Speckle methods for the display of motion paths

A. W. Lohmann and G. P. WeigeltPhysikalisches Institut der Universitdt, 8520 Erlangen, Federal Republic of Germany

(Received 12 April 1976)

We use speckle photography to measure the motion of a rough body. We show experimental results for lateraland longitudinal motions. The output of our experiments is a direct display of the motion path. Our methodsare based upon a mixture of concepts from speckle photography and from holography.

I. INTRODUCTION

Speckles occur when light interacts with a rough bodyor with an inhomogeneous transparent medium. Tenyears ago almost everybody would have said that specklesare a nuisance. But a speckle pattern is not chaotic. Itmay carry information about the diffusely reflectingbody, about the inhomogeneous transparent medium,and other interactions of the light as encountered on itsway from the source to the receiver. The specklepattern contains this information in encoded form.Many methods have been invented by Burch, Labeyrie,Tiziani, Francon, and many others to decode the in-formation from the speckle pattern. A complete surveywith extensive bibliography has been edited by Dainty.

The information we want to extract from specklepatterns describes the motion of the diffusely reflectingbody. We will describe several noncontact methods formeasuring the motion of the diffusely reflecting body.The output of our method is a direct graph of the motionof the diffuse body.

Our plan is to show the following: The process ofspeckle photography can be considered as a convolutionencoding process for the motion path. This encodingprocess is reversible if certain precautions are taken.These precautions are based on a concept, borrowedfrom incoherent holography. Next we want to show twoexperimental results, the one relating to lateral motionof the diffuse body, the other one to motion in depth,along the optical axis.

II. CODING THEORY OF SPECKLE PHOTOGRAPHY

To be specific we choose the setup as used byArchbold and Ennos. 2 A monochromatic plane wave hits

1271 J. Opt. Soc. Am., Vol. 66, No. 11, November 1976

the moving rough body. At a finite distance a photo-graphic plate records the speckle pattern in averagetime. No lens is needed.

Archbold and Ennos then take the speckle photographand perform a Fourier transform optically. TheFourier transform pattern contains some clues aboutthe motion of the rough body.

Now we want to describe mathematically the experi-ment by Archbold and Ennos. We call the static specklepattern intensity distribution I0(x). While the roughbody moves, the speckle pattern moves accordingly,without changing its detail structure, provided the mo-tion is lateral. If we call the path of a specific speckledot s(t), then the time integrated speckle intensity dis-tribution Is(x) is

Is (x) =f IJx - s(t)] dt. (1)

This speckle intensity Is (x) is governed by two contribu-tions: the geometrical structure of the rough bodysurface, and the motion of the rough body. We reshapeEq. (1) in order to separate these two contributions:

Iswx = ff IO(ie)6[x -x1 s(t)] dtdx' = JIO(x1) p(x- x')dx'

= IOW * p(x) , (2)

P(x)= f6[x-s(t)]dt.

We see that the time-integrated speckle intensity is theconvolution of the static speckle pattern and the motiontrajectory p(x). The second step of the Archbold-Ennosexperiment provides the spatial power spectrum I Is (v) 12

Copyright ( 1976 by the Optical Society of America 1271

of the speckle intensity [Eq. (3)]:

E[IS (V) 1 2 = I I2

(V) 12 * I AP6) 1 2+

E[ I I0(P) 21 A 6(Y)+ B

(E means expectation value, - means Fourier transfor-mation, and A and B are constants.) Since the powerspectrum of the static speckle pattern I0(x) consists ofa strong central peak and a fairly wide continuous back-ground with some fluctuations, the output of the Archbold-Ennos experiment IIs(v)12 shows the modulus square ofthe Fourier transform of the motion path p(x). Withsome experience in two-dimensional Fourier transformsit is sometimes possible to deduce the motion path p(x)from the observed Fourier transform of Fraunhoferpattern IP(,)12. However, in general, this deduction isnot possible, because the phase of the Fourier trans-form ;(v) is missing.

The phase information can be preserved, if we makeuse of the basic concept of holography, which is: Aknown reference signal has to be added to the unknownsignal before the modulus square process takes place.In our experiment, which is a modification of theArchbold-Ennos experiment, the reference signal con-sists of a static photograph of the speckle pattern, butshifted somewhat to the side. This shift is larger thanthe range of motion, but much smaller than the totalsize of the speckle pattern. The total intensity IT(X) re-corded on the photographic plate is now

IT(X) = S(x) + IR(X),

I, (x) =O(x - x,) = 1o&() *5 ( XR), (5)

IT(X) =I0(X) * [P(X) + 6(X -XR)].

We use a convolution description because it leads to asimple form of the Fourier transform pattern that isthe result of the second step, the Fraunhofer diffractionexperiment:

|I Y(V) 12 = I I0(P) 1 I * (P() + exp(- 27ripJ .XR) 1 2, (6)

I p + exp 1 2= 1+ Ip1 12 + p exp(27rip * x,) + h* exp(- * * * )

This intensity distribution is essentially the same asone would obtain in Fourier holography in the hologramplane, if the object would have the shape of p(x). Thefactor Ilo(v)12 does not change very much due to itssimple shape as expressed in Eq. (4). Hence we usethis Fraunhofer diffraction intensity recording like aFourier hologram, and reconstruct the image in addi-tion to a twin image as shown in Eq. (7),

f IP+expl 2 exp(27riv .x)dx- 6(x)+p(x+xR) +p* (-x+x,).

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Figure 1 illustrates the various steps of our experi-ments. The top picture contains to the left the motionpath p(x), which for simplicity consists of two separatedpoints only. The single dot to the right symbolizes theshifted position of the reference exposure. Of coursethis object does not exist directly. It is contained in-directly in the total intensity distribution IT(x), whichis shown partially in the second picture of Fig. 1. Inthe third picture the Fourier transform pattern of the

1272 J. Opt. Soc. Am., Vol. 66, No. 11, November 1976

A P FOURIER TR.

FIG. 1. Speckle photography in four steps. Upper sketch: themotion path p and the shift of the reference exposure (to theright); below: magnified small portion of the total specklephotograph; below: Fraunhofer diffraction pattern of the totalspeckle pattern (like a Fourier hologram); below: holographicreconstruction from the Fraunhofer pattern.

second picture is shown. Apparently it has the typicalstructure of a hologram. Hence the final Fouriertransform, or holographic reconstruction, yields, sur-rounding the central peak, an image of the motion pat-tern and the additional twin image to the right.

III. MEASUREMENT OF LATERAL MOTIONS

Up until now this project is similar to a previousshort communication about lateral motion measure-ments. 3 Now we modify our method by modulating theillumination according to the function M(t). Accord-ingly, the speckle intensity is

Is (X) =f M(t)ia[x - s(t)] dt. (8)

As a consequence the displayed motion pattern assumesthe form

p(x) = M(t)6[x - s(t)] dt. (9)

If the illumination consists of periodic short bursts, themotion pattern is subdivided into dots whereby the dotdensity is a measure for the motion velocity as is evi-dent from Eq. (10),

A. W. Lohmann and G. P. Weigelt 1272

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PmR - S(o-XR)

IT(R) =

(p(X S(Xx)) M I (X

IT ( X) @ 1T()

FIG. 2. Holographic reconstruction of a lateral motion path(over exposed) of a rough body. The motion path (like a S) wasreconstructed from a multiexposure speckle photograph.

M(t)= 6(t -fT),(n) (1 0)

p(x) =1j 6[x - s(nv)].(n)

We performed an experiment whereby the rough bodymoved along the path of the shape of the letter S. Theresults are shown in Fig. 2, where we see the recon-struction with the center peak, the image, and the twinimage. Here the exposure in reconstruction was rela-tively long. When using a shorter exposure in recon-struction the dots become much sharper, as shown inFig. 3. Figure 4 shows the "hologram" II(v)I 2 fromwhich Figs. 2 and 3 were reconstructed.

IV. MOTION IN DEPTH

In an earlier communication 4 we described threemethods for measuring motions in depth, along the op-tical axis. The first two of these methods were de-scribed in detail and experimentally verified. Briefly,in the first method we looked at the motion in depth

FIG. 3. Holographic reconstruction of the same motion path asshown in Fig. 2 (but proper exposure).

1273 J. Opt. Soc. Am., Vol. 66, No. 11, November 1976

FIG. 4. Fraunhofer diffraction pattern of the speckle photo-graph (acting as a Fourier hologram) from which Figs. 2. and3 were reconstructed.

from an oblique angle, thereby converting the depthmotion essentially into a lateral motion. Hence, es-sentially we could utilize our previous method for mea-suring lateral motions. However, there were somedifficulties encountered because the fine structure ofthe speckle pattern in the photographic observationplane changes when the distance from the rough body tothe plate changes by more than one focal depth XF2.The F means the aperture or stop number. Hence, theallowable depth variations are limited accordingly.

For the first method this depth dependance of thespeckle fine structure was a handicap. But for thesecond method this same depth dependance provided thebasis. By means of optical correlation we measuredthe similarity of two speckle patterns, photographed atdifferent depth locations. The amount of correlationwas a measure for the depth difference.

Now we want to describe the third method. Our goalis to have as output a display of the depth motion. Theabscissa of the display should represent the time, whilethe ordinate represents the depth location of the movingrough body. We want this method to be able to coverdepth ranges far in excess of the depth of focus XF2.Hence it is necessary to record not only one referencespeckle pattern, but several of them at different depthlocations separated by one depth of focus unit each. Inorder to distinguish these different contributions to thereference signal these individual reference exposuresare also shifted by equal amounts in the y direction.The signal intensity I,(x, y) is a stroboscopic record of

A. W. Lohmann and G. P. Weigelt 1273

30 z m t-

y vy t

-1m I v

A REF J =SIGNAL

11tmm

* X

T

, ' U t )

3mm x . t

FIG. 5. Holographic reconstruction of the depth motion graphz(t) of a rough body.

the speckle pattern while the rough object is moving indepth. Simultaneously, the photographic plate is mov-ing with constant velocity in x direction, which enablesus later to have time marks. The signal intensity isdescribed by Eqs. (11) and (12),

IS (X, y) = fM(t)1 0[x - vxt, y, z(t)] dt (11)

=EIo[x- nvxr) ye z (nT)]. (12)(n)

The reference exposure IR(x, y) is described by Eq. (13),

IR(X, y) =EI0(X +XR, Y + mVsY, mMVT). (13)(m)

The constant depth movement velocity during the refer-ence exposure is designed such that the different ex-posures are separated in depth by one focal depth unitaccording to vz- = XF2. The total intensity recorded onthe photographic plate is

IT(X, Y) =Is(XIY)+IR(XIy). (14)

In the second step of this method we produce theFraunhofer diffraction intensity 1I7TI(vx, vY) ,2 as in pre-vious methods. Another Fourier transform yields theoutput as shown in Eq. (15),

ffIYTI 2exp[27ri(xvx+yvy)] dvxdvy=ITOITr=ISO 'R+ ( 15

(1 5)

(o means correlation). The interesting part of the out-put signal is the cross correlation of the signal intensityand reference intensity. Both intensities containspeckle patterns photographed at different depth loca-tions from the rough body. Only those individual ex-posures can correlate that correspond to equal depth

FIG. 6. Geometricai visualizations of the various movementsduring signal and reference exposures for measuring the depthmotion z(t).

locations according to Eq. (16),

z(nT) =MVzT.

From this it follows that correlation peaks will occur atx and y coordinates as shown in Eqs. (17),

x = nVT + XR, (17)

y = mvT = z (nT)v 5 /vX.

In Fig. 5 we show an experimental result. The coordi-nate system for the depth z(t) and the time t have beenlined out. Figure 6 is intended to help in understandingthe multiexposure process leading to the total intensityI,(x, y). The various parts of these graphs describevarious lateral and longitudinal motions that contributeto the multitude of exposure in the total intensity.

In conclusion we can state that speckle patterns arenot necessarily chaotic. They contain information inencoded form as "random carrier modulation," wherethe modulation process consists of a convolution be-tween the motion path and the static speckle pattern.The demodulation process is a correlation of the time-.integrated speckle pattern and of a static speckle pat-tern as reference signal.

"'Laser Speckle and Related Phenomena," in Topics in AppliedPhysics, edited by J. C. Dainty (Springer, Berlin, 1975).

2 E. Archbold and A. E. Ennos, "Two-dimensional vibrationsanalysed by speckle photography," Opt. Laser Technol. 7,17-21 (1975).

3A. W. Lohmann and G. P. Weigelt, "The Measurement ofMotion Trajectories by Speckle Photography," Opt. Commun.14, 252-259 (1975).

4A. W. Lohmann and G. P. Weigelt, "The Measurement ofDepth Motion by Speckle Photography," Opt. Commun. 17,47-51 (19763).

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f l | * l * l | i * l * l l l -

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