUME 59, NUMBER 9 SEPTEM BER 1969

R. KEITH RANEY

analysis of an optical data-processing system,8 whichdoes not depart from the traditions of linear filtertheory.

More recent results have been reported that developcertain aspects of nonlinear processing in optics. 9' 10

This paper attempts to bring into conjunction theadvances of each of two mainstream developments inoptics-quadratic formulation and the methods oflinear-filter theory.

GENERAL PROPERTIES

This section summarizes basic properties of a quad-ratic filter. The approach is formal, although sufficientconditions have been imposed to assure the existenceof the required integrals. We proceed in the one-dimensional form, since extension to the two-dimen-sional format encountered in imaging (or to higher-dimensional input functions) may be triviallyaccomplished by use of a vector notation.

The Quadratic Filter Q

Let the observable output g of a nonlinear time-invariant filter be related to the input f by thetransformation

g(u) = ffQ(u-x, u-y)f(x)f* (y)dxdy,

T'

(1)

With the above definition, the method suggested inthis paper may be previewed. It is obvious that Eq. (1)represents a second-order (quadratic) nonlinear trans-formation. However, by a simple step, this equationmay be extended to the form

G(u,v)= ffQ( -x, v y)f(x)f* (y)dxdy. (6)

T

If the product ff* is viewed as a two-dimensional inputfunction, Eq. (6) relates G to ff* linearly, as a two-dimensional convolution with weighting Q. Equation(6) subsumes the original filter representation of Eq.(1) for u= v. These facts imply that many advantagesof linear methods may be enjoyed while obtaining anexact analysis of the quadratic nonlinearity.

Transforms of Q

A quadratic filter Q(x,y) is of course a function of twovariables. For convenience, we define the one-dimen-sional function

q(x) = Q(x,x) (7)

that represents the principal diagonal section of Q.We define the one-dimensional Fourier transform ofQ (x,x) as

where the function that characterizes the filter is Q.The asterisk denotes complex conjugate. The trans-formation of Eq. (1) is defined to be that of a quadraticfilter if Q(x,y) is an integrable correlation function.Thus, the two-dimensional impulse response Q whichrepresents the quadratic filter is complex (hermitian)symmetric

Q(x,y) = Q*(yx), (2)

positive definite, such that for any square integrabletest function s,

(8)

where unmarked limits on the integral imply the range(-_ , oo) as usual. Since q is integrable EEq. (5)], q

exists. The inverse transform, when it exists, is

q(x) =- fq()ejixdcw.27r

(9)

The two-dimensional Fourier transform of Q isdefined to be

I Q (x,y)s(x)s*(y)dxdy> 0,

T

is square integrable,

I fQ(xy)i2dxdy< ,

(3) Q(.,X) = Q(x,y)e-Jxe-iydxdy.

Existence of Q is assured in the Plancheral sense,because Q is square integrable on the plane [Eq. (4)].The inverse transform of Q exists, and is

(4)

and satisfies the condition

Q(x,y) =-- f Q(w,X)eiwxeixvdcdX.

A number of properties accrue to q and Q, individuallyand jointly. In particular, we have

f Q(x,x)dx< oo.

8 C. W. Helstrom, J. Opt. Soc. Am. 57, 297 (1967).9 B. R. Frieden, J. Opt. Soc. Am. 58, 1272 (1968).10 C. W. Helstrom, J. Opt. Soc. Am. 59, 164 (1969).

(5)(12)

which is a direct consequence of the symmetry ofQ(x,y), by which q(x)=q*(x), so that q is real. For the

(10)

(11)

Vol. 591150

q(w)= q(x)e-j1xdx,

VW@) = q* (- W),

PARTIALLY COHERENT OPTICAL SYSTEMIS

full two-dimensional filter,

Q(oX\)= Q*Q, -- ), (13)

which is a consequence of the hermitian symmetryof Q(x,y).

Input-Output Relationship

If a quadratic filter, as characterized by the two-dimensional impulse response Q (x,y) defined above,has as its input a known complex function f(x) (Fig. 1)then the observable output of the filter is g(u), givenby Eq. (1). This representation is equivalent (by alinear change of variables) to

g(u) = /JQ(s,) f(u-s) f* (u -t)dsdt, (14)

VI

where T' depends upon u.The frequency-domain description of the output of

Q is provided by the Fourier transformation of g. Theoutput Fourier transform g is (for infinite ranges)

1 r52 7rJ

which is a somewhat cumbersome relation. The develop-ment of this paper is motivated primarily by the desireto interpret this equation in terms of a more workableexpression.

The Linear Method

For a given quadratic filter Q, the relationshipbetween q and Q is obvious. That an equivalent corre-spondence may be established in the frequency domainis not obvious. In this section, we discuss association ofvariables, and in addition we introduce a comple-mentary operation, augmentation of variables." Thesemethods lead to an exact linear representation of aquadratic nonlinearity.

If Q(x,y) is a quadratic filter, and q(x)=Q(x,x),then q may be obtained from Q by association of vari-ables, an operation defined as

qQo)=- f Q(w-X, QdXA - / Q(n, W-r)dp. (16)27r 27r J

Equation (16) may be obtained formally from Eq. (11)by a change of variables.'2

The indicated operation of Eq. (16) is a convolutionintegral. This is not surprising, as the Fourier transformof a product of two functions is expressed in terms ofthe convolution (in the frequency domain) of theindividual transformations. Thus, the relationship be-

" R. K. Raney, Ph.D. thesis, Report No. 69-12, 218, UniversityMicrofilms, Inc., Ann Arbor, Mich.

12 A more precise development of Eq. (16) is available in Ref. It.

FIG. 1. Model of quadratic filter. f D g

tween q and Q is an extension of a basic property ofone-dimensional Fourier-transform theory. The sym-metry of Eq. (16) is consistent from this point of viewas well.

It is often more convenient to operate with a simplefunction of two variables than with a complicatedfunction of one variable. We therefore introduce anoperation that complements association. We defineaugmentation of variables as follows. Given a functionof one variable g(x), we extend it by augmentingvariables to a new function G(x,y) that has the proper-ties: (1) G is the weighting function of a quadratic filter;(2) G(x,x)=g(x).

Of course G is not unique; but when the function gto be augmented represents the output of a quadraticfilter, the appropriate choice is obvious, as discussedbelow.

We are now in a position to consider an alternativerepresentation of the Fourier transform of the output ofa quadratic filter. From Eq. (1), we extend the outputrepresentation g to a two-dimensional form by augment-ing variables,

g(u) -G(uv); (17)

from Eq. (14), an obvious choice for augmentingvariables is

G(uv) = f TQ(xy)AU (-x)f*(v-y)dxdy, (18)

which we define to be the canonic augmentation. Theresulting two-dimensional function G, we define to bethe dilinear extension of g."

The two-dimensional Fourier transform of thedilinear extension G is

G~csX)1(ci)J*(X)JJ Q(xy)e-jwxe-xydxdy,7,

which for infinite limits goes directly to

G(ckX) = Q(W,)J(W)fr(-)X),

(19)

(20)

which is an important result.14 For fixed X, this corre-sponds to the input-output transfer function in co of alinear filter, and conversely. With both X and X variable,this is a two-dimensional (linear) transfer-functionrelationship. The canonic augmentation leads to this

1 G is not bilinear in (u,v). The terminology "dilinear" isintroduced to represent the specific two-variable extension of asingle variable function that allows the quadratically nonlinearfilter of this discussion to be analyzed and synthesized by linear-transfer-function techniques.

"The linearity of Eq. (20) is not dependent on the infinite-limit assumption.

September 1969 1151

R. KEITH RANEY

U A GB

FIG. 2. Conceptual map of methods. Top row: two-dimensionallinear functions. Bottom row: one-dimensional quadraticallynonlinear functions. Left column: frequency space. Right column:state space.

linear extension of the quadratic filter, providing thesimplification we have been seeking.

We should verify that the two-dimensional frequency-plane product (Eq. 20) leads to the one-dimensionalFourier transform of the observable output. From Eq.(16), association of variables, we have

g(w) =- w-7)dn' (21)

which by substituting Eq. (20) becomes

I'2(=-J, w-v)f(X)r(ij-c)d\, (22)22r22J

which is identical to the result (Eq. 15) derived bythe direct calculation.

Summary

This formulation is summarized in Fig. 2. The one-dimensional observable output of a quadratic filtermay be augmented to two variables, which has theeffect of extending the quadratic nonlinearity to anequivalent two-dimensional linear filter. Subsequentinput-output operations may be expressed linearly intwo variables. Recovery of the one-dimensional non-linear form may be accomplished at any stage byassociation of variables.

APPLICATION TO PARTIALLY COHERENTOPTICAL SYSTEMS

The object of the following discussion is to show thatthe essential viewpoint developed above for quadraticfilters may be directly applied to the imaging equationof a partially coherent optical system. This equation,which is well known,4 -5" 5 we shall consider with thefollowing customary assumptions: (1) the illuminationis assumed to be quasimonochromatic; (2) the complex-amplitude impulse-response function h is spatially in-variant over the isoplanatic patch being considered;and (3) the complex-amplitude transmittance of theobject is represented by f.

With these assumptions, the imaging equation of anoptical system may be written in terms of the functions

1" M. Beran and G. B. Parrent, Jr., Theory of Partial Coherence(Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964).

fi and Jo of the object illumination and of the imagecomplex amplitude respectively, and is

JM(uv) ff (U-x)12* (v y)

Xf(x)f* (y)Jo(xy)dxdy, (23)

which is the one-dimensional form of the imagingequation for an object transilluminated by a partiallycoherent source.'

The function J(x,y) has been known as a mutualintensity function. It is the spatial part of the mutualcoherence function for quasi-monochromatic light.5 Inan attempt to be faithful to the concept of standardterminology and still retain a semblance of the earliernomenclature, we shall denote J(x,y) the mutualirradiance function.

If we formally let f= 1, then Eq. (23) represents thecorresponding imaging equation for a self-luminousobject whose mutual irradiance function is Jo. In theself-luminous case,

Ji(uv) =J h(u-x)h*(v-y)Jo(x,y)dxdy,

which has a two-dimensional Fourier transform

Ji(w,X) =Jo(w,X)k(c4A*(-A),

(24)

(25)

so that the mutual irradiance of a self-luminous (one-dimensional) body is propagated as by an equivalent-(two-dimensional) linear filter to the mutual irradianceof the image. This pregnant observation, previouslyconceived by Hopkins for self-luminous systems,'6 isextended below to the transilluminated case by use ofthe quadratic filter theory introduced above.

Transillumination and Quadratic Filters

In the transilluminated case, it is helpful to rewriteEq. (23) to group respectively the system terms andthe object terms. Thus, for spatially stationary illumi-nation, the mutual irradiance at the image plane is

Ji(uv) =fJ[ y(x-y)h(u-x)h1*(v-y)1

Xf(x)f*(y)dxdy, (26)

where h characterizes the optics, and f represents theobject. The correlation function y is the complexdegree of coherence, which, for the purposes of thispaper, is essentially the stationary normalized form ofthe mutual irradiance function Jo of the irradiance ofthe transilluminated object f.

Now we are interested in the Fourier transform ofthe output image. One procedure would be to take the

16 H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

1152 Vol, 59

PARTIALLY COHERENT OPTICAL SYSTEMS

two-dimensional Fourier transform of Ji(uv), just aswe did above, but this does not lead to a linear de-composition analogous to Eq. (25). The customarymethod is to calculate the one-dimensional Fouriertransform of Ji(u,u). This in turn leads to the Fouriertransform of the term in the square brackets in Eq.(26), which Wolf denotes the "transmission crosscoefficient," describing in the frequency domain thecombined effect of illumination and the optics.5 Thisdirect transformation, however, does not suggest aninterpretation in terms of a higher-dimensional linearfilter.

An alternative approach is to recognize at the outsetthat we are usually interested not in the full correlationfunction at the image plane, but in the irradiance.Quadratic-filter techniques may be applied to the outputirradiance, the result of which is to obtain an equivalentdilinear representation for the optical system. For atransilluminated object, the output irradiance (theobservable quantity) is

Ji(u,u) =ffE (X y)h(u-x)Iz*(u-Y)]

Xf(x)f* (y)dxdy. (27)

We denote the terms within the brackets in Eq. (27) as

Q(a,b)=-y(b-a)h(a)h*(b). (28)

Now Q is a two-dimensional function that representsthe optical system, including the complex-amplituderesponse h and the stationary spatial coherence -y ofthe illumination. The function Q is the weightingfunction of a quadratic filter as defined above; it ispositive definite and, for physically meaningful opticalsystems, it is integrable.

We substitute the quadratic-filter representation ofthe optics and illumination [Eq. (28)] for the bracketedterms in Eq. (27) so that the image irradiance may bewritten

g(u) =Ji(u,u) = ffQ(s-x, u-y)f(x)f* (y)dxdy, (29)

where for consistency we have let the image irradiancebe represented by g(u). We canonically augmentvariables in Eq. (29) to obtain the dilinear extensionon G of the image irradiance g, as in Eq. (18) above,yielding

G (u,v) JfQ(u-x, v-y)f(x)f*(y)dxdy. (30)

The two-dimensional Fourier transform of this is ofcourse

G(W,))= =(@,)f Mf*-X), (31)

which is the simplification we have been seeking forthe transilluminated system.' Note that this equation

is formally comparable to that of the self-luminouscase, Eq. (25), except that the roles of the one-dimen-sional and two-dimensional functions are reversed. Inthe self-luminous case, the object is represented byJo(wX), and in the transilluminated case the object isdescribed by J(o)f(- X. This is a consequence ofascribing the illumination correlation to the object inthe former, and to the optical system in the latter case.

Note also that the left-hand side of the transfer-function equations for the self-luminous case [Eq. (25)]and for the transilluminated case [Eq. (31)] are for-mally similar, but different in meaning. In the self-luminous case, it represents the two-dimensional spatialFourier transform of the output mutual irradiancefunction, whereas in the transilluminated case, it is thetwo-dimensional spatial Fourier transform of an ana-lytically convenient but physically artificial function.

The Fourier transform of the image irradiance maybe calculated from the dilinear extension [Eq. (31)] byassociation of variables. Thus, from Eq. (22),

1g(w)=- IQ%, X ?)J(n)J*(n1w)dn

2wJ(32)

is the Fourier transform of the observable object.In order to represent explicitly the roles of the

coherence and of the complex-amplitude response, theFourier transform of Q as defined by Eq. (28) is re-quired. Proceeding by the usual steps, from thedefinition

- O(SA)= ff y(y-x)h(x)h*(y)e-jrxe-Jydxdy, (33)

we may derive

Q(a1>,)=- / y27r

(34)

as one useful form of the transform of the quadratic-filter representation of an optical system, in terms ofthe transforms of the coherence, 5, and of the amplitudefunction, h. Equation (34) is the (one dimensional)transmission cross coefficient defined by Wolf.6

To complete characterizing the output of a partiallycoherent transilluminated system, we substitute Eq.(34) into Eq. (32) to express the output spectrum interms of It and y, yielding

1 JM(@ =-I f (q)7) (Xq-'d)

27r

x I (35)

This is the usual result4' 5"l 5 which normally is derivedby direct Fourier transformation of the image-irradiance

1153September 1969

R. KEITH RANEY

expression [Eq. (27)]. Thus, we have a check on theindirect calculations suggested here.

CONCLUSIONS

We have shown that for quadratically nonlinearsystems, there are two alternatives for representingthe frequency-domain input-output relationships (Fig.2). The usual method obtains the output Fouriertransformation directly, resulting in a nonlinear integralexpression [as in Eq. (16)]. The new method describedin this paper suggests that we should first retreat to ahigher-dimensional representation that is linear (thedilinear extension), for which Fourier transformationyields a true transfer-function input-output formula[as in Eq. (20)]. The usual nonlinear output relationmay be calculated from the dilinear extension byassociation of variables [Eq. (22)].

We have also shown that these general techniquesmay be applied directly to partially coherent opticalsystems. The illumination coherence and amplitude-

response functions of a transilluminated optical systemdetermine the quadratic-filter representation EEq.(28)] together with its Fourier transform [Eq. (34)].The input-output equation of the optical system, interms of the dilinear extension, is linear EEq. (31)],with the system transfer function being determinedby the optics and the illumination. The Fourier trans-form of the observable output may be calculated fromthe dilinear extension as above.

A biproduct of this result is that the nature of thetransmission cross-coefficient has been described. Wehave identified this cross coefficient as the Fouriertransform of the quadratic-filter representation of anoptical system EEqs. (28), (33), and (34)]. Hence, itmay be correctly described as the true-transfer functionfor the dilinear extension of the imaging system. Thisobservation, in addition to the analytical convenienceit affords, interprets its nonlinear convolutional rolein the conventional input-output equation of partiallycoherent imaging systems.

COPYRIGHT AND PERMISSION

This Journal is fully copyrighted, for the protection of the authors andtheir sponsors. Permission is hereby granted to any other authors to quotefrom this journal, provided that they make acknowledgment, includingthe authors' names, the Journal name, volume, page, and year. Reproduc-tion of figures and tables is likewise permitted in other articles and books,provided that the same information is printed with them. The best andmost economical way for the author and his sponsor to obtain copies is toorder the full number of reprints needed, at the time the article is printed,before the type is destroyed. However, the author, his organization, orhis government sponsor are hereby granted permission to reproduce partor all of his material. Other reproduction of this Journal in whole or inpart, or copying in commercially published books, periodicals, or leafletsrequires permission of the editor.

1154 Vol. 59