Abstract

A new technique for displacement measurement is proposed that makes use of phase singularities in the complex signal generated by a Laguerre-Gauss filter operation applied to a speckle pattern. The core structures of phase singularities are used as unique fingerprints attached to the object surface, and the displacement is determined by tracing the movement of registered phase singularities with their correspondence being identified by the fingerprints. Experimental results for translational and rotational displacement measurements are presented that demonstrate large dynamic range and high spatial resolution of the proposed optical vortex metrology.

©2006 Optical Society of America

1. Introduction

Electronic speckle photography has been studied for a long time since the advent of lasers [1], and various techniques have been developed for a wide range of applications such as those in experimental mechanics and bio-photonics [2–5]. Until now, the research in speckle photography has, for the vast majority, been based on the cross-correlation function of intensity of speckle patterns, and less attention has been paid to the applications based on the phase information. Recently, we proposed the use of pseudophase information of the complex signal obtained from partial Hilbert transform or Riesz transform of speckle patterns [5–8]. Although these complex representations of real-valued speckle patterns do not introduce new information, such transforms effectively exploit the existing information because the newly introduced pseudoamplitude and pseudophase associated with the complex analytic signal provide more effective means for analyzing, processing and understanding the available information from the recorded speckle pattern. Furthermore, due to its improved performance for micro-displacement measurement and the fact that the pseudophase can be detected without recourse to interferometry, the pseudophase information has the versatility that expands applications beyond those known for laser speckle metrology.

Meanwhile, the phase field has a unique structure known as phase singularities, which are the topological defects of the wave fronts [9]. From the optical information point of view, phase singularities are optimal encoders for position marking since the least photon noise is present where the amplitude is zero. On the other hand, phase singularities can also become the ideal markers because they are well-defined geometrical points, and can be localized with arbitrary accuracy, at least in principle. Observing many randomly distributed phase singularities in the analytic signal for a speckle pattern, we have explored new possibilities for making use of the phase singularities in optical metrology, and have proposed a method referred to as Optical Vortex Metrology (OVM) [6,8]. Though we have demonstrated a high-resolution measurement of nanometric displacement by OVM, the maximum measurable displacement was restricted because large displacements make difficult the unique search and identification of the registered marker phase singularity among many phase singularities that have the same topological charge.

The purpose of this paper is to propose a new technique that substantially improves the OVM technique in its ability of uniquely identifying the matching phase singularities under large displacements. We will experimentally demonstrate large displacement measurements that were not possible with our previous technique. They key idea of the newly proposed technique is that, in addition to the information about the location of the phase singularities, we also detect the core structures of the phase singularities. This core structure can serve as a “fingerprint” that uniquely characterizes an individual phase singularity, and can be used for the purpose of the unique identification of the registered phase singularity. Even a large displacement of an object can be determined by tracing the registered phase singularities from their fingerprints.

First, we briefly explain the two-dimensional (2-D) isotropic complex signal representation of a speckle pattern by using Lagurre-Gauss filtering and show the core structure of a phase singularity in the complex analytic signal. Then, we present the experiments of displacement measurements making use of the information about the location and the core structure of the phase singularities, and demonstrate the performance for nonuniform displacement measurement to show the validity of the proposed principle.

2. Principle

2.1 Laguerre-Gauss transform and 2-D isotropic complex signal representation

Before explaining the proposed improvement to optical vortex metrology, we first briefly review the two-dimensional isotropic complex signal representation of a speckle pattern.

It is common practice in physics and engineering to represent real-valued signals by the related complex-valued signals [10]. For one-dimensional (1-D) signals, the concept of analytic signals was introduced to communication theory by Gabor in the 1940s [11]. Usually, the real part of an analytic signal is the original signal with its mean value being subtracted, and the corresponding imaginary part is the Hilbert transform of the real part. When the Hilbert filter was applied to the 2-D speckle pattern, an obvious anisotropy was found in the generated analytic signal because the partial Hilbert filter introduces different spectral bandwidths in the fx -and fy -direction. To obtain a 2-D isotropic analytic signal for a 2-D speckle pattern, we have proposed to replace the partial Hilbert transform with the Riesz transform [12] or the vortex transform [13], and have demonstrated its validity experimentally [8]. On the other hand, noting that the Riesz transform is a pure spiral phase function with a vortex structure in the frequency domain and no use has been made of the amplitude information of the Fourier spectrum, we have proposed to replace the Riesz transform or the vortex transform with the Laguerre-Gauss transform to obtain stable and optimally distributed phase singularities by virtue of its band-pass filter characteristic [14]; more recently the use of Laguerre-Gauss spatial filter was also proposed independently by Guo et al. [15] for optical image processing, where the focus was placed on the amplitude information of the filtered image.

Let I(x,y)be the original intensity distribution of the speckle pattern, and let its Fourier spectrum be (fx,fy). We can relate I(x,y) to its isotropic complex signal Ĩ(x,y) through a Laguerre-Gauss (L-G) filter. Thus, our definition of a 2-D isotropic complex signal representation for the speckle pattern is

I˜(x,y)=++LG(fx,fy)·(fx,fy)exp[j2π(fxx+fyy)]dfxdfy,

where LG(fx,fy) is a Laguerre-Gauss filter in the frequency domain defined as follows:

LG(fx,fy)=(fx+jfy)exp[(fx2+fy2)/ω2]=ρexp(ρ2/ω2)exp().

where ρ=fx2+fy2, β = arctan(fy/fx) are the polar coordinates in the spatial frequency domain. In addition to the advantage of spatial isotropy common to the Riesz transform stemming from the spiral phase function with the unique property of a signum function along any section through the origin, the Laguerre-Gauss transform has the favorable characteristics to automatically exclude any DC component of the speckle pattern. It can also readily be seen from Eq. (2) that the doughnut-like amplitude for the L-G function serves as a band-pass filter by suppressing the high spatial frequency components that create unstable phase singularities. Therefore, the density of phase singularities in a complex signal can be controlled by choosing a proper bandwidth of the Laguerre-Gauss function, ω, to adjust the average speckle size. After straightforward algebra, we find that

I˜(x,y)=I˜(x,y)exp[(x,y)]=I(x,y)*(x,y),

where * denotes the convolution operation, and 𝕃𝔾(x, y) is again a Laguerre-Gauss function in the spatial signal domain,

(x,y)=1{LG(fx,fy)}=(jπ2ω4)(x+jy)exp[π2ω2(x2+y2)]
=(jπ2ω4)[rexp(π2r2ω2)exp()],

where, 𝔽 -1 is inverse Fourier transform, and r=x2+y2, α = arctan(y/x) are the spatial polar coordinates defined as usual. The phase θ(x, y) of the complex signal of a speckle pattern is referred to as the pseudophase to distinguish it from the true phase of the optical field responsible for the speckle pattern. Although it is not the true phase of the complex optical field, the pseudophase does provide useful information about the object, as will be shown in Section 3.

2.2 Core structure of phase singularity and its application to Optical Vortex Metrology

Just as a random speckle intensity pattern imprints marks on a coherently illuminated object surface, randomly distributed phase singularities in the pseudophase map of the speckle pattern imprint unique marks on the object surface. In our previous investigation [8], we have proposed a technique for nanometric displacement measurement, which makes use of the information about the locations of phase singularities before and after the displacement. To do this, we need to be able to identify the corresponding phase singularities between the pre- and post-displacement phase maps. If the displacement is known to be small a priori, we can restrict our search only to the closest neighbor phase singularities of the same topological charge. However, when the displacement is large and/or non-uniform and no information is given a priori, we cannot uniquely identify the corresponding phase singularities. This has been the drawback of our previous technique though it has the advantage of having a high resolution.

To solve this problem, we propose the use of additional information about the core structure of the phase singularities. Similarly to optical vortices in random laser speckle fields [16, 17], the changes of the pseudophase around the phase singularities are non-uniform, and the typical core structure around the phase singularities are strongly anisotropic. Figure 1 shows an example of the reconstructed amplitude contours and pseudophase structure of the complex signal representation of a speckle pattern in the neighborhood of a phase singularity. The phase singularity is located at the center of the elliptical contours of the amplitude, which is the intersection of the zero crossings of the real and imaginary parts of a complex signal Ĩ ; the phase has a characteristic feature of a 2π helical structure. We note that the eccentricity of the contour ellipse e and the zero crossing angle θRI between the real and imaginary parts, shown in Fig. 1(a), are invariant to the in-plane rigid-body motion of the object involving translation and rotation, and we use these two geometric parameters to describe the local properties of the phase singularities. In addition, each phase singularity has its own topological charge and vorticity defined by Ω⃗ ≡ ∇{Re[Ĩ(x,y)]}×∇{Im[Ĩ(x,y)]}, which we also assume to be invariant to the in-plane rigid-body displacement involving translation and rotation. Just as no fingers have exactly the same fingerprint patterns, no phase singularities have exactly the same local properties with identical eccentricity e, zero-crossing angle θRI, topological charge q, and vorticity Ω⃗. It is this uniqueness of the core structure that enables the correct identification and the tracking of the complicated movements of phase singularities.

Usually, the real and imaginary parts of the complex signal representation of a 2-D speckle pattern in the immediate vicinity of a phase singularity can be expressed as

Re[I˜(x,y)]=arx+bry+cr,Im[I˜(x,y)]=aix+biy+ci,

where the coefficients: ak,bk,ck (k = r,i)can be obtained by the least-square fitting method from the detected complex values at the pixel grids surrounding the phase singularity [8]. From the interpolated real and imaginary parts of the complex signal representation of the speckle pattern [17], we can obtain the detailed pseudophase profile around a phase singularity.

 figure: Fig. 1.

Fig. 1. Core structure around a phase singularity with zero crossings of real and imaginary parts of complex signal representation for a speckle pattern. (a) Amplitude contours and zero crossing lines; (b) Pseudophase structure.

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Based on their definitions, the geometrical and physical parameters characterizing the phase singularity can be expressed in terms of the fitting coefficients as

e=1(ar2+ai2+br2+bi2)(ar2+ai2br2bi2)2+4(arbr+aibi)2(ar2+ai2+br2+bi2)+(ar2+ai2br2bi2)2+4(arbr+aibi)2,
θRI={arctan[(arbiaibr)/(arai+brbi)]θRI<π/2πarctan[(arbiaibr)/(arai+brbi)]θRI>π/2,
Ω=arbiarbr,
q=sgn(Ω·ez)=sgn(arbiaibr).

Although the coefficients ak,bk,ck (k = r,i) may change with the displacement of the object, these parameters characterizing the local properties of the pseudophase singularities remain stable without significant changes because they are invariants to translation and rotation. When observed with focus on the object surface, the displacement of each phase singularity can be directly related to the local displacement of the object surface. After identifying corresponding phase singularities before and after displacement making use of their core structures as fingerprints, we can estimate the local displacement of an object by tracing the movements of the phase singularities. Meanwhile, it is interesting to note that the descriptions for the local structures of the anisotropic phase singularities are not unique. Besides the geometrical and physical parameters: q, e, θRI, and Ω⃗ used here, another set of parameters can also serve as a “fingerprint” for the purpose of the unique identification of the registered phase singularity [18–20]. To find the correctly matching phase singularities for the object after displacement, it is necessary to prescribe several conditions for the correct identification, which are expressed as follows:

q=q′,
Δe=ee′<ε1,
ΔΩ=(ΩΩ)/(Ω+Ω′)<ε2,
ΔθRI=θRIθ′RI<ε3,

where primed parameters are related to the phase singularity after displacement. After setting appropriate threshold values for ε 1,ε 2 and ε 3, we can efficiently eliminate most of the phase singularities with large differences in their core structures. Next, within a few survived candidates of phase singularities, the figure-of-merit for the best matching of phase singularities has been chosen as

E=(ee′)2+(ΩΩ′Ω+Ω′)2+[2π(θRIθ′RI)]2.

After calculating the merit function for each pair of phase singularities, we can identify the correct counterparts based on the minimum value of E. Thus, the in-plane displacement of an object can be estimated from the coordinate change (Δx, Δy) for each pseudophase singularity within the probing area.

3. Experiments

Experiments have been conducted to demonstrate the validity of the proposed principle. Since a white-light speckle pattern is known to be less prone to decorrelation [21], we generated a white-light speckle pattern by directly illuminating the stage surface of a microscope (Nikon ECLIPSE ME600) with a halogen lamp. Though we have referred to the observed random pattern as a white-light speckle pattern, strictly it is not a generic speckle pattern of interference nature but rather represents the microscopic texture of the stage surface. This means that our experiment also serves as a demonstration that the proposed technique can be applied to a wider class of random pattern including non-generic speckle patterns such as the random texture observed on an object illuminated by incoherent light. The white-light speckle pattern generated on the stage surface was imaged by a microscope (with a 20× objective lens and a 0.45× relay lens) onto an image sensor. Through introduction of controllable lateral and rotational micro-displacements with the precision mechanical stage of the microscope, we recorded the white-light speckle patterns with a CMOS camera (SILICON VIDEO 9M001) with the pixel size of 5.2μm×5.2μm. From these nominal magnifications and the pixel separation of the CMOS image sensor, the unit pixel displacement corresponds to an object displacement of 578nm. From the recorded white-light speckle pattern, we generated an isotropic complex signal by Laguerre-Gauss filtering and retrieved the pseudophase information. In the experiment, we adjusted the average speckle size and controlled the density of phase singularities carefully by choosing a proper bandwidth of the Laguerre-Gauss filter ω in Eq. 2, so that a single speckle includes about 40 pixels along a traversing line. After identifying the corresponding phase singularities for the object before and after displacement, we measured the given displacement by the proposed optical vortex metrology.

 figure: Fig. 2.

Fig. 2. (a). Displacement of phase singularities, ◦: before displacement; ■: after displacement; Histograms of coordinate changes of phase singularities, (b) x -direction;(c) y -direction.

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Figure 2(a) shows an example for the movements of phase singularities with a translational displacement, where the locations of phase singularities before and after displacement have been indicated by ◦ and ■, respectively. During this identification process to find the correct counterparts for phase singularities, the threshold values in Eqs. (11)–(13) have been selected empirically as: ε 1 = 0.05, ε 2 = 0.1 and ε 3 = 0.174 (rad), and the searching process has been performed over the whole probe area, since we have assumed no prior knowledge about the local displacement of the object. As expected, most of the phase singularities have found their counterparts in the upper-right direction as indicated with a short straight line connecting the correct pair of phase singularities, and the coordinate differences between the corresponding phase singularities gave a good estimation for the local displacement of the object as far as we use only those phase singularities. However, we also found that some phase singularities failed to find their correct counterparts as indicated with long straight lines stretching in random directions, and thus became the source of errors for the proposed optical vortex metrology. This phenomenon may be attributed to two different origins. One is that, because of the change in the illumination condition introduced by the displacement, the speckles begin to change their shapes; in other words, speckle decorrelation occurs when the displacement is increased. This gives rise to the distortions of the core structure of the phase singularities, and sometimes even introduces creation or annihilation of phase singularities in the pseudophase map. The distortions of the phase singularities enhance the difficulty of finding the correct counterparts for calculation of their coordinate difference. Furthermore, the newly created or annihilated phase singularities definitely have no counterparts in the pseudophase map. The other origin is the flow of the phase singularities across the boundary of the probe area. The large displacement of the object causes some of the phase singularities to move into (or out of) the probe area across the boundary, which also become another origin for the lack of counterparts in the other pseudophase map.

For the case of rigid-body displacement, this problem can be solved by the additional process to be described below. Figure 2(b) is the histogram for the x -coordinate changes of the phase singularities, and the histogram for the y -coordinate changes is shown in Fig. 2(c). From the locations of the histogram peaks, we can make initial estimate of the displacements of the object by simply calculating the mean values of coordinate differences in the x - and y -directions using only the samples belonging to these histogram peaks. In this measurement, the displacement along the x -direction has its mean value equal to 36.3 pixels (21.0μm) with the standard deviation σ = 17.9 pixels (10.3μm), which indicates a very large uncertainty in the estimation for one-dimensional displacement measurement based on a single singularity. Similarly, we get the mean value of Δy equal to 37.2 pixels (21.5μm) and the corresponding standard deviation equal to 17.4 pixels (10.1μm). Thus, the 2D displacements of the object can be roughly estimated as (Δx, Δy)=(36.3 pixels, 37.2 pixels) =(21.0μm, 21.5μm) in the initial step. Meanwhile, we have also calculated the standard deviations for Δe and ΔθRI, which serve as stability measures for the core structures of the pseudophase singularities during the object movement. In this measure, the standard deviation for Δe is 0.0148 and that of ΔθRI is equal to 0.066 (radian).

In both sides of the main histogram peaks, we can also observe very long tails (side-lobes) in both histograms, which indicate the region of errors in the displacement measurement arising from those phase singularities that failed to find their correct counterparts. Note that, in contrast to the small proportion of phase singularities spreading out in the histogram, the most coordinate differences for the phase singularities remain concentrated in the histogram with very narrow bar width. It is this high concentration that makes possible the precise initial estimate of the displacement.

Based on this initial displacement estimation and the a priori knowledge of the translational rigid-body displacement, we restricted our searching area to a small window of 20×20 pixels around the initial estimate of the displacement given by the rounded pixel coordinates (36,37). Figure 3(a) shows the results for the phase singularities displacements after the local search performed for the restricted area described above. The histograms for the coordinate changes of the phase singularities along the x - and the y - directions are shown in Fig. 3(b) and 3(c), respectively. As anticipated, all the phase singularities have been shifted with uniform displacement in the upper-right direction. From the location of the histogram peak, the displacement along the x -direction has its mean value equal to 33.9 pixels (19.6μm) with a standard deviation σ = 0.7 pixels (0.4μm), which indicates a significant improvement as compared with the result derived from the first preliminary estimation. Similarly, we can also get the mean value of Δy equal to 36.7 pixels (21.2μm), and the corresponding standard deviation equal to 0.5 pixels (0.3μm). In Fig. 3(a), the standard deviations for Δe and ΔθRI become 0.0159 and 0.0429 (radian), respectively. It is also important to note that the generated complex signal representation of a speckle pattern is indeed isotropic since the two displacement histograms for phase singularities have almost the same peak widths. From these two mean values, the object displacement distance can be obtained as ΔL = (< Δx >2 + < Δy >2)1/2 = 49.96 pixels (28.9μm), and the direction angle of the displacement is δ = arctan[< Δy > / < Δx >] = 0.825 (rad). As compared with our previous investigation [8], Fig. 3 serves as an experimental demonstration of the validity of the proposed technique for displacement measurement with a large dynamic range.

 figure: Fig. 3.

Fig. 3. Results of phase singularities after fine identification (a) Displacement of phase singularities; Histograms of coordinate changes of phase singularities, (b) x -direction;(c) y -direction.

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In many industrial control systems with various kinds of rotation mechanism, accurate measurement of angular displacement in dynamic environments is particularly important. Due to its inherent advantage of non-contact measurements, optical detections of angular displacement have been put forward mainly during the past decades [22, 23]. However, they are based exclusively on the intensity cross-correlation. Here, we extend the application of the proposed optical vortex metrology for the case of the rotational displacement measurement.

Figure 4 shows a schematic diagram for in-plane rotation measurement by using OVM. After the identification of the corresponding phase singularities in the pseudophase maps for the object before and after rotational displacement, the perpendicular bisector Li between this pair of phase singularities can be expressed as

Aix+Biy+Ci=0
Ai=2(xix′i),Bi=2(yiy′i),andCi=(x′i2+y′i2xi2yi2),

where (xi,yi) is the coordinate of the phase singularity before rotational displacement, and (xi,yi) of after a displacement.

 figure: Fig. 4.

Fig. 4. Schematic diagram for in-plane rotation measurement by optical vortex metrology

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Ideally, the center of rotation of the object, to be found at the intersections of all the perpendicular bisectors connecting each pair of the corresponding phase singularities, should be a single point. From Eq. (15) for each perpendicular bisector, we can determine the location of the center of rotation by applying the least-square fitting method given by

xc=(iBi2)(iAiCi)(iAiBi)(iBiCi)(iAi2)(iBi2)(iAiBi)2,
yc=(iAi2)(iBiCi)(iAiBi)(iAiCi)(iAi2)(iBi2)(iAiBi)2.

From the law of cosines, we can also obtain the rotation angle ϕi for each pair of phase singularities from the coordinates of the three points:(xi,yi) ,(x̂i,ŷi), and (xc,yc). Thus, the rotational displacement of an object can be estimated from the histogram of rotation angle for all phase singularities within the probing area.

By introducing the rotational displacement with a turntable of the microscope, we measured the rotation angle by using the proposed technique. After setting the threshold values as before, we performed the preliminary search for the phase singularity identification within the whole probe area. Figure 5(a) shows an example for the rotational displacements of phase singularities after this initial, unrestricted search. The histogram of rotation angles for the phase singularities is shown in Fig. 5(c). Just as what was the case for the translational displacement, most of the phase singularities have found their correct counterparts during this preliminary identification process. Based on the location of the maximum for the histogram, the rotation angle ϕ has its mean value equal to -0.527 radian with a standard deviation of σ = 0.082 radian. In addition, we can also obtain the coordinate of the rotation center (xc,yc) = (485.9,484.7) in units of pixels from Eq. (16) and Eq. (17). Based on these results derived from the initial rotation estimation, we introduced a location-dependent shift to the complex signal for the object after displacement, which is given by

x=xc+(xixc)cosϕ+(yiyc)sinϕ,
y=yc(xixc)sinϕ+(yiyc)cosϕ.

After choosing a small search window of 20×20 pixels, we performed the fine identification for phase singularities, and the results are shown in Fig. 5(b) and 5(d).

 figure: Fig. 5.

Fig. 5. Results of optical vortex metrology applied for rotational displacement. Left column: rough searching over whole probe area; Right column: local searching for fine identification. (a), (b) Displacement of phase singularities; (c), (d) Histogram of rotation angle of phase singularities (Note the difference in the scales.)

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As expected for the rotation of the rigid-body specimen, all the phase singularities have found their correct counterparts, and the local displacements in azimuth direction increase linearly with the radial distance from the center of rotation. From the location of the histogram peak, the rotation angle has its mean value equal to ϕ = -0.518 radian with the standard deviation σ = 0.002 radian, which also indicates a significant improvement as compared to the initial estimation. Figure 5 serves as an experimental demonstration of the validity of the proposed technique for speckle rotation measurement with 2 milliradian angular resolution.

4. Conclusions

We have proposed an improved technique of OVM for translational and rotational displacement measurements that makes use of the pseudophase singularities in the complex signal representation of a speckle pattern generated by Laguerre-Gauss filtering. As indicators of local spackle displacement, these pseudophase singularities can be traced through their core structures, and the displacement of an object can be estimated from the subsequent location displacement of the registered phase singularities. Experiments have been performed that demonstrate the validity of the proposed technique for non-uniform displacement with a large dynamic range. Furthermore, the registration of the phase singularities based on their anisotropic core structure make possible the characterization of the local properties of the speckle-like random pattern, and introduces new opportunities to explore other potential applications of the phase singularities in optical signal processing.

Acknowledgments

Part of this work was supported by Grant-in-Aid of JSPS B (2) No. 18360034, Grant-in-Aid of JSPS Fellow 15.52421, and by The 21st Century Center of Excellence (COE) Program on “Innovation of Coherent Optical Science” granted to The University of Electro-Communications.

References and Links

1. A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena,J. C. Dainty, ed. (Springer -Verlag, Berlin, 1976).

2. R. K. Erf, Speckle Metrology, (Academic Press, New York, 1978).

3. R. S. Sirohi, Speckle Metrology, (Marcel Dekker Inc., New York, 1993).

4. N. A. Fomin, Speckle Photography for Fluid Mechanics Measurements, (Springer-Verlag, Berlin, 1998).

5. S. J. Kirkpatrick and D. D. Duncan, “Optical assessment of tissue mechanics,” in Handbook of Optical Biomedical Diagnostics,V. V. Tuchin, ed., (SPIE Press, Bellingham, 2002).

6. W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005). [CrossRef]  

7. W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Pseudophase information from the complex analytic signal of speckle fields and its applications. Part I: Microdisplacement observation based on phase-only correlation in the signal domain,” Appl. Opt. 44, 4909–4915 (2005). [CrossRef]   [PubMed]  

8. W. Wang, T. Yokozeki, R. Ishijima, A. Wada, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120–127 (2006). [CrossRef]   [PubMed]  

9. J. F. Nye and M. V. Berry, “Dislocation in wave trains,” Proc. Roy. Soc. Lond. A 336165–190 (1974). [CrossRef]  

10. J. W. Goodman, Statistical Optics, (Wiley-Interscience, New York, 2000).

11. D. Gabor, “Theory of communications,” J. Inst. Electr. Eng. 93, 429–457 (1946).

12. M. Riesz, “Sur les fonctions conjuguees,” Math. Zeitschrift 27, 218–244 (1927). [CrossRef]  

13. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18, 1862–1870 (2001). [CrossRef]  

14. T. Yokozeki, “Displacement measurement using phase singularities in the Riesz transform of a speckle pattern,” Master Thesis, The University of Electro-Communications (January 30, 2006).

15. C. -S. Guo, Y. -J. Han, J. -B. Xu, and J. Ding, “Radial Hilbert transform with Laguerre-Gaussian spatial filters,” Opt. Lett. 31, 1394–1396 (2006). [CrossRef]   [PubMed]  

16. M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A , 456, 2059–2079 (2000). [CrossRef]  

17. W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902 (2005). [CrossRef]   [PubMed]  

18. M. R. Dennis, “Local structure of wave dislocation lines: twist and twirl,” J. Opt. A: Pure Appl. Opt. 6, S202–S208 (2004). [CrossRef]  

19. Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” J. Opt. A: Pure Appl. Opt. 6, S217–S228 (2004). [CrossRef]  

20. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002). [CrossRef]  

21. A. Asundi and H. North, “White-light speckle method- Current trends,” Opt. Laser Eng. 29, 159–169 (1998). [CrossRef]  

22. See, for example, X. Dai, O. Sasaki, J. E. Greivenkamp, and T. Suzuki, “Measurement of small rotation angles by a parallel interference pattern,” Appl. Opt. 34, 6380–6388 (1995). [CrossRef]   [PubMed]  

23. B. Rose, H. Imam, S. G. Hanson, H. Y. Yura, and R. S. Hansen, “Laser speckle angular displacement sensor: theoretical and experimental study” Appl. Opt. 37, 2119–2129 (1998). [CrossRef]  

References

  • View by:

  1. A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena,J. C. Dainty, ed. (Springer -Verlag, Berlin, 1976).
  2. R. K. Erf, Speckle Metrology, (Academic Press, New York, 1978).
  3. R. S. Sirohi, Speckle Metrology, (Marcel Dekker Inc., New York, 1993).
  4. N. A. Fomin, Speckle Photography for Fluid Mechanics Measurements, (Springer-Verlag, Berlin, 1998).
  5. S. J. Kirkpatrick and D. D. Duncan, “Optical assessment of tissue mechanics,” in Handbook of Optical Biomedical Diagnostics,V. V. Tuchin, ed., (SPIE Press, Bellingham, 2002).
  6. W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
    [Crossref]
  7. W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Pseudophase information from the complex analytic signal of speckle fields and its applications. Part I: Microdisplacement observation based on phase-only correlation in the signal domain,” Appl. Opt. 44, 4909–4915 (2005).
    [Crossref] [PubMed]
  8. W. Wang, T. Yokozeki, R. Ishijima, A. Wada, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120–127 (2006).
    [Crossref] [PubMed]
  9. J. F. Nye and M. V. Berry, “Dislocation in wave trains,” Proc. Roy. Soc. Lond. A 336165–190 (1974).
    [Crossref]
  10. J. W. Goodman, Statistical Optics, (Wiley-Interscience, New York, 2000).
  11. D. Gabor, “Theory of communications,” J. Inst. Electr. Eng. 93, 429–457 (1946).
  12. M. Riesz, “Sur les fonctions conjuguees,” Math. Zeitschrift 27, 218–244 (1927).
    [Crossref]
  13. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18, 1862–1870 (2001).
    [Crossref]
  14. T. Yokozeki, “Displacement measurement using phase singularities in the Riesz transform of a speckle pattern,” Master Thesis, The University of Electro-Communications (January 30, 2006).
  15. C. -S. Guo, Y. -J. Han, J. -B. Xu, and J. Ding, “Radial Hilbert transform with Laguerre-Gaussian spatial filters,” Opt. Lett. 31, 1394–1396 (2006).
    [Crossref] [PubMed]
  16. M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A,  456, 2059–2079 (2000).
    [Crossref]
  17. W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902 (2005).
    [Crossref] [PubMed]
  18. M. R. Dennis, “Local structure of wave dislocation lines: twist and twirl,” J. Opt. A: Pure Appl. Opt. 6, S202–S208 (2004).
    [Crossref]
  19. Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” J. Opt. A: Pure Appl. Opt. 6, S217–S228 (2004).
    [Crossref]
  20. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
    [Crossref]
  21. A. Asundi and H. North, “White-light speckle method- Current trends,” Opt. Laser Eng. 29, 159–169 (1998).
    [Crossref]
  22. See, for example, X. Dai, O. Sasaki, J. E. Greivenkamp, and T. Suzuki, “Measurement of small rotation angles by a parallel interference pattern,” Appl. Opt. 34, 6380–6388 (1995).
    [Crossref] [PubMed]
  23. B. Rose, H. Imam, S. G. Hanson, H. Y. Yura, and R. S. Hansen, “Laser speckle angular displacement sensor: theoretical and experimental study” Appl. Opt. 37, 2119–2129 (1998).
    [Crossref]

2006 (2)

2005 (3)

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[Crossref]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Pseudophase information from the complex analytic signal of speckle fields and its applications. Part I: Microdisplacement observation based on phase-only correlation in the signal domain,” Appl. Opt. 44, 4909–4915 (2005).
[Crossref] [PubMed]

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902 (2005).
[Crossref] [PubMed]

2004 (2)

M. R. Dennis, “Local structure of wave dislocation lines: twist and twirl,” J. Opt. A: Pure Appl. Opt. 6, S202–S208 (2004).
[Crossref]

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” J. Opt. A: Pure Appl. Opt. 6, S217–S228 (2004).
[Crossref]

2002 (1)

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
[Crossref]

2001 (1)

2000 (1)

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A,  456, 2059–2079 (2000).
[Crossref]

1998 (2)

1995 (1)

1974 (1)

J. F. Nye and M. V. Berry, “Dislocation in wave trains,” Proc. Roy. Soc. Lond. A 336165–190 (1974).
[Crossref]

1946 (1)

D. Gabor, “Theory of communications,” J. Inst. Electr. Eng. 93, 429–457 (1946).

1927 (1)

M. Riesz, “Sur les fonctions conjuguees,” Math. Zeitschrift 27, 218–244 (1927).
[Crossref]

Asundi, A.

A. Asundi and H. North, “White-light speckle method- Current trends,” Opt. Laser Eng. 29, 159–169 (1998).
[Crossref]

Berry, M. V.

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A,  456, 2059–2079 (2000).
[Crossref]

J. F. Nye and M. V. Berry, “Dislocation in wave trains,” Proc. Roy. Soc. Lond. A 336165–190 (1974).
[Crossref]

Bone, D. J.

Dai, X.

Dennis, M. R.

M. R. Dennis, “Local structure of wave dislocation lines: twist and twirl,” J. Opt. A: Pure Appl. Opt. 6, S202–S208 (2004).
[Crossref]

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
[Crossref]

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A,  456, 2059–2079 (2000).
[Crossref]

Ding, J.

Duncan, D. D.

S. J. Kirkpatrick and D. D. Duncan, “Optical assessment of tissue mechanics,” in Handbook of Optical Biomedical Diagnostics,V. V. Tuchin, ed., (SPIE Press, Bellingham, 2002).

Egorov, Y. A.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” J. Opt. A: Pure Appl. Opt. 6, S217–S228 (2004).
[Crossref]

Ennos, A. E.

A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena,J. C. Dainty, ed. (Springer -Verlag, Berlin, 1976).

Erf, R. K.

R. K. Erf, Speckle Metrology, (Academic Press, New York, 1978).

Fadeyeva, T. A.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” J. Opt. A: Pure Appl. Opt. 6, S217–S228 (2004).
[Crossref]

Fomin, N. A.

N. A. Fomin, Speckle Photography for Fluid Mechanics Measurements, (Springer-Verlag, Berlin, 1998).

Gabor, D.

D. Gabor, “Theory of communications,” J. Inst. Electr. Eng. 93, 429–457 (1946).

Goodman, J. W.

J. W. Goodman, Statistical Optics, (Wiley-Interscience, New York, 2000).

Greivenkamp, J. E.

Guo, C. -S.

Han, Y. -J.

Hansen, R. S.

Hanson, S. G.

Imam, H.

Ishii, N.

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[Crossref]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Pseudophase information from the complex analytic signal of speckle fields and its applications. Part I: Microdisplacement observation based on phase-only correlation in the signal domain,” Appl. Opt. 44, 4909–4915 (2005).
[Crossref] [PubMed]

Ishijima, R.

Kirkpatrick, S. J.

S. J. Kirkpatrick and D. D. Duncan, “Optical assessment of tissue mechanics,” in Handbook of Optical Biomedical Diagnostics,V. V. Tuchin, ed., (SPIE Press, Bellingham, 2002).

Larkin, K. G.

Miyamoto, Y.

W. Wang, T. Yokozeki, R. Ishijima, A. Wada, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120–127 (2006).
[Crossref] [PubMed]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[Crossref]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Pseudophase information from the complex analytic signal of speckle fields and its applications. Part I: Microdisplacement observation based on phase-only correlation in the signal domain,” Appl. Opt. 44, 4909–4915 (2005).
[Crossref] [PubMed]

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902 (2005).
[Crossref] [PubMed]

North, H.

A. Asundi and H. North, “White-light speckle method- Current trends,” Opt. Laser Eng. 29, 159–169 (1998).
[Crossref]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocation in wave trains,” Proc. Roy. Soc. Lond. A 336165–190 (1974).
[Crossref]

Oldfield, M. A.

Riesz, M.

M. Riesz, “Sur les fonctions conjuguees,” Math. Zeitschrift 27, 218–244 (1927).
[Crossref]

Rose, B.

Sasaki, O.

Sirohi, R. S.

R. S. Sirohi, Speckle Metrology, (Marcel Dekker Inc., New York, 1993).

Suzuki, T.

Takeda, M.

W. Wang, T. Yokozeki, R. Ishijima, A. Wada, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120–127 (2006).
[Crossref] [PubMed]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[Crossref]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Pseudophase information from the complex analytic signal of speckle fields and its applications. Part I: Microdisplacement observation based on phase-only correlation in the signal domain,” Appl. Opt. 44, 4909–4915 (2005).
[Crossref] [PubMed]

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902 (2005).
[Crossref] [PubMed]

Volyar, A. V.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” J. Opt. A: Pure Appl. Opt. 6, S217–S228 (2004).
[Crossref]

Wada, A.

Wang, W.

W. Wang, T. Yokozeki, R. Ishijima, A. Wada, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120–127 (2006).
[Crossref] [PubMed]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[Crossref]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Pseudophase information from the complex analytic signal of speckle fields and its applications. Part I: Microdisplacement observation based on phase-only correlation in the signal domain,” Appl. Opt. 44, 4909–4915 (2005).
[Crossref] [PubMed]

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902 (2005).
[Crossref] [PubMed]

Xu, J. -B.

Yokozeki, T.

W. Wang, T. Yokozeki, R. Ishijima, A. Wada, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120–127 (2006).
[Crossref] [PubMed]

T. Yokozeki, “Displacement measurement using phase singularities in the Riesz transform of a speckle pattern,” Master Thesis, The University of Electro-Communications (January 30, 2006).

Yura, H. Y.

Appl. Opt. (3)

J. Inst. Electr. Eng. (1)

D. Gabor, “Theory of communications,” J. Inst. Electr. Eng. 93, 429–457 (1946).

J. Opt. A: Pure Appl. Opt. (2)

M. R. Dennis, “Local structure of wave dislocation lines: twist and twirl,” J. Opt. A: Pure Appl. Opt. 6, S202–S208 (2004).
[Crossref]

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” J. Opt. A: Pure Appl. Opt. 6, S217–S228 (2004).
[Crossref]

J. Opt. Soc. Am. A (1)

Math. Zeitschrift (1)

M. Riesz, “Sur les fonctions conjuguees,” Math. Zeitschrift 27, 218–244 (1927).
[Crossref]

Opt. Commun. (2)

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[Crossref]

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
[Crossref]

Opt. Express (1)

Opt. Laser Eng. (1)

A. Asundi and H. North, “White-light speckle method- Current trends,” Opt. Laser Eng. 29, 159–169 (1998).
[Crossref]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902 (2005).
[Crossref] [PubMed]

Proc. R. Soc. Lond. A (1)

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A,  456, 2059–2079 (2000).
[Crossref]

Proc. Roy. Soc. Lond. A (1)

J. F. Nye and M. V. Berry, “Dislocation in wave trains,” Proc. Roy. Soc. Lond. A 336165–190 (1974).
[Crossref]

Other (7)

J. W. Goodman, Statistical Optics, (Wiley-Interscience, New York, 2000).

A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena,J. C. Dainty, ed. (Springer -Verlag, Berlin, 1976).

R. K. Erf, Speckle Metrology, (Academic Press, New York, 1978).

R. S. Sirohi, Speckle Metrology, (Marcel Dekker Inc., New York, 1993).

N. A. Fomin, Speckle Photography for Fluid Mechanics Measurements, (Springer-Verlag, Berlin, 1998).

S. J. Kirkpatrick and D. D. Duncan, “Optical assessment of tissue mechanics,” in Handbook of Optical Biomedical Diagnostics,V. V. Tuchin, ed., (SPIE Press, Bellingham, 2002).

T. Yokozeki, “Displacement measurement using phase singularities in the Riesz transform of a speckle pattern,” Master Thesis, The University of Electro-Communications (January 30, 2006).

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Figures (5)

Fig. 1.
Fig. 1. Core structure around a phase singularity with zero crossings of real and imaginary parts of complex signal representation for a speckle pattern. (a) Amplitude contours and zero crossing lines; (b) Pseudophase structure.
Fig. 2.
Fig. 2. (a). Displacement of phase singularities, ◦: before displacement; ■: after displacement; Histograms of coordinate changes of phase singularities, (b) x -direction;(c) y -direction.
Fig. 3.
Fig. 3. Results of phase singularities after fine identification (a) Displacement of phase singularities; Histograms of coordinate changes of phase singularities, (b) x -direction;(c) y -direction.
Fig. 4.
Fig. 4. Schematic diagram for in-plane rotation measurement by optical vortex metrology
Fig. 5.
Fig. 5. Results of optical vortex metrology applied for rotational displacement. Left column: rough searching over whole probe area; Right column: local searching for fine identification. (a), (b) Displacement of phase singularities; (c), (d) Histogram of rotation angle of phase singularities (Note the difference in the scales.)

Equations (21)

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I ˜ ( x , y ) = + + LG ( f x , f y ) · ( f x , f y ) exp [ j 2 π ( f x x + f y y ) ] d f x d f y ,
LG ( f x , f y ) = ( f x + j f y ) exp [ ( f x 2 + f y 2 ) / ω 2 ] = ρ exp ( ρ 2 / ω 2 ) exp ( ) .
I ˜ ( x , y ) = I ˜ ( x , y ) exp [ ( x , y ) ] = I ( x , y ) * ( x , y ) ,
( x , y ) = 1 { LG ( f x , f y ) } = ( j π 2 ω 4 ) ( x + jy ) exp [ π 2 ω 2 ( x 2 + y 2 ) ]
= ( j π 2 ω 4 ) [ r exp ( π 2 r 2 ω 2 ) exp ( ) ] ,
Re [ I ˜ ( x , y ) ] = a r x + b r y + c r , Im [ I ˜ ( x , y ) ] = a i x + b i y + c i ,
e = 1 ( a r 2 + a i 2 + b r 2 + b i 2 ) ( a r 2 + a i 2 b r 2 b i 2 ) 2 + 4 ( a r b r + a i b i ) 2 ( a r 2 + a i 2 + b r 2 + b i 2 ) + ( a r 2 + a i 2 b r 2 b i 2 ) 2 + 4 ( a r b r + a i b i ) 2 ,
θ RI = { arctan [ ( a r b i a i b r ) / ( a r a i + b r b i ) ] θ RI < π / 2 π arctan [ ( a r b i a i b r ) / ( a r a i + b r b i ) ] θ RI > π / 2 ,
Ω = a r b i a r b r ,
q = sgn ( Ω · e z ) = sgn ( a r b i a i b r ) .
q = q′ ,
Δ e = e e′ < ε 1 ,
ΔΩ = ( Ω Ω ) / ( Ω + Ω′ ) < ε 2 ,
Δ θ RI = θ RI θ′ RI < ε 3 ,
E = ( e e′ ) 2 + ( Ω Ω′ Ω + Ω′ ) 2 + [ 2 π ( θ RI θ′ RI ) ] 2 .
A i x + B i y + C i = 0
A i = 2 ( x i x′ i ) , B i = 2 ( y i y′ i ) , and C i = ( x′ i 2 + y′ i 2 x i 2 y i 2 ) ,
x c = ( i B i 2 ) ( i A i C i ) ( i A i B i ) ( i B i C i ) ( i A i 2 ) ( i B i 2 ) ( i A i B i ) 2 ,
y c = ( i A i 2 ) ( i B i C i ) ( i A i B i ) ( i A i C i ) ( i A i 2 ) ( i B i 2 ) ( i A i B i ) 2 .
x = x c + ( x i x c ) cos ϕ + ( y i y c ) sin ϕ ,
y = y c ( x i x c ) sin ϕ + ( y i y c ) cos ϕ .

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