Automated optical analysis of young's fringes— optical autocorrelation

Optics and Lasers in Engineering 14 (1991) 351—361

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Automated Optical Analysis of Young’s Fringes— Optical Autocorrelation Jeremy M. Coupland* Institute of Sound and Vibration Research, Southampton University, Highfield, Southampton S09 5NH, UK

& Neil A. Halliwell Department of Mechanical Engineering, Loughborough University of Technology, Loughborough, Leicestershire LE11 3TU, UK

ABSTRACT Extraction of displacement or velocity data from laser speckle photography or particle image velocimetry transparencies requires measurement of the spacing and orientation of several thousand Young’s fringe patterns. In contrast with classical interferometry, fringes formed in this way have poor visibility and it is generally desirable to compute the two-dimensional Fourier transform in order to extract the fringe frequency. The large amount of data produced makes digital acquisition and image processing computationally intensive and time consuming. This paper introduces an automated processor which utilises a photorefractive crystal of bismuth silicon oxide to perform this analysis in a fast and cost-effective manner.

INTRODUCTION Laser speckle photography’ (LSP) is now an established method for two-dimensional measurements of in-plane surface displacements. This technique uses double exposure photography to record the specklemodulated images observed when the surface of interest is illuminated with coherent light. The fundamental principle of LSP is that for an *

Present address: Department of Mechanical Engineering Loughborough University of

Technology, Loughborough, Leicestershire LE11 3TU, UK. 351

Optics and Lasers in Engineering 0143-81661911$0350 © 1991 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland

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in-plane displacement, both the surface and its speckle modulated image move in sympathy with one another. Therefore, if the surface is displaced between exposures, the resulting transparency consists of two superimposed images modulated by identical but laterally displaced speckle patterns. For a complex in-plane distortion, a two-dimensional surface displacement or strain map can be constructed from pointwise measurement of the local speckle displacement. Particle image velocimetry2 (PIV) is a technique derived from LSP for the measurement of two-dimensional velocity vector fields from a plane of interest within a fluid flow. In this case double exposure photography is used to record the position of seeding particles within a thin sheet of light illuminating a section of the flow. The displacement of particle images recorded in a small region of the transparency can be related to the local fluid velocity provided the time interval between exposures and the magnification of the recording optics are known. Analysis of both LSP and PIV transparencies essentially involves mapping the displacement of a random pattern. In the past, this information has been recovered by examining the far field or Fraunhoffer diffraction pattern observed when each transparency is interrogated using a low power laser beam.3 The paired nature of the speckle or particle image distribution within the illuminated region results in the formation of parallel fringes with spacing and orientation proportional to the magnitude and direction of the required displacement. By analogy with the fringe pattern obtained using parallel slits. this method is usually referred to as Young’s fringe analysis. Since, typically, several thousand displacement measurements are required from each LSP or PIV transparency, it is highly desirable to automate the analysis procedure. However, Young’s fringe patterns generated from both LSP and PIV transparencies are generally heavily modulated by noise as shown in Fig. 1(a) and (b). As a result, automated measurement of the spacing and orientation of the underlying fringe pattern is problematic. One approach to automatic transparency analysis is to interrogate digitised fringe patterns using suitable measurement algorithms implemented on image processing hardware. In a recent publication, Huntley4 has assessed the relative performance of several algorithms with reference to both ability to suppress the effect of noise, and measurement errors associated with each technique. It can be concluded from this work that two-dimensional Fourier transformation and subsequent measurement of the position of the most significant peaks in the resulting distribution is superior to other methods in these respects. Mathematically, it should be noted that this operation is equivalent to

Automated optical analysis of Young’s fringes

(a)

(b) Fig. I.

Young’s tringe’. obtained from (a) LSP and (h) PIV transparencies.

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the computation of the two-dimensional spatial autocorrelation of the amplitude transmitted by the illuminated region.5 The disadvantage of processing the fringe pattern in this manner is that Fourier transformation is computationally intensive and, consequently, is either prohibitively time consuming or prohibitively expensive for large scale studies. As a solution to this problem, in the following we demonstrate an automated analysis system which uses rapid optical processing techniques to compute two-dimensional spatial autocorrelation. Accordingly, the ability of this type of analysis to suppress the influence of noise, and provide high accuracy data, is exploited in a fast and cost effective manner. OPTICAL AUTOCORRELATION A major advantage of image processing using optical methods, rather than conventional digital computing techniques, is a substantial gain in processing speed. This is mainly due to the inherent ability of optical processors to perform parallel operations, but also because these operations are performed on the data in its initial form, i.e. a spatial distribution of light. There is, therefore, no need to transform this data into an ordered array of digital values (pixels) before processing can commence. The design of the processor described in this report, originates from the work of Vander Lugt6 who demonstrated the synthesis of an optical matched filter using holographic methods. Although this work was published in 1964 it has only met very specialist applications, since processing and accurate repositioning of the filter are necessary. The recent development of photorefractive materials has renewed interest in optical processing since holograms can now be recorded and erased in situ and several designs of photorefractive correlators and convolvers have been published.7~8 Figure 2 shows a new design of optical autocorrelator first reported in Applied Optics,9 which is simple to construct and particularly suitable for the analysis of LSP and PIV transparencies. A crystal of bismuth silicon oxide (BSO) was chosen for this work since it is readily available and displays a large photorefractive coefficient when an external electric field is applied. Transparency analysis is split into two distinct operations. Firstly a Vander Lugt filter is recorded as a Fourier plane hologram of the light amplitude immediately behind the illuminated interrogation region. The reference beam is then shuttered off and the autocorrelation function of this region is subsequently formed in the output plane.

Automated optical analysis of Young’s fringes

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THEORY OF OPERATION The holographic grating formed within the optically thick, BSO crystal, is described in full by the coupled mode equations of Kogelnik.’° However, provided the Bragg condition is satisfied, this theory is reduced to that of thin holographic gratings, which is presented here. For a discussion of the Bragg condition with reference to optical processing the reader is referred to the work of Pepper et al.” With reference to Fig. 2, according to the Fourier transforming properties of convex lenses,5 the complex amplitude, U 2(x2, Y2), of the field in the crystal plane, can be written as the sum of two components, U2(x2, Y2) = k1A(k1x2, k1y2) + B(x2, Y2) where A(k1x2, k1y2) is the Fourier transform of the input field a(x1, y’) and B(x2, Y2) is the reference field. The constant k1 is given by k1 = 1/AF1, where A. is the wavelength of the laser illumination. This field distribution gives rise to a refractive index change in the crystal proportional to the intensity, ‘2, given by, 2 ‘2 = Ik1A + B1 = k~Al + IBI + k 1AB* + k1A*B The resulting filter is described by the complex transmission function, T(x2, y2), given by, T = exp [11312] where 13 is a constant describing the photorefractive efficiency of the crystal. Since the photoinduced index modulation is small, i.e. I~I2<< 1, the transmission function can be written, T = 1 + j1312

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When the reference beam is switched off, the light incident on the crystal, the optical field immediately after the filter is given by. kAT= k1A[l +j/3k~Al2 +j/3 1B11 +j13k~AAB*+jf3k~Al2 B It is the last term which is of interest since it describes a reconstruction of the reference beam modulated by the squared modulus of the Fourier transformed input fIeld. In fact, fulfillment of the Bragg condition dictates that this is the dominant diffracted held. Accordingly. the output field of interest U.~(x 3,vi), is this term transformed by Lens 2 to give, U3(x3, ~,V3) =jfik2a(xj’~/1~,vJ~/1~) ® a(x2J~/F..vj~/E~) ~ b(xj’/I’~, %‘~i’~/F)

where ~ and 0 denote convolution and correlation respectively, and the constant k2 is given by k2 = l/AI~.If the reference beam is a plane wave, b(x3, ~ is a delta function on which the autocorrelation of a(x3, y3) is centred. Choosing the position of this delta function as the origin of a new coordinate system, we can write the intensity distribution in the output plane, [~(x~,vt). as. 2 f~(x~, y~) la(x~1’~/F~, ~v~I~/r;)0 a(x~I~/!~, ~‘F/F)l Figure 3 shows an isometric plot of the output intensity obtained from a typical PIV transparency. It can he seen that the autocorrelation consists of a central peak and two ‘signal’ peaks situated diametrically

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Automated optical analysis of Young’s fringes

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opposite each other and surrounded by randomly positioned ‘noise’ peaks. The position of the signal peaks can easily be measured to find the required surface displacement or particle velocity vector.

AUTOMATED TRANSPARENCY ANALYSIS The optical processor has been incorporated into the analysis system shown in Fig. 4. A microcomputer is used to control the advance of an X—Y traverse upon which the transparency of interest is mounted, and provides the timing signals for the two shutters. An exposure meter and variable density filter form a control circuit used to maintain the correct holographic exposure and to ensure equal intensity scaling of each autocorrelation. The output field is captured by the CCD array within a solid state television camera and passed to a 512 x 512 pixel framestore within the microcomputer via a standard video interface and digitising card. At present a relatively simple algorithm is used to locate and measure the position of the relevant peaks in each autocorrelation. This process is divided into three main operations as follows. First, the diameter of the central peak is estimated from the central horizontal line of pixels using a thresholding operation. Subtracting the area occupied by this peak, the top half of the digitised image is then sorted to find the location of the brightest pixel, and hence, the position of the first signal

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peak. Finally, the brightest pixel within an area equal to that occupied by the central peak but located diametrically opposite the first signal peak is found, giving the position of the second signal peak. Using this method the repeatability of each position measurement was found to he of the order of ±2pixels resulting in a measurement error of less than 1% of full scale. Since the photorefractive effect saturates when the crystal has absorbed sufficient energy, there is a trade off between illumination intensity and response time. In this case, the total laser power was of the order of 10 mW (A = 514~5nm) giving a response time of approximately 1 s. It should be noted that immediately after the reference beam has been shuttered off, the remaining optical field incident on the crystal begins to erase the Vander Lugt filter. However, this detrimental effect is generally several orders of magnitude slower than the camera frame rate and consequently, is not a problem. The automated system has been used successfully to analyse transparencies recording an essentially two-dimensional flow past a step. This flow is approximately potential and was chosen for this work since an exact analytical solution is possible via the Schwarz—Christoffel transformation.’2 Figure 5 shows 1524 velocity vector measurements taken from this flow. It is clear from these results that the majority (98%) of the velocity vectors have been correctly identified and good agreement is observed with theoretical streamlines. A total analysis time of approximately 1 h 20 mm was taken, which was principally governed by the speed of the hardware used to advance the traverse and control the holographic exposure.

CONCLUSION An optical method using a photorefractive crystal of bismuth silicon oxide to perform full two-dimensional Fourier transformation of Young’s fringe patterns, formed during the analysis of LSP and PIV transparencies,

has been demonstrated.

This method, which is equiv-

alent to calculating the two-dimensional spatial autocorrelation of the amplitudc transmitted by each illuminated region, has been incorporated into an automated transparency analysis system. A total processing time of approximately 4 s was required to make a single displacement measurement, although this figure was principally governed by the speed of the hardware used to control the system. Further work with the objective of both determining the specification, and increasing the processing speed of the transparency analysis

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system is presently underway. It is intended that by developing dedicated hardware to measure the position of peaks in the autocorrelation plane and paying particular attention to details such as the computer control functions, video rate analysis of LSP and PIV transparencies will be realised. Finally, it should be noted that the rapid computation of spatial autocorrelation afforded by optical processing techniques results in further advantages over conventional PIV transparency analysis methods. First, by interrogating overlapping areas of the PIV transparency. the position of the correlation peak could potentially be ‘tracked’ in real time as the transparency is scanned, thus avoiding ambiguity problems.’3 Secondly, the random nature of the seeding particle distribution within the fluid means that some data dropout is to be expected. However, by measuring the relative magnitude of the signal and noise peaks in the autocorrelation function,’4 it is relatively easy to attach a confidence flag to each measurement and in this way, low confidence data can be replaced by interpolated results. REFERENCES I. Ennos, A. C., Speckle interferometry. Prog. in Opt., 16 (1977) 235—243. 2. Pickering, C. J. D. & Halliwell, N. A., Particle image velocimetry: A new field measurement technique. In Optical Measurement in Fluid Mechanics, (Inst. Phys. Conf. Series). Adam Huger, Bristol, 1985, pp. 147—152. 3. Pickering, C. J. D. & Halliwell, N. A., PIV: Fringe visibility and pedestal removal. App!. Opt., 24 (1985) 2474. 4. Huntley, J. M., Speckle photography fringe analysis: Assessment of current algorithms. App!. Opt., 28 (1989) 4316—4322. 5. Goodman, J. W. Introduction to Fourier Optics. McGraw-Hill, New York, 1968. 6. Vander Lugt, A., Signal detection by complex spatial filtering. IEEE Transaction on Information Theory, IT—b, 1964, 139—146. 7. White, J. 0. & Yariv, A., Real-time image processing via four-wave mixing in a photorefractive medium. App!. Phys. Lett., 37 (1980) 5—7. 8. Pichon, L. & Huignard, J. P.. Dynamic joint-Fourier-transform correlator

by Bragg diffraction in photorefractive Bi,

2Si020 crystals. Opt. Comm., 36 (1981) 277—280. 9. Coupland, J. M. & Halliwell, N. A., Particle image velocimetry: Rapid transparency analysis using optical correlation. App!. Opt., 27 (1988)

19 19—1921. 10. Kogelnik, H., Coupled wave theory for thick hologram gratings. Bell Syst. Tech. J., 4.8 (1969) 2909—2947. 11. Pepper, D. M., AuYeung, J., Fekete, D. & Yariv, A., Spatial convolution and correlation of optical fields via degenerate four-wave mixing. Opt. Lett.,

3 (1978) 7—9.

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12. Milne-Thomson, L. M. Theoretical Hydrodynamics. Macmillan, London,

1968. 13. Coupland, J. M., Pickering, C. J. D. & Halliwell, N. A., Particle image velocimetry: The ambiguity problem. Opt. Eng., 27 (1988) 193—197.

14. Coupland, J. M. & Pickering, C. J. D., Particle image velocimetry: Estimation of measurement confidence at low seeding density, Optics and Lasers in Engng, 9 (1988) 201—210.