Optics and Lasers in Engineering 50 (2012) 399–404

Contents lists available at SciVerse ScienceDirect

Optics and Lasers in Engineering

0143-81

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/optlaseng

Displacement measurement based on joint fractional Fouriertransform correlator

Peng Ge, Qi Li n, Huajun Feng, Zhihai Xu

State Key Lab of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, China

a r t i c l e i n f o

Article history:

Received 27 March 2011

Received in revised form

13 October 2011

Accepted 21 October 2011Available online 12 November 2011

Keywords:

Image processing

Displacement measurement

Fractional Fourier transform

Joint fractional Fourier transform correlator

66/$ - see front matter & 2011 Elsevier Ltd. A

016/j.optlaseng.2011.10.016

esponding author.

ail address: liqi@zju.edu.cn (Q. Li).

a b s t r a c t

A displacement measurement technology based on joint fractional Fourier transform is firstly proposed.

Contrast to conventional displacement measurement based on joint Fourier transform correlator, the

position of cross correlation peak in the proposed technology could be fixed arbitrarily according to the

order of fractional Fourier transform. The optical setup in the proposed technology is more flexible and

easier to implement. Simulation and experiment results are given out to verify the analysis.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The joint Fourier transform correlator (JFTC) was proposed byWeaver and Goodman in 1966 [1]. It is a 4f system and uses twoFourier lenses to correlate two related images, and has beenwidely used in pattern recognition and motion tracking [2–5].Janschek and et al. proposed using JFTC to detect the offsetbetween imaging devices and principle axis in order to compen-sate the blurred images and navigate satellites, monitor spaces inreal-time [6–10]. They made use of JFTC to find out the correlationpeaks related to the displacement vector. Once the space separa-tion between the reference image and the target image is chosen,the positions of two cross correlation peaks are fixed. If the spaceseparation and displacement between the reference image andthe target image is too small, the cross correlation may bedispersed in the auto-correlation signal. On the other hand, ifthe space separation and the displacement between them is solarge that the cross correlation peaks may fall outside the viewfield of detecting devices. These will both result in failuredetection of cross correlation. Thus it is necessary to control thepositions of the cross correlation peaks if possible.

We proposed using joint fractional Fourier transform correla-tor (JFrFTC) to solve this problem. Fractional Fourier transform isemployed instead of common Fourier transform. The referenceimage and the target image are fractional Fourier transformedand are captured by a square law device. Through anotherfractional Fourier transform the cross correlation peaks, which

ll rights reserved.

relate to the displacement would be positioned according to thefractional order that we choose.

The completed concept of Fractional Fourier transform (FrFT) wasproposed by Namias in 1980 [11]. In 1993 Lohmann introduced theFrFT to optics for the first time [12] and gave two optical implemen-tation setups [13,14]. Then in 1995 Mendlovic, Ozaktas and Lohmannproposed the concept of joint fractional Fourier transform correlator(JFrFTC) [14]. Since then JFrFTC has been further studied andachievements have been expressed in lists [15,16].

2. Displacement measurement based on JFrFTC

The fractional Fourier transform (FrFT) of image g(x0,y0) withthe order p is defined as [11]

Fp gðx0,y0Þ� �

¼ Cp

Z Z 1�1

exp jx2

0þx21þy2

0þy21

2tana�j

x0x1þy0y1

sina

� ��gðx0,y0Þdx0dy0, ð1Þ

where Cp is a constant and a¼pp/2.The shift rules of FrFT [11,17–19] are different from conven-

tional Fourier transform and are expressed as:

Fp expðjbx0Þgðx0Þ� �

¼ exp jb cos a x1�b

2sin a

� �� ��Fpfgðx0Þgðx1�b sin aÞ, ð2Þ

Fp gðx0þaÞ� �

¼ exp ja sin a x1þa

2cos a

� h iFp gðx0Þ� �

ðx1þa cos aÞ,

ð3Þ

where a,b are constants in Eqs. (2) and (3).

P. Ge et al. / Optics and Lasers in Engineering 50 (2012) 399–404400

In our analysis, the reference image is expressed as f(x0,y0), andthe target image has a displacement of (a,c) relative to the referenceimage, so the target image could be expressed as g(x0þa,y0þc). Thereference image and the target image are displayed symmetrically atthe position (�b,0),(þb,0) on the input plane. Thus the joint inputimage is expressed by:

uðx0,y0Þ ¼ f ðx0�b,y0Þþgðx0þbþa,y0þcÞ: ð4Þ

Appling the first FrFT with an order p1 on Eq. (4), we get thespectrum of u(x0,y0) expressed as:

Fp1 uðx0,y0Þ� �

¼ exp �jb sin a1 x1�b

2cos a1

� �� �Fp1 f ðx0,y0Þ� �

�ðx1�b cos a1,y1Þ

þexp jðaþbÞsin a1 x1þaþb

2cos a1

� �� �

�exp jc sin a1 y1þc

2cos a1

� h i�Fp1 fgðx0,y0Þgðx1þðaþbÞcosa1,y1þccosa1Þ: ð5Þ

The intensity (power spectrum) of u(x0,y0) is:

I¼ 9Fp1 uðx0,y0Þ� �

92¼ 9Fp1 f ðx0,y0Þ

� �ðx1�b cos a1,y1Þ9

2

þ9Fp1 gðx0,y0Þ� �

ðx1þðaþbÞcos a1,y1þc cos a1Þ92

þFp1 f ðx0,y0Þ� �

ðx1�b cos a1,y1ÞFp1n gðx0,y0Þ� �

�ðx1þðaþbÞcos a1,y1þc cos a1Þ

�exp �j sin a1 ðaþ2bÞx1�a2þ2ab

2cos a1

�� �

�exp �jc sin a1 y1þc

2cos a1

� h iþFp1n f ðx0,y0Þ

� �ðx1�b cos a1,y1ÞF

p1 gðx0,y0Þ� �

�ðx1þðaþbÞcos a1,y1þc cos a1Þ

�exp j sin a1 ðaþ2bÞx1þa2þ2ab

2cos a1

�� �

�exp jc sin a1 y1þc

2cos a1

� h i: ð6Þ

Due to the shift variant property of FrFT, Eq. (6) has itemsðx1þðaþbÞcos a1,y1þc cos a1Þ and ðx1�b cos a1,y1Þ. If we apply asecond FrFT with the order p2 on Eq. (6), it is hard to get the rightresult. So in this step, we let p1¼1, a1¼p1p/2, cosa1 ¼ 0. Eq. (6)can be simplified to

I¼ fFp1 ðuÞg2

¼ 9Fðx1,y1Þexpð�jbx1ÞþGðx1,y1Þexp½jbðbþaÞx1þ jcy1�92

¼ 9Fðx1,y1Þ92þ9Gðx1,y1Þ9

2þFnðx1,y1Þexpðjbx1ÞGðx1,y1Þexp½jðaþbÞx1þ jcy1�

Fðx1,y1Þexpð�jbx1ÞGnðx1,y1Þexp½�jðaþbÞx1�jcy1�: ð7Þ

Fig. 1. (a) Schematic architecture of JF

The first and second items in Eq. (7) are auto-correlation,which will produce a huge DC item existing in the correlationplane. This term is not beneficial to the displacement measure-ment. The third and fourth items in Eq. (7) are cross correlationand its conjugate item. We consider only the fourth item andapply the second FrFT with the order p2 on it,

Fp2 Fðx1,y1Þexpð�jbx1ÞGnðx1,y1Þexp �jðaþbÞx1�jcy1

� � �¼ Fp2 Fðx1,y1ÞG

nðx1,y1Þexpð�jðaþ2bÞx1�jcy1Þ

� �¼ exp �jðaþ2bÞcos a2 x2þ

aþ2b

2sin a2

� �� �dexp �jc cos a2 y2þ

c

2sin a2

� h i

�Cp1 ,p2ðx2þðaþ2bÞsin a2,y2þc sin a2Þ, ð8Þ

where

Cp1 ,p2¼ Fp2 fFp1 ½f ðx0,y0Þ�F

pn

1 ½gðx0,y0Þ�g: ð9Þ

This is the cross correlation, and it will be a bright peak whencaptured by a square law device. First we need to fix the idealposition when the target image has no displacement relative tothe reference image, that is, the target image is exactly the sameas the reference image. In this situation, a¼0, c¼0. ThroughJFrFTC the coordinates of cross correlation peak (xi,yi) is detectedby:

ðaþ2bÞsin a2 ¼ xi

c sin a2 ¼ yi

a¼ 0

c¼ 0

:

8>>><>>>:

ð10Þ

Then consider the target image that has nonzero displacement(a,c) relative to the reference image. The coordinates of crosscorrelation peak (xp,yp) is detected by:

ðaþ2bÞsin a2 ¼ xp

c sin a2 ¼ yp

(ð11Þ

Unite Eqs. (10) and (11), we get the displacement (a,c) as:

a¼ xp�xi

sin a2¼

xp�xi

sinp2p

2

c¼yp�yi

sin a2¼

yp�yi

sinp2p

2

:

8><>: ð12Þ

Contrast to the conventional JFTC, the correlation peaks’coordinates are not only relevant to the separation b anddisplacement a or c between the target and the reference image,but also relate to the fractional order p2.

According to Ref. [14], the schematic optical setup of jointfractional Fourier transform correlator (JFrFTC) is shown in Fig. 1.

rFTC; (b) Real experimental setup.

Fig. 2. (a) Input images; (b) Cross correlation of JFrFTC when p2¼0.1; (c) Cross correlation of JFTC.

Fig. 3. 3D plot of JFrFTC with (a) p2¼1; (b) p2¼0.95; (c) p2¼0.8; (d) p2¼0.1.

P. Ge et al. / Optics and Lasers in Engineering 50 (2012) 399–404 401

In Fig. 1,

d¼ f ð1�cos a2Þ ¼ f 1�cosp2p

2

� : ð13Þ

3. Simulation and experiment results

Simulations are performed according to Fig. 1. The targetimage has displacement (5,5) pixels relative to the referenceimage. They form joint input images as shown in Fig. 2 (a). Weuse digital fractional Fourier transform instead of optical frac-tional Fourier transform to calculate the power spectrum of thejoint input images, and then apply digital fractional Fourier

transform to the power spectrum to obtain the correlation output.Distance d is chosen to determine the fractional order. The DCitem in Eq. (7) is subtracted. The cross correlation of Fig. 2(a) byJFrFTC with the order p2¼0.1 is shown in Fig. 2(b).And the crosscorrelation of Fig. 2(a) by JFTC is shown in Fig. 2(c). It is apparentthat the distance between the two correlation peaks depends onthe order of p2. We can control the position of cross peaks bychoosing different p2. The 3D plot of correlation with different p2

is shown in Fig. 3. The displacement detected with different p2 isshown in Table 1. We used centriod algorithm to acquire sub-pixel precision. From Table 1, we can see that as p2 decreases, theerrors become larger, because sin a2 decreases more rapidly, andthe small region around the integral pixel coordinates of cross

Table 1Simulation results.

p2 a(pixel) c(pixel) d(mm) Errors on x axis Errors on y axis

1 4.9999 4.9708 300 �0.0001 �0.0292

0.95 5.0124 4.7572 276.3963 0.0124 �0.2428

0.9 5.0503 5.0319 253.0918 0.0503 0.0319

0.85 5.126 5.117 230.004 0.126 0.117

0.8 4.8417 4.7255 207.3351 �0.1583 �0.2745

0.75 4.5436 4.3249 185.2098 �0.4564 �0.6751

Fig. 4. (a) Two same reference image and (b) the reference image and the 10th

target image.

Fig. 5. Output when d¼300 mm, p2¼1: (a) total correlation output of Fig. 4(a); (b) the l

(b) and (c) are captured in the same position.

Fig. 6. Output when d¼280 mm, p2¼0.96: (a) total correlation output of Fig. 4(a); (b

Fig. 4(b), (b) and (c) are captured in the same position.

Fig. 7. Output when d¼250 mm, p2¼0.89: (a) total correlation output of Fig. 4(a); (b

Fig. 4(b), (b) and (c) are captured in the same position.

P. Ge et al. / Optics and Lasers in Engineering 50 (2012) 399–404402

correlation peak is vital to get the sub-pixel precision. The size ofthe region selected to interpolate for sub-pixel precision willaffect detection precision after dividing sin a2 when p2 is toosmall. The distance d between Fourier lens 2 and output plane 2,could be shortened when p2 decreases. So the total length ofstructure can be adjusted when choosing different p2.

Experiments are accomplished according to Fig. 1. The wave-length of laser1 and laser2 is 650 nm. The focal length of FT lens1and lens 2 is 300 mm. The SLM is made by Xi’an GongchuangGuangdian corp. Its maximum resolution is 1920�1080 pixels.The resolution of SLM1 and SLM2 is set to 1024�768 pixels. Theresolution of CCD, A602f we use here, is 656�491 pixels, madeby Basler Corp. and the resolution of reference image is 200�200pixels. We move the reference image by (2, 2) pixels each stepand repeat it by 10 times to form the target images. So the 10thtarget has displacement (20, 20) relative to the reference image.The input image is shown in Fig. 4. Fig. 4(a) is used to figure outthe ideal cross correlation peaks because two images have nodisplacement. Fig. 4(b) is composed of the reference image andthe 10th target image. The joint input image is displayed on SLM1through COM1.

After being collimated by collimator lens and turned with901by mirror 1, the optical beams generated by laser 1 passesthrough cubic polarization beam splitter 1 (PBS1) and enters intoinput SLM1. The reflected light modulated by SLM1 takes along

eft cross-correlation peak of Fig. 4(a); (c) the left cross-correlation peak of Fig. 4(b),

) the left cross-correlation peak of Fig. 4(a); (c) the left cross-correlation peak of

) the left cross-correlation peak of Fig. 4(a); (c) the left cross-correlation peak of

Fig. 8. Output when d¼220 mm, p2¼0.83: (a) total correlation output of Fig. 4(a); (b) the left cross-correlation peak of Fig. 4(a); (c) the left cross-correlation peak of

Fig. 4(b), (b) and (c) are captured in the same position.

Fig. 9. Output when d¼150 mm, p2¼0.67: (a) total correlation output of Fig. 4(a); (b) the left cross-correlation peak of Fig. 4(a); (c) the left cross-correlation peak of

Fig.4(b), (b) and (c) are captured in the same position.

Fig. 10. Output when d¼100 mm, p2¼0.54: (a) total correlation output of Fig. 4(a); (b) total correlation of the reference image and the 5th target image; (c) total

correlation output of Fig.4(b), (a), (b) and (c) are captured in the same position.

Fig. 11. Errors on x axis when p2 chooses different values.

P. Ge et al. / Optics and Lasers in Engineering 50 (2012) 399–404 403

the information of the reference and the target image, succes-sively passes through PBS1, FT lens 1, and finally enters intoCCD1. CCD1 captures the power spectrum of the input images andtransports it to COM2. Then the spectrum is imported to SLM2 byCOM2. SLM2 transforms the digital spectrum signals into opticalsignals. After another fractional Fourier transformation proces-sing, the cross-correlation peaks are captured by CCD2. By findingthe cross-correlation’s peak and utilizing Eq. (12), the imagedisplacement is obtained. Experimental results with different p2

are given from Figs. 5–10.From Figs. 5 to 10, it is apparent that the distance of two cross-

correlation peaks shortens when p2 declines. It is useful forcorrelation detection when the view field of detecting device isnot large enough to cover the whole view field. And we can seewhen p2 decreases from 1 to 0.67, the intensity of cross correla-tion becomes very weak and it is too obscure to discriminate.Detection precisions are shown in Figs. 11 and 12. The errorsbecome larger when p2declines. So it is not wise enough to assignp2 a too small value. Fig. 5(a), 6(a), 7(a), 8(a), 9(a) and 10(a) arecaptured in the full view field. The differences ranging fromFig. 5(a) to 10(a) are caused by different p2. Fig. 5(b) and (c) arecaptured in the half view field, in the same position that we do notmove the position of CCD2. So are the Fig. 6(b) and (c), Fig. 7(b) and(c), Fig. 8(b) and (c), Fig. 9(b) and (c), Fig. 10(b) and (c). Fig. 5(b) is

captured when the reference and the target image are the sameand have no displacement, to fix the ideal position. Fig. 5(c) iscaptured when the reference and the 10th target image have arelative displacement.

Fig. 12. Errors on y axis when p2 chooses different values.

P. Ge et al. / Optics and Lasers in Engineering 50 (2012) 399–404404

4. Conclusion

A displacement measurement based on JFrFTC is proposed inthis paper. It is very convenient to assign p2, the order of thesecond fractional Fourier transform to determine the distance ofthe two cross-correlation peaks. It can solve the problem whenthe field of detecting device is not large enough to cover thewhole correlation plane, and can shorten the total length of thewhole structure. Simulation and experiments are given out toverify the analysis. The detection precisions will decrease when p2

is assigned a very small value. So it is not suggested to assign p2 atoo small value.

Acknowledgment

This research was supported by The National Natural ScienceFoundation of China (Grant no. 61178064) and National BasicResearch Program (973) of China (Grant no. 2009CB724002).

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