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Moore et al. Vol. 27, No. 8 /August 2010 /J. Opt. Soc. Am. A 1863

Imaging of vibrating objects usingspeckle subtraction

Thomas R. Moore,1,* Ashley E. Cannaday,1 and Sarah A. Zietlow2

1Department of Physics, Rollins College, Winter Park, Florida 32789, USA2Paul J. Hagerty High School, Oviedo, Florida 32765, USA

*Corresponding author: tmoore@rollins.edu

Received April 20, 2010; revised June 16, 2010; accepted July 1, 2010;posted July 6, 2010 (Doc. ID 127192); published July 28, 2010

A simple method for imaging vibrational motion is proposed. The process consists of capturing two speckledimages of a region illuminated by coherent radiation. One of the images is captured before the onset of motionand the other during motion. If the mean speckle intensity is below the threshold for detection or above thesaturation intensity of the detector, subtracting the two images produces a high contrast image of the movingregion. A theory is shown to agree well with experimental data. © 2010 Optical Society of America

OCIS codes: 120.7280, 120.6150, 110.6150.

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. INTRODUCTIONhe scientific and engineering communities have been in-erested in imaging small vibrations since the 18th cen-ury, when Chladni first placed sand on a vibrating plateo make the modal patterns visible. Other methods wereeveloped in the latter half of the 20th century that madet possible to optically image sub-micrometer vibrations,ncluding laser Doppler vibrometry, holographic interfer-metry, and electronic speckle pattern interferometry [1].ach of these methods is still in common use today, and

hey have found wide application in situations requiringisualization of small-amplitude motion. Unfortunately,ach of these processes requires expensive equipment andsually requires isolating the object of interest from am-ient vibrations. Laser speckle contrast analysis (LASCA)vercomes these problems by calculating the difference inpeckle contrast that results from the relative motion ofifferent parts of an imaged area, and it can be used inome cases to differentiate between moving and non-oving regions in an image [2,3]. However, while LASCA

oes not require an extensive investment in equipment, itoes require significant image processing and the reso-ution of the resulting image is at best 25% that of the res-lution of the detector.

Here we introduce another method of visualizing vibra-ions that fundamentally differs from those mentionedbove. This method provides large-area imaging oficrometer-sized motion in real time without the need for

xpensive equipment, extensive signal processing, or iso-ating the object from ambient vibrations. The necessaryondition to apply this method is that the motion of thebject of interest can be induced on a time scale that ishorter than that which characterizes the ambient mo-ion. Surprisingly, this requirement is met in many in-tances where the more common methods of detecting vi-rations do not work well. We have tentatively termed therocess speckle subtraction imaging (SSI).To image vibrations using speckle subtraction imaging,

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he area of interest must be illuminated by light with aarrow enough linewidth that the speckle pattern that re-ults from imaging the area of interest has a high con-rast. The region of interest is then imaged onto a detec-or and two images are captured. One image is capturedhile the motion of interest is ongoing, and the other is

aptured either before the motion begins or after it hastopped. The two images are then subtracted in real timeo produce a difference image. As expected, subtractingwo images of the region will produce a completely blackifference image in regions where no motion was induced.owever, since some motion in the imaged area was in-uced during the process of capturing one of the images,he portion subject to motion will produce a speckle pat-ern with lower contrast than will be seen in regions thatere not moving.This reasoning may lead one to believe that merely im-

ging an area that is illuminated by coherent radiationnd subtracting subsequent images captured before andfter the onset of motion will result in areas of motion be-ng visible in the image. Unfortunately, this is not gener-lly the case because ambient motion usually changes thepeckle pattern slightly during the time needed to recordhe two images, or the speckle is not completely resolvedn the detector. Since the mean value of the speckle in-ensity does not normally change with induced motion, ei-her of these situations will result in an inability to easilyifferentiate between moving and non-moving regions.One way to take advantage of the change in the speckle

ontrast to unambiguously detect motion within the im-ged area is to introduce a nonlinearity into the detectionrocess. In speckle subtraction imaging, it is the fact thatll detectors have a minimum intensity below which theyill not record the image that allows regions of motion toecome visible. A detection threshold is a characteristic ofll detectors, so it is not necessary to specify the type ofetector a priori.When the mean intensity of the speckle is below the

010 Optical Society of America

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1864 J. Opt. Soc. Am. A/Vol. 27, No. 8 /August 2010 Moore et al.

hreshold for detection (or above the saturation intensity),nly speckle with an intensity that is in the tail of the in-ensity distribution will be detected. Since a reduction inpeckle contrast occurs when the imaged object is set intootion, because of a reduction in the standard deviation

f the intensity distribution, a change in speckle contrastill result in a change in the mean detected intensity.his will occur even though the mean intensity of thepeckle does not change. Therefore, subtracting an imageith little motion on the time scale of the integration timef the detector from one where some portion of the images moving during the collection process will result in anmage that differentiates the moving portion by a measur-ble change in the recorded intensity. In this way, areas ofittle motion (high speckle contrast) can be differentiatedrom areas where motion occurs (low speckle contrast)ith only minimal signal processing.

. THEORYo investigate the process of SSI theoretically we intro-uce two correlation times. We refer to the correlationime of light scattered from the region during the inducedotion as �1, and the longer correlation time that is mea-

ured either before the motion has begun or after it haseased as �2. The speckle contrast K1 and K2 will be dif-erent for the two cases, but the mean intensity of thepeckle �I� will be the same. The contrast is defined as

K =�

�I�, �1�

here � represents the standard deviation of the inten-ity distribution; therefore, in the case considered herehe contrast is linearly proportional to �.

If the mean value of the speckle intensity is below thehreshold for detection, simply subtracting two imagesill produce an image with nonzero intensity if the sta-

istics of the scattered light have changed between thewo images. Defining the integration time of the detectors T and the time between capturing the two images ast, and assuming that 2T+�t��2, the speckle pattern

rom the region where no motion was induced will nothange significantly between the time that the two im-ges are captured. This condition implies that if there iso onset of motion between two images then they are

dentical, and the subtraction process will result in a newmage with no variation in intensity.

The intensity of any region of interest that is imagednto a detector can be approximated by the mean value ofhe speckle intensity, which is defined as usual by

�I� =�0

�

IP�I�dI, �2�

here P�I� is the probability density function for the in-ensity and I is the intensity. The probability densityunction for the intensity of fully developed speckle of fi-ite bandwith can be written in terms of the speckle con-rast, and this can be used to approximate the probabilityensity in the case considered here. In its approximateorm this function is [4]

PM�I� =�M�MIM−1

�I�M��M�e−MI/�I�, �3�

here ��M� is the gamma function with argument M, and=K−2.As noted above, since the mean intensity of fully devel-

ped speckle will be the same for all values of M, calcu-ating the mean value of the speckle intensity of the re-ion will result in the same value regardless of theorrelation time of the scattered light. In this case a re-ion that is in motion will not contrast sharply with thether areas of the image. However, the detector does notave an infinite response; therefore, the limits of integra-ion in Eq. (2) do not extend from zero to infinity if oneishes to calculate the mean value of the intensity re-

orded by the detector. Rather, the intensity measured byhe detector is calculated by assuming a lower limit thatas some value, IL, that is greater than zero. All detectorslso have a saturation intensity, so the upper limit of thentegration must be some value, IU, that is less than in-nity. Therefore, while the mean value of the speckle in-ident on the detector is given by Eq. (2), the value that isecorded by the detector is not.

The intensity of the SSI image, which results from tak-ng the absolute value of the difference between two im-ges of the same area with different scattering statistics,s given by

ISSI�M1,M2� = ���IL

IU

�I − IL�PM1�I�dI +�

IU

�

�IU�PM1�I�dI�

−��IL

IU

�I − IL�PM2�I�dI

+�IU

�

�IU�PM2�I�dI�� , �4�

here Mn=Kn−2 and n is either 1 or 2 depending on which

ase is under consideration. More succinctly, Eq. (4) cane written as

ISSI�M1,M2� = ��IL

IU

�I − IL�PM1�I� − PM2

�I�dI

+ �IU��IU

�

PM1�I� − PM2

�I�dI� . �5�

If the mean value of the speckle intensity falls betweenL and IU, the effect of having a threshold of detection and

saturation intensity will be minimal. However, if �I�IL, then when M1�M2 the result will be that

SSI�M1 ,M2��0. A plot of the mean recorded intensity asfunction of speckle contrast is shown in Fig. 1 for three

ifferent values of IL. One can conceivably pick any valuef IL to achieve the desired slope of image intensity versuspeckle contrast; however, if �I��IL the number of speck-es with a detectable intensity becomes vanishingly small.n practice, the second term in Eq. (5) can usually be ig-ored and the intensity of the SSI image can be expresseds

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Moore et al. Vol. 27, No. 8 /August 2010 /J. Opt. Soc. Am. A 1865

ISSI�M1,M2� � ��IL

IU

�I − IL�PM1�I� − PM2

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similar analysis can be performed for the case whereI��IU.

To theoretically investigate the utility of SSI in imagingibrations, it is useful to assume a probability densityunction that represents a common situation. Therefore,e assume that the area being imaged consists of a largeumber of independent particles, and that the incident

ight is scattered randomly from numerous particles be-ore being detected. In this case the instantaneous field is

complex circular Gaussian process. We further assumehat at some point in time the motion of the particles cane induced, and that the motion of these particles at thatime can be described as being stochastic. This situationas been studied by Duncan and Kirkpatrick, who havehown that the speckle contrast of the scattered light isiven by [2]

Kn =�n −n

2

2�1 − e−2/n�, �7�

here the normalized correlation time is defined as

n =�n

T. �8�

In practice, it is the visibility of the moving region withespect to a non-moving region that is of interest. The vis-bility is defined as

V = � I1 − I2

I1 + I2� , �9�

here I1 and I2 are the intensities of the SSI image in thereas that represent the moving and static regions, re-pectively. That is, I1=ISSI�M1 ,M2� and I2=ISSI�M2 ,M2�.t is evident that the visibility of the part of an SSI imagehat is in motion is always unity provided 1�2, because

is always identically zero. Unfortunately, this is only

ig. 1. (Color online) Plot of the mean recorded intensity of thepeckle as a function of speckle contrast predicted by Eq. (5) forhree different values of the detector threshold.

2

rue in practice if 2T+�t�2

1. While this can be arranged inome cases, it is more instructive to assume that there isome minimal change in the speckle over the time 2T�t that does not result in a change in the contrast or theean recorded intensity, but will generally result in

hanges in the details of the speckle. This change cansually be attributed to ambient motion. In this case theoint by point subtraction of the two images will result in2 having some small, non-zero value.

For the purpose of theoretically investigating the vis-bility of an SSI image, we will assume that in the ab-ence of any induced motion in the region, the SSI processroduces an image with an average intensity on the orderf one-tenth the mean value recorded by the detector be-ore subtraction. This value appears consistent with thexperiments described below. The visibility of SSI as aunction of 1 is plotted for several different values of 2n Fig. 2 using this assumption. The threshold for detec-ion is assumed to be twice the value of the mean inten-ity of the speckle.

There are two surprising conclusions that can be drawnrom the results shown in Fig. 2. First, the shape of theurves indicates that regions of motion will be visible inhe SSI image even when the value of 1 approaches thatf 2. There is a nonlinear relationship between the vis-bility and 1, but the visibility remains high until 1

2. The second interesting result is that it is possible toetect the motion of an object using SSI even when theres significant ambient motion. Even when the integrationime of the detector is twice the correlation time of theight scattered from the region with no induced motioni.e., 2=0.5), the visibility of a moving region within anSI image can be as high as 0.5.

. EXPERIMENTSo demonstrate that the theory described above accu-ately describes the process of SSI, a 2 cm diameter piezo-lectric disk was covered with approximately 0.5 cm ofne sand. The surface of the sand was illuminated with

ight from a laser with variable output power having aavelength of 532 nm, and the image was projected ontoCCD array using a commercially available camera lens.he integration time of the array was specified by theanufacturer as being 0.0313 s. The piezoelectric diskas driven by a 295 Hz sinusoidal signal with an ampli-

ude that varied from 0–10 V. The light from the laseras expanded using a short focal length lens and pro-

ected onto the area at an angle of approximately 10°. Therea of interest was imaged from directly above; however,here were no discernible effects attributable to imaginghe area at an oblique angle, except for the fact that therea of interest covered a smaller region on the detector.To ensure maximum contrast, the mean diameter of the

peckle was chosen to be close to the pixel size of the digi-ized image, which corresponded to approximately 13 �mn the CCD array. The mean speckle diameter on the ar-ay is given by

d = 1.22�1 + m��f, �10�

here m is the magnification of the image, � is the wave-ength of the light, and f is the aperture ratio of the im-

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1866 J. Opt. Soc. Am. A/Vol. 27, No. 8 /August 2010 Moore et al.

ging lens. The calculated speckle size was approximately2 �m.The amplitude of the oscillation of the piezoelectric disk

s a function of driving potential was determined before-and by viewing the disk through an electronic speckleattern interferometer while it was driven with a sinu-oidal signal of several different amplitudes. Measure-ents of the amplitude of oscillation were recorded as a

unction of driving voltage, and a linear regression of theata revealed the slope to be 297±6 nm/V. In all caseshe light reaching the detector was from the surface of theand, not the piezoelectric disk, and although the drivingotion was harmonic it is reasonable to expect that on av-

rage the motion of the top region of the sand can be ap-roximated as Brownian.The decay time of the motion of the sand was measured

y recording the speckle pattern reflected from the areaith a high-speed camera having a frame rate of 1 kHz.he individual images were studied to determine how

ong it took for the speckle pattern to become static after

ig. 2. (Color online) Plot of the visibility of SSI as a function of1 for three different values of 2. The threshold for detection isssumed to be twice the mean intensity of the speckle �IL=2�I��.

ig. 3. (Color online) Plot of the normalized intensity of SSI im-ge as a function of contrast for three different threshold values.he curves represent the predictions of Eq. (5).

he piezoelectric disk stopped moving. The motion of theand ceased within 8±1 ms. The time between capture ofhe two images was approximately 300 ms, ensuring thatmages taken after the motion had stopped did not exhibithe effects of residual motion of the sand.

In each experiment an image of the surface of the sandas stored before the piezoelectric disk was activated. An-ther image was captured after the disk began to vibrate.he mean and standard deviation of the intensity of theegion was calculated for each case (before motion, duringotion, and after image subtraction) and stored for later

nalysis. The region used for analysis encompassed ap-roximately 57,000 pixels and the results from 50 imagesere averaged for each different value of the driving po-

ential.Once the data had been collected, the visibility of the

SI image for each driving potential was calculated usingq. (9). The intensity I2 was derived from the SSI imagebtained by subtracting two different images of the regionith no induced motion. The intensity I1 in each case was

he average of the SSI intensity in the region that re-ulted from subtracting an image captured during theime that the motion was induced from one captured dur-ng a time with no induced motion. The speckle contrastecorded by the detector, which is defined by Eq. (1), waslso calculated for each driving potential. Using the sameegion to determine both I1 and I2, rather than usingeparate moving and non-moving regions in a single im-ge, eliminated the effects attributable to uneven illumi-ation and ensured that the true value of the visibilityas measured.The results of these experiments are plotted in Fig. 3

long with the predictions derived from Eq. (5) using therobability density function shown in Eq. (3). The onlyree parameter in the fitting process was the ratio of thehreshold intensity IL to the mean value of the speckle in-ensity �I�.

Note that to effectively apply SSI to the study of vibrat-ng objects it is not necessary to know the statistics of thepeckle. All that is required to image a vibrating region inn image is that the contrast of the speckle change whenhe object being imaged is set into motion. Therefore, SSI

ig. 4. Plot of the visibility of the piezoelectric disk as a functionf the amplitude of vibration.

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Moore et al. Vol. 27, No. 8 /August 2010 /J. Opt. Soc. Am. A 1867

an be used in many situations for detection and analysisf vibrations. To demonstrate this, the sand was removedrom above the piezoelectric disk, which was then paintedhite. The visibility of the vibrating surface alone waseasured and is plotted as a function of amplitude of vi-

ration in Fig. 4.

. CONCLUSIONShe data presented above indicate that speckle subtrac-ion imaging may be useful in situations where imagingibrations is necessary, but the more common imagingechniques are difficult to implement. As can be seen inig. 2, the integration time of the detector can exceed theecorrelation time attributable to ambient motion by asuch as a factor of two and still result in high visibility.

n the experiments reported here, no effort was made toontrol the integration time of the detector. However, theddition of a camera with a variable integration time willrovide considerable flexibility in controlling the normal-zed correlation times , and will expand the applicabilityignificantly.

Speckle subtraction imaging is a simple, inexpensiveethod that requires minimal equipment and limited ex-

erience to implement. Yet it can provide high-resolution

mages of micrometer-sized motion of objects under aide variety of conditions in near real time. Although SSIoes not provide a precise measurement of the amplitudef vibration, the visibility provides an estimate of the am-litude. Furthermore, the motion under study need not beontinuous or harmonic, and therefore SSI can be used tomage transient motion and differential flow.

CKNOWLEDGMENTShis work was supported by National Science Foundation

NSF) grant no.0551310 and the generosity of Christineenore Barensfeld.

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measurements,” in Handbook of Acoustics, T. Rossing, ed.(Springer, 2007), pp. 1101–1123.

2. D. D. Duncan and S. J. Kirkpatrick, “Can laser speckleflowmetry be made a quantitative tool?” J. Opt. Soc. Am. A25, 2088–2094 (2008).

3. A. R. Fercher, “Velocity measurement by first order statis-tics of time-differentiated laser speckles,” Opt. Commun.33, 129–135 (1980).

4. G. Parry, “Speckle patterns in partially coherent light,” in-Laser Speckle and Related Phenomena, J. C. Dainty, ed.

(Springer, 1984), pp. 77–122.