Problem of defocusing in speckle photography, its connection to holograminterferometry, and its solutions

Karl A. StetsonInstrumentation Laboratory, United Technologies Research Center, East Hartford, Connecticut 06108

(Received 19 April 1976)

The movements, in three-dimensional space, of laser speckles observed from diffusely reflecting objects aredescribed. The description connects these motions to relationships between fringe vectors and fringelocalization in hologram interferometry. The results indicate the conditions under which defocusing leads tofalse measurements with conventional techniques, and methods are suggested that could solve the problemencountered due to defocusing when small displacements are to be measured on highly three-dimensionalobjects.

If a double-exposure photograph is made of an object,before and after some deformation, and laser light isused to illuminate the object, then a small displace-ment of the object can be determined by relative dis-placements of the laser speckles observed in the twoexposures. This is often called speckel photography.Since any individual speckle photograph resolves onlythe two components of motion perpendicular to theline of sight, at least two are required to obtain com-plete vectorial displacements. Consequently, the depthof field of both photographs must extend over the entireobject, and this can be done only by reducing the nu-merical apertures of the lenses used. This, however,increases speckle size and reduces the capacity of thetechnique to measure small displacements of the object.Figure I shows a plot of resolution versus depth offield for red light, and indicates the trade-off involved.Hologram interferometry can be expected to yield vec-torial displacements up to about 25 tm. If a depth offocus of 100 mm is necessary, then the resolution limitfor speckle photography would be 250 /im. Measure-ments within the remaining interval (25-250 tam) arelikely to give trouble, and, unfortunately, this is a common

1267 J. Opt. Soc. Am., Vol. 66, No. 11, November 1976

region of interest in engineering (i.e., 0.001-0.01 in. ).

Unfortunately, speckle displacements are observablewith the object out of focus in the same way as when

1000

E

cc 100

101 i 10 100

Depth of Field (mm)

FIG. 1. Plot of resolution vs depth of field in speckle pho-tography.

Copyright ( 1976 by the Optical Society of America 1267

FIG. 2. Geometry used inthe analysis of speckle mo-tions.

K 2

tion comes, in fact, from the requirement that the totalfield and the grating should have the same periodicityat the object surface. Thus, we may define a period-icity vector P by G= MIx P and obtain

h xfP ix K. (2)

Taking the first-order differential of this equation weobtain

the object is. 1 Consequently, defocusing creates falsedata where it would be better to have none. When anobject, which is not in focus, is deformed, it is de-sirable to understand how observed speckles move.Only by this means can we learn both the severity ofthe problem and how to deal with it most effectively.

FIELD PROPAGATION

Figure 2 illustrates the geometry that will be usedin this analysis. A region of the object is illuminatedby laser light with propagation vector K, and is ob-served with a camera from a direction correspondingto a propagation vector K2. It is considered to be lo-cally flat with surface normal ft. The camera is as-sumed to be focused at a distance D0 b away from thesurface of the object. The field illuminating the objectand the field propagating toward the camera are as-sumed to be connected by a diffracting structure in thesurface of the object rather than a scattering one. Thesubject of interest, which will be related to specklemovements, is the change in the propagation directionof the field traveling toward the camera that resultsfrom displacements, rotations, and deformations of theobject. All of the various physical mechanisms thatcould result in diffraction of light from the direction ofK, to that of K2 (first-order diffraction, higher-orderdiffraction, etc., from single gratings, or diffractionfrom intermodulation that arises from summations ofgratings) respond to object changes in the same way asdoes the first-order diffraction from a simple grating.Thus, we shall connect these two fields by a single sim-ple grating on the surface of the object. It is assumedthat the propagation directions are not so altered thatthe portion of the field scattered by the object and ac-cepted by the camera lens changes very much when theobject moves. This condition is essential if the specklepattern before the object moves is to be correlated withthe one after. It would seem that larger camera aper-tures would help this.

We wish to know the change in propagation directionAK 2 of the field propagating in the direction of K2. Thederivation of this in terms of general object deforma-tions presented previously 2 was unnecessarily compli-cated and contained an error of sign. A simpler der-ivation is presented below. The basic, vectorial,grating equation is

G= nx(K2 -KK) ixK, (1)

where G is a grating vector pointing along the grooveswith a magnitude equal to 2ir divided by the groovespacing, and K is the sensitivity vector. This equa-

1268 J. Opt. Soc. Am., Vol. 66, No. 11. November 1976

AnXP+h X AP ft X AK 2,+ A x K,

where we have assumed that AK, = 0.

By factoring we obtain

h XAK 2, f>xAP+Ahx (P-K).

(3)

(4)

P has the same components as K in the object surfaceso that their difference consists only of the componentof K perpendicular to the surface, i, e., (P - K)

t(ii*.K). Thus,

pXAK 2 J=3lXAP+Aflxft(i? K). (5)

To solve for AK2 , take the cross product of K. withboth sides of Eq. (5) and note that AK2 - K2 0 becausethose two vectors are perpendicular. This leads, afterexpansion of the triple vector products, and factoring, to

AK2 =AP -Ail ( * K) - [(AP - (h * K)A.)- K2 , jl/(f * K,). (6)

For the moment let V= AP - Ah!(?i K) so that

AK2 = V-(V K2 )ft/(h K2 ). (7)

We shall now solve for the vector V, which is ex-pressed in terms of the change in periodicity vectorAP, and the change in surface normal Af. To solvefor the change in periodicity vector, consider the grat-ing as a scalar function F, defined by

F=P P R. (8)

This function moves with the object surface when it de-forms, and the deformation gives a new periodicityvector P+AP, and shifts all points on the surface fromP to RixAR. The scalar product of the new periodicityvector and the new space vector must generate the samevalue of the function F. Thus,

F =P- R= (P+AP) (R+AhR). (9)

To first order,

APR= -P. AR. (10)

The vector AR is the vectorial displacement of theobject minus the bulk translation of the coordinate sys-tem. Thus,

AR - R,

where

1=LX LX Lx

VLX L t j

(11)

(12)

Lx, Ly, and L. are the components of the vectorial dis-placement of the object L, and the superscripts denotepartial differentiation with respect to that variable.Substituting Eq. (11) into Eq. (10) gives

AP*R= -PfR= - (Pf) -R.

From Eq. (13) we may conclude that

AP = - Pf.

(13)

(14)

Karl A. Stetson 1268

I

At this point, it is helpful to consider the surface aslying in the x, y plane so that the surface normal 7Pmay be replaced by the unit vector in the z direction, k.From this we find that

Pt= kx + 7 jk(15)

Also, the change in surface normal can be written di-rectly as

n =- -1 Y(16)

since L4 and L4 are rotations of the surface about they and x axes, respectively.

Substituting into the definition of V gives

V = -%(kL, + k Ly + kzLz) - 9'(kLAL + kvLy + kLz). (17)

By inspection,

V= - VR(K. L), (18)

where VR is the gradient operator, i. e.,^ a a a

VR a + a + a

From this

AK, = - VR(K. L) + Vi/G K,)]VR(K. L) . K2]. (19)

Note that, from the definition of V, variations in il-lumination direction AK,, due to object motion and cur-vature of illumination, are included in Eq. (19).

SPECKLE DISPLACEMENTS

We are now in a position to describe the motion ofspeckles in the general three-dimensioqal imageformed by a lens when the object is a diffusely reflect-ing surface illuminated by laser light. It is well estab-lished that where the surface is in focus, the specklesmoves as though attached to the surface, for reasonablysmall motions of the object. 3 The three-dimensionalspace surrounding the image plane of the object surfacecan be thought of, however, as an array of imageplanes corresponding to surfaces in front of and behindthe object surface. At any of these planes, of course,fields from the lens aperture interfere to form speckles.Any set of speckles corresponds to a particular angularensemble of fields. As long as the individual compo-nents of the ensemble retain their relative phases, am-plitudes, and angles, the speckle pattern will remainintact regardless of whether the whole pattern trans-lates or whether the central angle of propagation changes.If the object surface does not translate, but the fieldsleaving the surface propagate in a slightly different di-rection, they will pass through the image plane at acorrespondingly different angle. Thus, rotation of thefield relative to the object surface will correspond torotation of the field about the image surface. Thethree-dimensional array of speckles in the image spacewill shear relative to the image plane, and they willdo so in a manner directly related to the change in prop-agation directions of the fields leaving the object sur-face. When the change in propagation direction is com-bined with a translation of the surface, the resultant speck-le motion will be the vector sum of a translation due to ob-ject translation and a stewing due to the field rotation.

The foregoing has been necessary because the onlyobservable is the motion of the speckles in the image

1269 J. Opt. Soc. Am., Vol. 66, No. 11, November 1976

field, and these speckles do not exist in the field com-ing from the object except in a virtual sense, It ispractical, however, to talk about the motions of theseimage speckles projected back into the object space,mainly to be rid of the problems of magnification anddistortion. Thus, in the following, we shall talk aboutthe motion of speckles relative to the surface of theobject with the understanding that these are virtualspeckles corresponding to real ones in the image plane.We shall refer to the motion of the object speckles per-pendicular to the optical axis as the observed specklemotion, and we shall denote this by the vector Hobo Asstated already, this vector is the sum of the observedobject motion Lb,, and a displacement resulting fromthe stewing of the field. This latter displacement isproportional to the distance between the observationplane and the object surface Dob, and the magnitude ofthe angular change in propagation direction AK2 , and itspoints in the AK2 direction. This leads to the equation

Hub = Lob + AK 2ADob/2 T. (20)

We may, of course, substitute Eq. (19) into Eq. (20),but before we do this, it is helpful to reformulate Eq.(19). The scalar product of K and L is a function, con-stant values of which will define fringe loci in a holo-graphic interferogram. Recently, however, the con-cept of a fringe vector K, has been introduced to de-scribe the formation and appearance of fringes on three-dimensional objects. 4'5 For homogeneous deformationsand rotations of an object, the fringes seen on its sur-face will appear to lie on the intersection of the surfaceand a set of equally spaced planes. These planes maybe characterized by a vector perpendicular to them andwith a magnitude inversely proportional to their spac-ing. Consequently, the scalar product of the space vec-tor, describing points on the object surface, and thefringe vector, is defined as being equal to the fringe-locus function:

K *L = K, * Ro. (21)We may see, by inspection, that the gradient of thefringe-locus function on the surface of the object issimply the fringe vector:

VR(K L) = VR(K, RO) = KfJ. (22)The fringe vector has recently been defined in terms ofthe homogeneous components of object deformation androtation, a 3X 3 matrix we denote by f in Eq. (12), anda similar matrix of first-order variations in the sen-sitivity vector g (Ref. 6):

K, = Kf + Lg.

In Eq. (23), K and L are 1 x 3 row matrices.

(23)

Substitution of Eq. (22) into Eq. (19) yields the interest-ing result that the change inpropagationdirection AK2 isequal to the negative of the observed fringe vector Kfob:

AK2 = - Kf + h(Kf K2)/(f K2) = - Knobs (24)The observed fringe vector is that vector which is per-pendicular to the direction of observation that has thesame projection onto the object surface as the truefringe vector Kf. 5 The observed fringe vector could beextracted directly from the fringes seen on the photo-graph of an object. The interpretation of Eq. (24) isstraightforward. Fringes are seen in the observed

Karl A. Stetson 1269

field as a result of rotations imparted to the field bythe motions of the object. The direction and magnitudeof the field rotations relate directly to the directionand spacing of the fringes. In short, the field rotatesin a direction perpendicular to the fringes and by anamount inversely proportional to the fringe spacing.If we now substitute Eq. (24) into Eq. (20) we have

Hob L Lb - Kfb (XD0b/2DT). (25)

DISCUSSION

Equation (25), together with Eqs. (24) and (23), allowcomplete determination of speckle motions (in the ob-ject space) for arbitrary motions and deformations ofan object, viewed from any perspective and illuminatedwith any analytic wave front. A curious result is ob-tained when Eq. (25) is used to locate those regions inspace where speckles do not move. These are definedby the vector equation

Lob = Kfob (ADb/2T), (26)

which is precisely the equation obtained for fringe lo-calization in hologram interferometry. This leads tothe very logical conclusion that, in hologram inter-ferometry, fringes are localized to those regions inspace where speckles do not move.

The foregoing analysis does indeed confirm that themovements of speckles when the object is not in focusmay be quite complicated. Clearly, the vector Kfob

would have to be determined and its effect subtractedout if Eq. (25) were to be used for measuring objectdisplacements. This vector would be obtained, strictlyspeaking, only from a holographic interferogram of thetwo fields formed before and after the object deforma-tion. For the object displacements usually measurableby speckle photography, however, the fringes wouldvery likely be too close together to be resolvable in aholographic interferogram. Consequently, other ap-proaches must be taken.

One method to deal with the defocusing problem wouldbe to record the speckle fields as holograms and, thus,to preserve their three-dimensional character. Somework along this line has been reported by Adams7 whohas observed transform-plane fringes (also called halofringes) from photographs of reconstructions from con-ventional double-exposure holograms. He reports aloss in contrast of the halo fringes due to the fact thatthe two fields interfere. A logical development wouldbe to record the two speckle fields, one before and oneafter object deformation, either on two separate holo-grams or on the same hologram with widely separatedreference beams. Then, in the photographing of thespecklegram, the two holographic images would be re-constructed sequentially so as to avoid the interferenceof the two fields. The advantage, of course, would bethat a number of different specklegrams could be re-corded with the regions of interest on the object in fo-cus, and with as high a numerical aperture as requiredfor resolving the object motion. This method has thedefect that it does return to the stability problems ofholography, and in the end it does not make strain in-formation so easily available as in the following method.

1270 J. Opt. Soc. Am., Vol. 66, No. 11, November 1976

A simpler, but more elegant, method is available iftwo specklegrams (i. e. , speckle photographs) are re-corded at separate focal planes for the same object dis-placement. The common method of determining thespeckle displacements is to illuminate a small regionof the specklegram with a convergent beam of laserlight. Fringes will appear in the halo of light diffractedaround the point where the beam comes tofocus. Thespacing of these fringes is inversely proportional tospeckle displacement, and they lie at right angles tothe direction of displacement. They are equivalent tothe fringes that would be obtained from two mutuallycoherent point sources separated by a distance equalto the speckle displacement, e. g., in Young's experi-ment. This method lends itself to a determination ofthe speckle displacements at corresponding points onthe two specklegrams, and, after correction of mag-nifications, these would yield two vectors H1 and H2b,in the object space. This would give two equations ofthe form of Eq. (25) for two distances from the objectsurface D' and D%,. Subtracting these two vectorialequations gives

AH, = - Kfob( /2T)D 0 5b- (27)

Thus, with D0 b and AHS known, Kf1 b can be calculated.If either D1 or D2b are known, then Lob may be com-puted. Thus, given two or more independent views ofthe object, each with tandem specklegrams, the vectorial

displacements of the object points could be determined.

One difficulty may occur because there is no way topresume the sign of H 1 relative to that of H2,. Themethod of determining speckle displacements givesmagnitude and direction, but it leaves an ambiguity ofplus or minus in that direction. The sign of one of thevectors must be set arbitrarily, but the vector AHobwill depend greatly upon the sign of the second relativeto the first. With a small AD0b, the large disparity inmagnitude of the two possibilities in AH0, could be usedto rule one out. Otherwise, three tandem specklegramswould be needed to assure the proper choice of relativesigns. After this is resolved, all other relative signsare accounted for if it is kept in mind that De, is posi-tive going toward the observer.

A further benefit comes from this method. We nowhave a method of determining observed fringe vectorsfrom specklegrams and at levels of object displace-ments beyond what hologram interferometry could toler-ate. Recent work with the analysis of hologram inter-ference fringes 6'8 indicates that, with three or morenoncoplanar views of the object, these observed fringevectors could be processed to obtain object strains,shears, and rotations directly. This would appear tobe an extension, and possibly an improvement, of themethods presented by Tiziani for making independentmeasurements of object displacements and tilts. 9

CONCLUSION

The motion of speckles in a photograph of an objectwhich is not in focus can be described by the sum ofthe vectorial displacement of the object and that vectorthat would generate fringes in an equivalent holographicinterferogram. Recording specklegrams in tandem, at

Karl A. Stetson 1270

different focal planes in the image space, provides amethod for not only extracting object displacements butalso extracting strains, shears, and rotations. Record-ing speckle fields in a three-dimensional form withholograms is also a method for solving the problem ofdefocusing in speckle photography.

1E. Archbold and A. E. Ennos, "Laser Photography to Mea-sure Deformation of Weld Cracks Under Load," Nondestruc-tive Testing, pp. 181-184 (1975).

2K. A. Stetson, "The Argument of the Fringe Function inHologram Interferometry of General Deformations, ") Optik31, 576-591 (1970).

SE. Archbold and A. E. Ennos, "Displacement Measurementfrom Double Exposure Laser Photographs," Opt. Acta 19,

253 (1972).4K. A. Stetson, "Fringe Interpretation for hologram Inter-ferometry of Rigid-Body Motions and Homogeneous Deforma-tions, " J. Opt. Soc. Am. 64, 1-10 (1974).

'K. A. Stetson, "Fringe Vectors and Observed-Fringe Vectorsin Hologram Interferometry, " Appl. Opt. 14, 272-273 (1975).

6R. Pryputniewicz and K. A. Stetson, "Holographic StrainAnalysis: Extension of Fringe-Vector Method to Include Per-

spective, " Appl. Opt. 15, 725-728 (1976).7F. D. Adams, "In-Plane Displacement Measurements UsingSpeckle Photographs of Holographic Images, " Report No.AFFDL-TR-75-45 (unpublished) (Wright-Patterson Air Force

Base, Dayton, Ohio 45433).8K. A. Stetson, "Hologram Interferometry," Appl. Opt. 14,

2256-2259 (1975).311. J. Tiziani, "Analysis of Mechanical Oscillations by Speck-

ling, " Appl. Opt. 11, 2911-2917 (1972).

Copyright © 1976 by the Optical Society of America 127111-71 J. Opt. Soc. Am., Vol. 66, No. I 1, November 1976