Use of projection matrices in hologram interferometryKarl A. Stetson

Optics and Acoustics Group (81), United Technologies Research Center, East Hartford, Connecticut 06108(Received 14 June 1979)

Projection matrices that project vectors onto surfaces from given directions are defined. The use ofthese matrices in hologram interferometry is discussed. They are shown to systematize the solutionof a number of computation problems and to clarify the relationship between many of the vectorialparameters.

INTRODUCTION

Of the various aspects of holography, few have withstoodas much analytical assault as the fringes of hologram inter-ferometry. The reason for this may lie in the vectorial natureof the parameters involved: object displacement, illuminationand observation propagations, object surface normals, normalsto fringes, etc. These parameters are often difficult to visu-alize in three dimensions unless one has solved the problemof fringe formation and localization oneself.

There exists, however, a more likely cause for the persistentreworking of holographic fringe analysis. Most of the im-portant relationships between the vectorial parameters aremost clearly described by matrix transformations. Trans-formations that map fields of three-dimensional vectors intocorresponding fields of three-dimensional vectors are oftenparticularly hard to visualize. The introduction of thesetransformations six years ago by Schumann and Dubas", 2 wasperhaps the single greatest advance in the theory of holograminterferometry, and yet, the corresponding concepts are stillnot widely used. It is the purpose of this paper to reiteratesome of the formulations of Schumann and Dubas in thecontext of my own writings with the hope that both lines ofwork may become better understood.

PROJECTION MATRICES: DEFINITIONS ANDPROPERTIES

The matrix transformation, which is the primary concernof this paper, is that which transforms a vector into its shadowon a surface. This may be called a projection matrix, and itmay fall into either of two categories. If the direction fromwhich the shadow is cast is parallel to the surface normal, wemay call the operation a normal projection; if it is not, we maycall it an oblique projection. Naturally, the first is a specialcase of the second, but it is important enough to justify specialdesignation. Some confusion may arise from the term, theprojection of a vector, because it could logically be applied toboth the operation or the result of the operation, i.e., thecasting of the shadow or the shadow itself. The distinctionshould be clear from the context; however, use will be madeof the word shadow in a number of cases to avoid the ambi-guity. With this in mind, let us begin with a mathematicaldescription of normal projections and generalize to obliqueprojections.

Refer to Fig. 1 where A is a vector, 6 is a unit vector normalto a surface, and Ab is the projection of A onto the surfacenormal to 6. Clearly, all that is required to obtain Ab is tosubtract from A its component in the direction 6. The mag-nitude of that component is given by the scalar product L-A

and its direction is that of 6, so the result is

Ab = A - 6(b. A). (1)

We may recognize the right-hand side of Eq. (1) as the ex-pansion of the triple vector product

Ab = -b X (6 x A). (2)

Because the vector product of two vectors yields a thirdvector, vector multiplication may be described as a transfor-mation and represented by a matrix. This is achieved byarranging the elements of one vector as the antisymmetricmatrix

I0 -b, +byb = +b[ 0 -b::

-by +b, 0

Using this matrix, we may rewrite Eq. (2) as

Ab = -bbA = PbA,

(3)

(4)

where P6 = - 66 is the matrix transformation that projectsthe vector A onto the surface which is normal to 6 to form theshadow Ab.

With the foregoing in mind, let us examine Eq. (1) to see ifan alternative form for Pb may be derived. Let us rewrite Eq.(1) in matrix form.

bx 0 0 by b b, ja, 0 0Ab = IA- by 00 10 0 0 ay (U 0

b 0.0- 0 00 [az 0 0(5)

where ax, ay, a, are the components of A, and bx, by, b2 are thecomponents of 6. I is the identity matrix. The product ofthe first two matrices in the second term of Eq. (5) yields amatrix of all nine possible products of three components of6. This operation defines a third type of product between twovectors, in addition to the well known scalar and vectorproducts. It is referred to as a dyadic product by Schumann';however, the word dyadic is often used to denote the result ofthe product, i.e., the tensor or the matrix operator.3A4 It maybe preferable to call the product a matric product of twovectors.3 (The word matric means of or pertaining to a ma-trix.) In any case, we shall adopt Schumann's symbol for it,a cross in a circle ®. Thus, we may rewrite Eq. (5) as

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It is important to keep in mind that in these sequences thedirection along which the projection is made does not change.Finally, let us observe with caution that the sequence of twonormal projections, first onto one surface then onto another,is not the same as an oblique projection. Thus,

(13)

FIG. 1. Normal projection of the vector A onto the surface that is normalto unit vector b.

Ab -(I-6 )A =PbA (6)

from which

Pb=I-6 6 . (7)

It is easy to verify that I - 6 e 6 = £f.

Let us assume that we now wish to project along a direction6 onto a surface that is normal to c. Referring to Fig. 2, wesee that we must subtract from A a component in the directionof 6 such that the result has no component in the direction ofc. The magnitude of that component is greater than c-A bya factor (c-6)-1 so we have

Abc = A-6 (c * A)/(6. ), (8)

where Abc is the projection of A from the direction of 6 ontoa plane perpendicular to c. By the previous reasoning, weobtain

= I - (6 0 c0 * c), (9)

where Pbc is the transformation that projects a vector alongthe direction of 6 onto a plane normal to c. (Note the orderof subscripts b and c is that of 6 ® 6.) Clearly, this equals Eq.(7) when 6 = 6.

It is obvious that the order of multiplication is not inter-changeable for matric products because c ® 6 yields a matrixnot equal to 6 ® 6, but equal instead to its transpose. Fromthis, we may deduce that the transpose of an oblique projec-tion matrix simply exchanges the role of the two unit vectors.Therefore,

P = Pcb, (10)

where Pcb is the transformation that projects a vector alongthe direction of 8 onto a plane normal to 6. All projectionmatrices are singular and, therefore, do not possess inverses.There is one property that they do have that will be exploitedlater. If you project a vector from a given direction onto oneplane, and then project the result from the same directiononto a second plane, the final result is the same as havingprojected the original vector onto the second plane directly.Thus, for example,

IbPbc = ?b

APPLICATIONS

Determination of displacements from multiple hologramsIt is well known that fringe localization, observed with or-

thogonal slit apertures, can be used to determine two com-ponents of object displacement transverse to the observationdirection (e.g., Ref. 5). The vector sum of these two compo-nents may be referred to as the observed object displacement,Lob. If fringe parallax is used, and absolute fringe order isunknown (as with double-exposure holograms), the accuracyfor determination of the component of object displacementin the viewing direction may be poor. This is especially trueif the hologram is not very large. 6' 7 In this case also, knowl-edge of the object displacement may be limited to its compo-nents normal to the average viewing direction. The totaldisplacement of the object L may be obtained, however, if asecond, independent hologram can be analyzed to yield theobserved object displacement from a different direction.Projection transformations allow a systematic solution to suchproblems.

Let a and 6 be unit vectors in two viewing directions foreach of which we have determined the two correspondingtransverse components of object displacement, Lobl and Lob2.Each of these is related to the total displacement vector, L,by a projection matrix:

Lobl = PaL,

and

Lob2 = VbL.

Neither equation by itself can be inverted to yield L becauseof the singularity of the projection matrices. The equationsmay be combined, however, to yield an overdetermined set ofequations

[L~bll = [PalLbl v. L.

LLob2l = [?ib(14)

(11)

or

PbaPbc Pb.N

FIG. 2. Oblique projection of the vector A, along the direction of b, onto(12) the plane normal to c.

1706 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979

1' Pb 0l Pab-

Karl A. Stetson 1706

Multiplying both sides of Eq. (14) by the matrix [PTP b] re-duces the system to a set of three equations with three un-knowns. Solution of that system yields a value for L whosetwo projections onto planes normal to d and 6 have the leastsquare error with respect to Lob, and Lob2. If we note that thetwo projection matrices Pa and Pb are symmetric, and,therefore, Pa Pa = Pa Pa, we may write the solution as

L = [PaPa + PbPebY(PaL 0bl + PbLob2). (15)

Applying the results of Eqs. (11) and (12), we see that Pa Pa=Pa and PaLob1 = Lbi. Equation (15) simplifies, therefore,to

L = [P + Pbh'((Labl + L06 2). (16)

where Ro is the vector from the origin to a nearby point on theobject surface, LT is the vector translation of the object at theorigin, Kf is the fringe vector, and VR is the gradient operatorin real space. Thus, we may formally identify the fringevector as the gradient of the fringe locus function.

Kf = VR(K- L). (22)

Examination of the gradient of the scalar product in Eq. (22)shows that it may be written compactly with the aid of twomatrices, f and g, composed of the nine first derivatives of thecomponents of L and K, respectively. The result is

where

The process may be extended to any number of views by

L = [ZPi]-1(YLabi) (17)

Physically, 2Lobi is the vector sum of all the observed objectmotions. What physical significance if any can be given to2Pi, the sum of the projection matrices, as yet eludes thisauthor.

Definition of fringe vectorsProjection matrices are quite useful in the definition of

fringe vectors. To the extent that the deformations in someregions of an object may be considered homogeneous, thefringes on that region of the object surface appear to be in-tersections of the object with fringe lamina that are equallyspaced and perpendicular to some vector. If we give thatvector a magnitude equal to 7r divided by the interlaminaspacing (for cosine fringes), we may define it as a fringe vector.The scalar product of the fringe vector with the space vectorthat defines points on the object surface generates the functionwhose constant values define fringe loci. Let us review theformalism and notation involved. Let Iohs be the irradianceof the image observed through the hologram and Iobj be theirradiance of the original object. Then

Iobs = M2 (Q)Iobj (18)

where M(O) is the characteristic fringe function that mul-tiplies the field reconstructed by the hologram. The fringelocus function, Q2, is the scalar function, constant values ofwhich define fringe loci. It is definable as the scalar productof the object displacement, L, and the sensitivity vector K,i.e.,

Q = K - L. (19)

The sensitivity vector is the vector difference between theobservation propagation vector, K2 , and the illuminationpropagation vector, K1, i.e.,

K = K2 - K1. (20)

Let us locate the origin of a coordinate system at some pointon the surface of the object and expand the fringe locusfunction as a Taylor series about that point. The result canbe shown to be8

Q = Q(0) + VR(K * L) * Ro = K * LT + Kf - Ro, (21)

1707 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979

and

(23)Kf = Kf+Lg,

[Lx LY Lz

=Lx LY Lz

X kY kZLx LY Lz1

g = kx ky k1kx ky kz

(24)

As shown by Schumann and Dubas,1 2' these matrices may bewritten with the matric product notation as

fT = VR 0 L,

and

gT= VR a K. (25)

(Note that the matrices used here are transposes of those usedby Schumann and Dubas. This arises from the conventionused in the original definition of the fringe vector. 5)

The matrix f describes the homogenous components ofstrain and rotation of the object via its symmetric and skewsymmetric parts. The components of the matrix g can beevaluated directly 8' 9 or may be described by the more elegantnotation of Schumann and Dubas, using matric products,as

g = (k/Rob)[I - A2 A2] - (k/Ri1 1 )[I - A1 0 A1]. (26)

k is the wave number, 27r/X, Ro is the radius of the observer'sspherical perspective, Rill is the radius of the spherical illu-mination, and A2 and A 1 are unit vectors in the observation andillumination directions. The two matrices in Eq. (26) maybe recognized as projection matrices, which operate on theobject displacement via Eq. (23). Thus, we may write thefringe vector as

Kf = Kf + h(Lob/RQb -Lil/Rili). (27)

The radii require a sign convention and will be consideredpositive if the light diverges from the illumination source andconverges toward the observer.

The last term of Eq. (27) offers a very convenient way todescribe the component of the fringe vector due to objecttranslation under spherical illumination and perspective. Inparticular, let us consider the two radii to be equal to R and

Karl A. Stetson 1707

FIG. 3. Projections involved in complete fringe localization. Both theobject displacement L and the fringe vector Kf are projected onto an ob-servation surface that is normal to the viewing direction, k2 . Whereas theobject displacement is projected normally, the fringe vector is projectedobliquely along the normal to the object surface. Complete localizationcan occur only if the resulting projections are colinear.

let us consider the strains and rotations to be zero. Equation(27) reduces to

KJ = (k/I?)f/(kQ - L) - k2G 2 L)]. (28)

If a translation is to be used to generate contour fringes on theobject,10 it would be desirable to know in what direction totranslate the object so as to make the fringes correspond todepth from the observer. For this condition, Kf must beparallel to k2, and Eq. (28) shows that this condition is met ifthe translation is perpendicular to the illumination. Withunequal radii, or with added rotations, this would not betrue.

Conditions for complete localizationProjection matrices may be used to describe the condition

that must be fulfilled in order to have complete localizationin hologram interferometry. Simply stated, that conditionis that the observed object displacement, Lob, times the wavenumber k equals the observed fringe vector, Kfob, times thedistance to the object surface from the localization region D.(D is positive in front of the object surface.) The observedobject displacement has been defined earlier. The observedfringe vector has been defined in a previous paper,' but quitesimply, if the fringes formed on the object are projected ontoa plane normal to the observation direction, the observedfringe vector lies perpendicular to them and has a magnitudeequal to 7r divided by the observed fringe spacing (for cosinefringes). Inspection of the relationship between the truefringe vector (defined in the previous section) and the ob-served fringe vector" shows that they are related by obliqueprojection along the surface normal onto the plane perpen-dicular to the observation direction. Using projection ma-trices, we may write the condition for complete localizationas

kPk2 L = DPrk 2KfJ (29)

in terms of the actual object displacement and the true fringevector. Figure 3 shows a view of these projections.

1708 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979

Determination of true fringe vectors from observed fringevectors

Fringe vectors can be used to extract strains and rotationsfrom the fringes formed on objects that have undergone ho-mogeneous deformations. ' 2 8 The fringes seen in the photo-graphs of such objects may be easily characterized by observedfringe vectors, whereas the corresponding true fringe vectorsare required for strain analysis. These may be obtained onlyif the object is suitably three-dimensional, and methods havebeen published for this in Refs. 8 and 12. These methodsrequire, however, knowledge of the spatial coordinates of thefringes, in three dimensions, on the surface of the object. Ifthese are not readily obtainable, it is possible to work directlywith observed fringe vectors and the normals of the objectsurfaces on which they were observed. Let us determine twoobserved fringe vectors, Kfobl observed on a surface perpen-dicular to A and Kfob2 observed on a surface perpendicularto 42.

Both are related to the true fringe vector by correspondingoblique projections, and from these relationships we may formthe overdetermined system of equations

Kf 1 b Kf.1Kf~b21 = 1?fllk2(30)

The solution for Kf that gives least square error in fitting theobserved fringe vectors can be written as

Kf [1P2n1Pn1k2 + ?k2n2Vn2k2]

X (Ph2nl1 Kfob1 + Pk2n2Kfob2), (31)where we have substituted Ph2n = PTk2

A somewhat more interesting relationship results frommultiplying each observed fringe vector by the matrix thatprojects it onto the object surface. By definition, both theobserved fringe vector and the true fringe vector have acommon projection onto the object surface, Kp. Thus, wehave

PnKfob = K, = PNKf. (32)

Observation on two or more object surfaces leads to an over-determined set of equations that has a least-square-error so-lution in the same form as Eq. (17),

(33)

Physically, ZKpi is the vector -sum of the fringe vectors on theobserved surfaces of the object.

Determination of surface strainsWhen the object deformations are not homogeneous over

the entire body under study, they may nonetheless be ap-proximately so for small sections of the object. Because onlythe surface can be observed with conventional hologram in-terferometry, this approximation will restrict us to determi-nation of the average surface strains in a small region of theobject surface. Projection matrices are very helpful in for-mulating the solution to this problem.

Let us review the general procedure and then specialize itto the solution of surface strains. For any particular illumi-

Karl A. Stetson 1708

Kf = [1 'nJ_1(FKpi)-

nation and observation direction (i.e., for any sensitivityvector) we may determine a fringe vector by the methods citedabove. We may also determine fringe order number for thepoint on the object chosen for the origin of the coordinatesystem, to within an additive constant. (For vibrating objects,which yield Bessel function fringes, the additive constant maybe zero, and this will simplify some of the following calcula-tions.) We set up a system of equations of the form of Eq. (27)and solve for the matrix of coefficients f. To reduce mea-surement errors, we usually use an overdetermined set andsolve for f as a best fit for least-square-error.

To set up the system of equations, however, we must de-termine the object displacement at the origin of the coordinatesystem. This proceeds from the combinations of fringe ordernumber and sensitivity vectors. Let us call the additiveconstant QO and bear in mind that it comes from the lack ofknowledge of the absolute fringe order. We now express theobserved fringe order as the scalar product of two four-di-mensional vectors,

Q = (kx, ky, kz, 1) (Lx, Ly, L2, o). (34)

We must make four or more determinations of relative fringeorder, Q, corresponding to four or more independent sensi-tivity vectors (i.e., the tips of the sensitivity vectors must notlie on a single plane). We arrange the values of Q into a vector,Q, and the four-dimensional sensitivity vectors into a matrix,[K,1] = G. The solution for the vector (LQo) is

(LQo) = [GTG]-1GTQ. (35)

(Note that the usual methods eliminate the unknown QO fromthe system of equations by subtracting one member of the setfrom the rest, or by subtracting pairs of equations.1 3 Themethod given above is preferable because, unless care is ex-ercised in the subtraction process, the effects of some mea-surement errors can be escalated.' 4)

Once the object displacement has been determined, it ispossible to form, for each sensitivity vector, the projectionsappearing in the last term of Eq. (27), and to arrange them inmatrix form. The sensitivity vectors and the fringe vectorsmay also be arranged in matrix form, and the solution for thestrain rotation matrix can be written as

f = [KT K'-1KT[Kf - (k/Rob) ob + (k/Rii)LinI. (36)

(Note that the fringe vectors, sensitivity vectors, and objectdisplacement projections form rows in their respective ma-trices.) If Eq. (36) is expanded, it yields f as the sum of threematrices. The first is an apparent strain rotation matrixresulting directly from the fringe vectors, and the sum of thesecond and third yields a correction due to the object's dis-placement within the curved illumination and perspective.Thus, we may write

f = fKf- fLg, (37)

where

fKf= [%TK -1KTKf

and

fLg = I]KTKI-1KT[(k/Rob)Lob-(k/RiY!Li,,I.

When we wish to reduce the problem to a solution for sur-

1709 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979

face strains, we must first determine the object displacementin the center of the surface segment considered. Next, let usconsider the relationship between the surface strain-rotationmatrix, fs, and the complete strain-rotation matrix, f. Thefirst is obtained from the second by setting all derivatives inthe out-of-plane direction equal to zero. This is identical tomultiplying f by the projection matrix, P,. Therefore,

fC = fPn. (38)

Let us consider the effect of this multiplication on the firstterm of the right-hand side of Eq. (37)

fKf pn = [K T K -'K T Kf Pn. (39)

If we examine the final product in Eq. (39), and recall Eq. (32),we see that we may make the substitution

(40)K-fPn = KfobbPn;

that is, we may replace the matrix of true fringe vectors by amatrix of corresponding observed fringe vectors. Thissuggests defining a quantity called the observed strain-rota-

tion matrix, fob,

(41)fob = [KT

K]-KT

Kfob.

The surface strain-rotation matrix may be written in termsof the observed strain-rotation matrix as

(42)Cfs [fob - fLg]Pn.

In the absence of the correction term due to the combinationof object displacement and perspective, the observed strain-rotation matrix and the true strain-rotation matrix have acommon projection onto the object surface. This is a directconsequence of the relationship between true and observedfringe vectors.

SUMMARY

We have defined both normal and oblique projection ma-trices in a systematic way, by making use of matric productsof vectors. They have been shown to be useful in the deter-mination of object displacements, fringe vectors, fringe lo-calization, and surface strains. Aside from some helpfulcodifying of the relationships between the vector parametersof hologram interferometry, the use of these matrices aidsconsiderably in the numerical analysis that must be employedto solve practical problems in this field.

'W. Schumann, "Some aspects of the optical techniques for straindetermination," Exp. Mech., 13, 225-231 (1973).

2M. Dubas and W. Schumann, "Sur la determination holographiquede l'etat de deformation a la surface d'un corps non transparent,"Opt. Acta. 21, 547-562 (1964).

3L. Brand, Vector and Tensor Analysis (Wiley, New York, 1947), p.170.

4H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass.1950), p. 146-149.

5K. A. Stetson, "Fringe interpretation for hologram interferometryof rigid-body motions and homogeneous deformations," J. Opt. Soc.Am. 64, 1-10 (1974).

6E. Ek and K. Biedermann, "Analysis of a system for hologram in-terferometry with a continuously scanning reconstruction beam,"Appl. Opt. 16, 2535-2542 (1977).

7D. Noblis and C. M. Vest, "Statistical analysis of errors in holographicinterferometry," Appl. Opt. 17, 2198-2204 (1978).

8R. Pryputniewicz and K. A. Stetson, "Holographic strain analysis:Extension of fringe vector method to include perspective," Appl.

Karl A. Stetson 1709

Opt. 15, 725-728 (1976).9C. M. Vest, Holographic Interferometry (Wiley, New York, 1979),

p. 112.'ON. Abramson, "Holographic contouring by translation," Appl. Opt.

15, 1018-1022 (1976)."K. A. Stetson, "Fringe vectors and observed fringe vectors in holo-

gram interferometry," Appl. Opt. 14, 272-273 (1975).12K. A. Steston, "Homogeneous deformations: Determination by

fringe vectors in hologram interferometry," Appl. Opt. 14, 2256-2259 (1975).

"Ref. 9, p. 76.'L. Ek, (Private communication).

Wigner distribution function and its application to first-order opticsM. J. Bastiaans

Technische Hogeschool Eindhouen, Afdeling der Elektrotechniek, Postbus 513, 5600 MB Eindhoven, The Netherlands(Received 27 March 1979)

The Wigner distribution function of optical signals and systems has been introduced. The conceptof such functions is not restricted to deterministic signals, but can be applied to partially coherentlight as well. Although derived from Fourier optics, the description of signals and systems by meansof Wigner distribution functions can be interpreted directly in terms of geometrical optics: (i) forquadratic-phase signals (and, if complex rays are allowed to appear, for Gaussian signals, too), it leadsimmediately to the curvature matrix of the signal; (ii) for Luneburg's first-order system, it directlyyields the ray transformation matrix of the system; (iii) for the propagation of quadratic-phase sig-nals through first-order systems, it results in the well-known bilinear transformation of the signal'scurvature matrix. The zeroth-, first-, and second-order moments of the Wigner distribution functionhave been interpreted in terms of the energy, the center of gravity, and the effective width of thesignal, respectively. The propagation of these moments through first-order systems has been derived.Since a Gaussian signal is completely described by its three lowest-order moments, the propagation ofsuch a signal through first-order systems is known as well.

INTRODUCTION

Optical system theory has been well developed in termsof Fourier optics. Among many other papers on this subject,we would like to refer to Butterweck.' On the other hand,geometrical optics, and especially Luneburg's first-order op-tics,2 provides a well-developed system theory, too. The aimof this paper is to present a link between Fourier optics andLuneburg's first-order optics. The link between these twodisciplines is formed by the Wigner distribution function.>-6

In Sec. I we start with a Fourier-optics description of asignal in terms of its complex amplitude, and introduce itsWigner distribution function. We notice a close resemblancebetween the Wigner distribution function and the ray conceptin geometrical optics. For instance, the Wigner distributionfunction of a quadratic-phase signal, i.e., a signal whosecomplex amplitude has a quadratic phase, can be interpreteddirectly in terms of the curvature matrix7 of such a signal.

In Fourier-optics terms, a system can be described by itspoint-spread function. From this point-spread function wederive the double Wigner distribution function of the system,which allows us to formulate the input-output relation com-pletely in terms of Wigner distribution functions.

The concept of the Wigner distribution function is not re-stricted to deterministic signals; it applies to partially coherentlight as well. In the latter case, the Wigner distributionfunction is similar to the generalized radiance, introduced byWalther.8' 9

In Sec. II we turn our attention to a first-order system.2

Such a system can be described in Fourier-optics terms bymeans of a point-spread function having a quadratic phase;in geometrical-optics terms it can be described by means ofa ray transformation matrix. 7 Although derived from Fourieroptics, the double Wigner distribution function of a first-ordersystem can be interpreted directly in terms of geometricaloptics, i.e., in terms of the ray transformation matrix. Forinstance, the description of the propagation of a quadratic-phase signal through a first-order system in terms of Wignerdistribution functions immediately yields the well-knownpropagation law for the curvature matrix of such a signal, inwhich law the curvature matrix in the output plane is pre-sented as a bilinear transformation of the one in the inputplane.

In Sec. III we determine the zeroth-, first-, and second-ordermoments of the Wigner distribution function. The propa-'gation of these moments through a first-order system caneasily be derived. Since the Wigner distribution function andthe ambiguity function1 0 "' are dual representations of a sig-nal,' 2 the moments of the former are strongly related to thederivatives of the latter.

A Gaussian signal, i.e., a signal whose Wigner distributionfunction has a Gaussian form, is completely described by itszeroth-, first-, and second-order moments. Since we knowthe propagation of these moments through a first-order sys-tem, we know exactly how Gaussian signals propagate throughsuch a system. A Gaussian signal can be considered as atwo-dimensional cross section of a Gaussian beam. The re-sults of this paper may, therefore, be helpful when one studies

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