Electronic speckle pattern interferometry ~ESPI! is awell-known, nondestructive, whole-field techniquefor measuring displacements. It is based on televi-sion camera recording of speckle interference pat-terns that provide correlation fringes, which can befurther related to deformations or vibrations of a testobject, for example.1The recent inclusion in ESPI of pulsed lasers with
high repetition rates ~up to 60 Hz! has extended therange of engineering problems that can be solvedwith this technique. Perhaps one of the most flex-ible of such lasers is the twin-cavity, diode-seededNd:YAG laser working at either 50 or 60 Hz. Thislaser is composed of two identical laser cavities thatemit pulses that can be synchronized with thetelevision-frame rate2 and with the vibrating testobject in the case of harmonic-vibration analysis.For studying vibrations, two modes of laser oper-
ation are possible: namely subtraction and addi-tion. In the first mode the laser is usuallyoperated with one cavity in the single-pulse mode,and the speckle interferograms are recorded on sep-arate television frames and then are subtracted in aframe grabber. The resultant correlation fringesare displayed on a television monitor. The sub-
traction operation eliminates most of the stationaryoptical noise, for example, diffraction patterns fromdust particles on the optics, which facilitates theuse of temporal phase-shift techniques to automatedata analysis.3In the subtraction mode the time separation be-
tween the correlated exposures is approximately 16.7ms, that is, one television-frame period, and the tech-nique remains sensitive to environmental instabili-ties, especially when measurements are to be carriedout in an industrial environment. This limitationcan be overcome with the laser in the twin-pulsemode, in which it fires twin pulses with a very shortpulse separation within a single television frame.The twin interferograms are added on the televisioncamera sensor. The resultant addition correlationfringes contain stationary optical noise that reducesthe fringe visibility and the signal-to-noise ratio. Toovercome this problem, some authors have proposedthat fringes of improved visibility can be generated bysubtracting two or more succesive addition fringepatterns.4 However, as for the single-pulse subtrac-tion mode, the proposed technique remains sensitiveto environmental disturbances, although continuous-updating subtraction schemes can be used succes-fully when the environmental disturbance developsslowly with respect to the television-frame rate.5We present the results of an investigation into themethod by which addition-fringe visibility can be in-creased by digitally filtering an addition interfero-gram. A comparison is made with an existinganalogue filtering method.
1 May 1997 y Vol. 36, No. 13 y APPLIED OPTICS 2783
Fig. 1. Computer-generated addition-fringe pattern and enhancements: ~a! speckle pattern, where a 5 0 and w 5 2p~2! ~x2 1 y2!; ~b!threshold; ~c! local averaging; ~d! Sobel; ~e! variance filters.
2. Basic Theory
Let us consider a pulsed ESPI system, which is usedto study an object undergoing harmonically inducedvibration. The intensity of the interferogram gen-erated at the television-camera faceplate by thefirst laser pulse, fired at t 5 t1, can be approxi-
2784 APPLIED OPTICS y Vol. 36, No. 13 y 1 May 1997
mated by1
I1~x, y! 5 Io~x, y! 1 Ir~x, y!
1 2ÎIo~x, y!Ir~x, y!cos@c~x, y!#, (1)
where Io and Ir are the intensities of the object andthe reference beams, respectively, and c is the
Table 1. Fringe Contrast for the Methods Described: Threshold, Local Average, Gradient, and Variance Filtersa
Filters
Simulation Experimental
Without Low-Pass Filter With Low-Pass Filter Without Low-Pass Filter With Low-Pass Filter
Original A 0.01 0.00 0.00 0.00Threshold Aa 0.13 0.23 0.13 0.15Local average As 0.15 0.21 0.23 0.24Gradient Ag 0.18 0.24 0.19 0.20Variance Av — 0.34 — 0.39Electronic — — 0.30 0.26
aFringe contrast after electronic filtering is shown for experimental results only.
random-speckle phase. At t 5 t1 the vibration am-plitude and rigid-body motion may be set to zero.If a second pulse is fired at time t 5 t2 within the
same television frame, the interference relation willbe
I2 5 Io 1 Ir 1 2ÎIoIrcos~c 1 w 1 a!, (2)
where w~x, y! is a phase term that describes the har-monically induced deformation and a~x, y! is a phaseterm produced by the rigid body motion of the target.The ~x, y! dependence of each term has been droppedfrom Eq. ~2! and subsequent equations. The rigidbody motion is assumed to be sufficiently small so asnot to decorrelate the two interferograms. AddingEqs. ~1! and ~2! yields
A5 2~Io 1 Ir! 1 4ÎIoIrcosSc 1w
21
a
2DcosSw2 1a
2D . (3)
The term 2~Io 1 Ir! in Eq. ~3! represents stationaryoptical noise and reduces the fringe visibility. Anexample of such pattern when a 5 0 is shown in Fig.1~a!. The target is a rectangular metal plateclamped at its edges and excited in its fundamentalmode of vibration at 320 Hz.
3. Noise Reduction of Addition ESPI Fringes
In this section we describe four filtering techniquesused to enhance addition visibility by reducing theterm 2~Io 1 Ir!. To test the performance of the fil-ters, we use the fringe contrast, C, defined as1
C 5^Imax& 2 ^Imin&^Imax& 1 ^Imin&
, (4)
where ^Imax& and ^Imin& are average intensity valuescalculated along fringe maxima and minima, respec-tively, in the addition-fringe pattern. The positionsof fringe minima and maxima were determined froma fringe pattern calculated without speckle noise, i.e.,Io 5 Ir 5 constant, and c 5 0 in Eq. ~3!. The filter isconsidered to be better as C approaches unity.The fringe patterns were simulated for Eq. ~3!, with
c, Io and Ir taken as random variables with valuesuniformly distributed over the intervals @2p, p#, @0,Im#, and @0, rIm#, respectively, where Im is a constantvalue and r is a normalized visibility parameter. Imand rIm are related to the visibility of the specklefringe pattern, with its maxima when r 5 1.
As an example, Fig. 1~a! shows a simulated specklepattern for Eq. ~3!, with a 5 0, r 5 0.2, and w~x, y!calculated from
w~x, y! 5 2p~2!~x2 1 y2!, uxu , 1, uyu , 1. (5)
This fringe pattern was used to test various filteringmethods. The tests were implemented on a 486 per-sonal computer, at 33 MHz with a frame grabber of256 gray levels. Images of 256 pixels 3 240 pixelswere used. The four different filtering methods aredescribed in the following paragraphs.
A. Threshold
The dc-level 2~Io 1 Ir! may be reduced by threshold-ing A in Eq. ~3! as follows:
Aa 5 HA 2 a0
if A . aif A # a , (6)
where a is a constant specified by the operator. Thehistogram of the resulting image Aa is stretched overthe dynamic range of the frame grabber, i.e., 256 greylevels. A constant value for a does not allow forillumination variations over the image, e.g., a Gauss-ian illumination-beam profile. The image will alsohave many pixels with zero intensity that contain nouseful information. Figure 1~b! shows the result ofthresholding Fig. 1~a!. The value a was found bytaking the average value ^A& over the whole image.
B. Local Averaging
In the local averaging method, a local neighborhoodaverage of the speckle fringe pattern calculated overanm 3 n pixel window, Am3n, is subtracted from theoriginal fringe pattern A. A global mean intensity,a 5 ^A&, no longer needs to be determined. Theabsolute value is taken after subtraction to reducethe number of zero-intensity pixels. The operationis represented by
As 5 uA 2 ^Am3n&u, (7)
where As is the resulting enhanced image. Figure1~c! shows the enhanced image obtained using a 3pixel 3 3 pixel window when the method is applied toFig. 1~a!. Equation ~7! may be recognized as a high-pass filter followed by rectification, a process com-monly implemented in electronic hardware fortime-averaged vibration and addition ESPI fringes.6A comparison of the digital filters discussed here with
1 May 1997 y Vol. 36, No. 13 y APPLIED OPTICS 2785
such an electronic filter is presented with the exper-imental results.
C. Gradient
The gradient of image A~x, y! is defined by
Ag 5]A]x
î1]A]y
Ê, (8)
2786 APPLIED OPTICS y Vol. 36, No. 13 y 1 May 1997
and its magnitude uAgu, calculated by
uAgu 5 FS]A]xD
2
1 S]A]yD
2G1y2
. (9)
Equation ~9! can be evaluated by a variety of meth-ods; Roberts and Sobel filters are the most commonlyused.7 Figure 1~d! shows the results obtained by the
application of a Sobel filter to Fig. 1~a! with a windowof 3 pixels 3 3 pixels.
D. Variance
At each pixel ~x, y! we can define a local variancegiven by
Av~x, y! 5 (2m
m
(2n
n
uA~x 1 m, y 1 n! 2 ^Am3n&u, (10a)
or, alternatively,
Av~x, y! 5 (2m
m
(2n
n @A~x 1 m, y 1 n! 2 ^Am3n
^Am3n&, (10b)
where Eq. ~10b! is the normalized variance measuredwithin the window. Equation ~10a! is similar to thehigh-pass filter of Eq. ~7!. However, the mean valueused at each pixel differs slightly between the meth-ods, and the result is averaged over an m 3 n pixelwindow. The results of applying Eq. ~10b! to Fig.1~a! are shown in Fig. 1~e! for a 3-pixel 3 3-pixelwindow.Quantitative results, with the fringe contrast
defined in Eq. ~4!, that compare the four filtered im-ages, Figs. 1~b!–1~e!, calculated from the originalcomputer-generated image are presented in Table 1.The first three filters are all seen to improve fringecontrast. The final filter ~gradient! includes a low-pass filter. To permit a more fair comparison amongthe techniques, a 3-pixel3 3-pixel local-neighborhoodlow-pass filter was applied to the results of the firstthree filters. These results are also shown in Table1. The variance filter, Av, gives superior contrast,even after the results are low-pass filtered from thefirst three filters.
4. Experimental Results
The techniques were then applied to experimentallyobtained addition fringes. Figure 2~a! shows an out-of-plane addition-fringe pattern obtained from ametal plate vibrating at 320 Hz.6 It consists of cir-cular fringes that correspond to the first-resonantmode. Spurious interference fringes can be seenand are due to multiple reflections in the CCD sensorcover.Figures 2~b!–2~e! show the results of applying the
four filters, described previously, to Fig. 2~a!. A3-pixel 3 3-pixel window was used again in the lasttwo filters—gradient and variance—which require awindow to be defined. Figure 2~f ! shows the resultof applying an electronic analogue filter.6 Table 1shows the numerical results of the contrast enhance-
ment that correspond to each of the filtered images ofFig. 2. In this case, fringemaxima andminimaweremarked manually before the fringe contrast was cal-culated. It can be seen that the values of fringecontrast obtained from the computer simulation arein reasonable agreement with the experimental val-ues. Once again the variance filter, Av, gives supe-rior results. The variance filter also gives betterfringe contrast than the electronic filter, which is tobe expected: The electronic filter is working in realtime, that is, video-frame rates, and so cannot per-form a spatial average over a window as it can for thedigital technique.
5. Conclusions
We have examined the contrast enhancement of asingle twin-pulsed ESPI addition-fringe pattern.We have shown that a variance filter enhances fringecontrast significantly. Results from a computer sim-ulation and an experiment, in which the test objectwas vibrating harmonically, were presented. It isclear that addition fringes obtained in otherexperiments—for example, an object undergoingtransient deformations—may be enhanced as well.
Noe Alcala acknowledges financial supportthrough a scholarship from Consejo Nacional deCiencia y Technologia, Mexico.
References1. R. Jones and C. Wykes, Holographic and Speckle Interferome-
try, 2nd ed. ~Cambridge U. Press, Cambridge, UK, 1989!.2. F. Mendoza Santoyo, D. Kerr, J. R. Tyrer, and T. C. West, “A
novel approach to whole field vibration analysis using a pulsedlaser system,” in Holographic Optics II: Principles and Appli-cations, G. M. Morris, ed., Proc. SPIE 1136, 335–345 ~1989!.
3. D. Kerr, F. Mendoza Santoyo, and J. R. Tyrer, “Extraction ofphase data from electronic speckle pattern interferometricfringes using a single-phase-step method: a novel approach,”J. Opt. Soc. Am. A 7, 820–826 ~1990!.
4. M. C. Shellabear, F. Mendoza Santoyo, and J. R. Tyrer, “Pro-cessing of addition and subtraction fringes from pulsed ESPI forthe study of vibrations,” in Proceedings of the Society for Exper-imental Mechanics Conference on Hologram Interferometry andSpeckle Metrology ~Society for Experimental Mechanics, Balti-more, Md., 1990!, pp. 238–244.
5. A. J. Moore and C. Perez Lopez, “Fringe visibility enhancementand phase calculation in double pulsed addition ESPI,” J. Mod.Opt. 43, 1829–1844 ~1996!.
6. A. J. Moore and C. Perez Lopez, “Low frequency harmonic vi-bration analysis with double-pulsed addition ESPI,” Opt. Eng.35, 2641–2650 ~1996!.
7. R. Gonzalez and P. C. Wintz, Digital Image Processing~Addison-Wesley, Reading, Mass., 1987!, pp. 398–402.
1 May 1997 y Vol. 36, No. 13 y APPLIED OPTICS 2787
