- Email: [email protected]
Optical distance sensing using digital speckle shearing interferometry
Optics and Lasers in Engineering 26 (1997) 449-460 Copyright 0 1996 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0143-8166/97/$15~00 OX%-8166(95)00143-3
ELSEVIER
Optical Distance Sensing Using Digital Speckle Shearing Interferometry T. W. Ng Department
(Received
of Mechanical & Production Engineering, National University 10, Kent Ridge Crescent, Singapore 0511 15 September
of Singapore,
1994; revised version received and accepted 13 March 1995)
ABSTRACT A novel distance sensing technique using digital speckle shearing interferometry is outlined in this paper. The primary incentive for the development of this technique is of one that can be: (i) incorporated into any TV imaging system; (ii) used as a focusing mechanism; (iii) suited to measure from diffuse targets; and (iv) relatively insensitive towards the environment. Results obtained reveal the feasibility of further research and development of this technique for practical applications. Copyright 0 1996 Elsevier Science Ltd.
1 INTRODUCTION Distance is one of the fundamental measurements in metrology. Although it is possible to measure spatial distance in 3D space, inevitably it is the horizontal equivalent that is required. The advent of optical measurement techniques has rendered physical methods like taping obsolete for all base-line measurements. Currently, there is a wide range of optical distance measuring techniques. These techniques can be grouped into four different classes and are conveniently named the time-of-flight, interferometric, geometric and diffraction techniques.’ Time-of-flight techniques operate on the simple principle of measuring the time delay between the propagation and receipt of pulsed or sinusoidial optical waves which are travelling with known velocity. Interferometric techniques involve splitting a coherent signal beam into two components, whereby one component is sent to the target and back whereas the other is used as the reference. The shift in phase is 449
T. W. Ng
used to sense the distance between the target and detector in this class of technique. Geometric techniques, in essence, function on the principle of triangulation, whereby a pair of similar triangles, one between the target and the measurement fulcrum, and the other between the image and measurement fulcrum, are established to sense distance. Diffraction techniques basically involve measuring the contrast properties of coherent illumination and relating them to determine the target distance. A general comparison of the effectiveness of individual techniques in each class will reveal various levels of strengths and weaknesses. For example, interferometric techniques generally give higher accuracy measurements as compared with the other class of techniques. Nevertheless, it is also more prone to noise arising from environmental vibration. Very often, the distance sensing technique that is adopted is dependent on the needs of the measurement scheme itself. The primary incentives for the development of the distance measurement technique outlined in this work is of one that can be (i) incorporated into any TV imaging system; (ii) used as a focusing mechanism; (iii) suited to measure from diffuse targets; and (iv) relatively insensitive towards the environment. These requirements restrict the direct use of distance measurement techniques that have been previously reported. The novel distance measuring technique outlined in this paper is based on digital speckle shearing interferometry (DSSI). Speckle shearing interferometry is a laser interferometric technique that has been used widely in the determination of flexural* and in-plane3 strains of diffuse object surfaces subjected to deformation. The optical configuration of speckle shearing interferometry renders it relatively insensitive towards environmental noise. Coupled with the incorporation of video camera recording and digital processing schemes,- which yield the fringe pattern almost instantaneously, DSSI has proliferated as a tool for non-destructive inspection. That the output signal from DSSI is digital in nature, facilitates post-processing. In this paper, image-processing techniques are incorporated to enhance the distance measurement procedure. A Fourier transform technique is shown to accurately measure the fringe period, which is an important parameter in the distance measurement procedure, despite the presence of speckle noise. A secondary fringe-sharpening technique is demonstrated to improve the accuracy of fringe-period determination.
2 TECHNIQUE
DESCRIPTION
The setup used to measure distance via DSSI is illustrated in Fig. 1. The target, which comprises any flat diffuse object, is illuminated by a laser
Optical distance sensing
451 Laser
Ta
Video
Image
Fig. 1.
ProcesIor
Personal
Digital speckle shearing interferometry
Camera
Computer
set-up for optical distance measurement.
light source expanded using a lens. The first step involves focusing the imaging system. In DSSI, intensity fringe-patterns are obtained when the light source is translated between two recordings and the two speckle images subsequently differenced. When the imaging system is out-offocus, fringes with low to zero contrast are obtained whereas high contrast fringes are obtained when the system is in-focus. By doing a successive number of double recordings and monitoring the fringe contrast, it is possible to bring the imaging system to the condition of best focus. After the imaging system is in-focus, measurement of the fringe distance in relation to expander source displacement will reveal the desired distance measurement.
3 FRINGE FORMING PRINCIPLES AND THEIR RELATIONSHIP TO DISTANCE MEASUREMENT Since speckle images are recorded before and after the expander lens is translated along the direction of illumination by an amount SR, the intensities obtained using the interferometric system are described by z1= a’, + a’b+ 244
cos (4)
Z2= a’, + a’b+ 24~~ cos (4 + A)
(la) (lb)
T. W. Ng
4.52
where a,, and ab are the amplitudes of the interfering wavefronts, 4 the random phase angle, and A’ the phase change due to the expander lens translation. Differencing the two speckle images yield a third speckle image described by
(2) When the light source is shifted from S, to S,, the relative phase shift A between the two sheared wavefronts induced by the relative optical path change Sd (see the Appendix) is described by A=ySd=
29@R)(&)X AR2
(3)
where A is the wavelength of the coherent light, R the distance between the light source and test target, and SX the magnitude of shearing. The fringe pattern is hence related to the phase change by A=(2n
+ l)n=
27r(SR)(6x)x hR2
where n is the fringe order. If the fringe period is denoted can be rewritten using
as x’, eqn (4)
Ii2 I From eqn (4), it is obvious that the fringe pattern basically comprises equally-spaced bands orientated normal to the direction of shear. Based on eqn (5), R can be measured provided that SR, 6x, A, and x’ are known,
4 FOURIER
TRANSFORM METHOD OF FRINGE DETERMINATION
SPACING
An important parameter to be determined in the distance measurement technique is the fringe period x’. The amount of speckle noise in the fringe pattern often does not allow this parameter to be accurately determined. The incorporation of image processing schemes is therefore necessary in order to extract this parameter precisely. Since the fringe pattern obtained comprises equally-spaced bands orientated normal to
Optical distance sensing
453
Amplitude
I\
* -ve
Fig. 2.
w
+ve
Frequency
Separated
Fourier spectra of a fringe pattern.
the x direction, the spatial intensity function d(x, y), described in eqn (2), may be rewritten as 44 y) = 4x7 y) + &, Y > cos [27cf,xl
(6)
where a(_~,y) describes the background variation, a(~, y) is related to the local contrast of the pattern, and fo the frequency of the equally-spaced fringes. Expressing eqn (6) in a more convenient form, we obtain i(-&y) = a(% y ) + 0(x, y ) + 6*(x, Y)
(7)
whereby 44 Y > = (MT and * denotes a complex respect to x gives
conjugate.
Y > exp
W7G4
Fourier
(8)
transforming
eqn (6) with
JUY)=J4f,Y)+~W%Y)+~*(f+f,,Y)
(9)
where the capital letters denote the Fourier spectra and f is the spatial frequency in the x direction. Since the spatial variations of a and 6 are slow compared with fo, the Fourier spectra is as shown in Fig. 2. The frequency values corresponding to either % or %‘*, which appear as peaks in the spectra apart from the DC term, depict f0 in the speckle image. The fringe period is basically determined by taking the inverse of fo.
5 FRINGE
SHARPENING
OF FRINGE
PATTERNS
Fringe sharpening has been previously demonstrated to enhance the visibility of speckle fringe patterns. ’ In this paper, fringe sharpening is incorporated to enhance the fringe period determination procedure using the Fourier transform technique mentioned in the previous section. This
T. W. Ng
454
feature is important in cases where the reflectance of illumination from the target surface is low; leading to fringes with low contrast. From eqn (3), it is obvious that A is directly proportional to the expander lens translation SR. If SR is incremented in equal progressive steps, the intensity values described in eqns (la) and (lb) can be generalized using z, = af + a’b+ 2a,ab cos (4 + (m - l)A)
(10)
where m = 1,2, 3, . . . is the number of data frames used. By analysing the binomial coefficients, it is shown that operations on three, four, five and six frames, respectively, produce fringe patterns given by I(Z1+ 1,)/2 - 121= (4u,ab cos (4 - A) sin’ t ( 11 l(Z1- Z,)/3 - Z2+ Z31= ((~)a,a,
cos (4 -
36/2) sin3 ($) /
/(I1 + Z,)/6 - 2(Zz+ Z,)/3 + Z3[= (y)a,a,, cos (4 - 2A) sin4 $ ( )I
(11) (12) (13)
](I1- Z,)/lO - (Z2- ~,)/2 + Z3- &I = /(~)a,a, cos (4 - SA/2) sin5 (g) I (14)
It is obvious that the fringe function is directly proportional to sin (A/2) raised to the power of (m - 1). As the number of data frames increase, the visibility of the fringe patterns improves, since the bright fringes become spatially narrower. Since the information to derive the fringe period is found in the Fourier spectra, it is necessary to study the accompanying effect in this domain. According to the multiplication property of Fourier transforms, g(W@) = G(f) @H(f) where Cl9denotes the convolution
(15)
of two functions which is defined by
G(f)eH(f)-l+=g(t)h(t-2)dz --m
(16)
It is easy to see that when more data frames are used, the spectrum described in Fig. 2 is convolved with itself since g(t) equals h(t) and G(f) equals H(f). The convolution operation of a function with itself produces a resultant function that is more defined at the peaks. The fringe sharpening process therefore also sharply defines the peaks of % and %* in the Fourier spectra; thereby allowing the frequency value of f0 to be more easily determined.
Optical distance sensing
6 EXPERIMENTAL
4.55
DEMONSTRATION
An experimental demonstration using the proposed technique is performed on a flat diffuse test target located 56.5 mm (measured using a tape) from the expander lens. In the test, A = 6.32 X lOA mm and Sx = 7.67 mm. Six speckle images are obtained for 6R = O-O-5 mm at equal steps of 0.1 mm. Fringe patterns are obtained using the operations described in eqns (2) and (ll)-(16). A cross-sectional intensity distribution is taken in the x direction from each fringe pattern. These distributions are Fourier transformed in order to determine the fringe period. To reduce the effect of spectral leakage at the ends, a Hanning window is multiplied over each intensity distribution before it is Fourier-transformed.
7 RESULTS
AND DISCUSSION
Figure 3(a)-(e) gives digital speckle shearing interferometry fringe patterns obtained by processing two, three, four, five and six frames of speckle pattern data, respectively. It can be seen that as increased frames of data are used, the fringe patterns progressively become more visible due to the narrowing of the bright fringes. Figure 4(a)-(c) gives the cross-sectional intensity distributions in the x direction of the fringe
Fig. 3.
Digital speckle shearing interferometry fringe pattern obtained by processing: (a) two; (b) three; (c) four; (d) five; and (e) six frames of speckle pattern data.
456
T. W. Ng
patterns processed using two, four, six frames of speckle data, respectively. It is obvious that when more frames of data are used, the fringes becomes more clearly defined despite the presence of speckle noise. However, even with the improved definition of the intensity distribution using six frames of speckle data, as shown in Fig. 4(c), it is difficult to determine the fringe period with certainty due to the effect of speckle noise. Figure 5(a)-(c) provides the respective Fourier spectra of the intensity distributions given in Fig. 4(a)-(c). With increasing frames of data, it can be seen that peaks of Ce and q* in eqn (9) increase in magnitude accordingly. With six frames of data, as shown in Fig. 5(c), these peaks are very pronounced and there is little trouble in determining fo from the spectrum. The fringe sharpening technique therefore enables improved measurements of the fringe period in order to determine distance. The fringe period is found to be 26*7mm using the proposed DSSI technique of distance measurement. Incorporation of this value and values of 6R, A and SX already determined in eqn (5) yields a distance value of 56.9 mm, which is barely 0.71% off the distance of 56.5 mm measured using tape. As with all physical measurement techniques, it is necessary to study the various influencing factors involved with the proposed technique in order to draw solid conclusions on the limits of its practical application. The list of possible contributing factors which may affect performance include beam divergence, speckle noise, electronic noise and decorrelation between each recorded frame. In this work, the focus is to demonstrate the working principles behind this technique as well as to present a set of measurement results to validate the technique. The results obtained, however, do reveal the feasibility of pursuing further research and development work on the technique. 8 CONCLUSION In this paper, a novel technique adapted from digital speckle shearing interferometry, is demonstrated to be able to sense distance optically. By performing a Fourier transform on the speckle fringe pattern, it is possible to extract the fringe period-which is an importrant parameter to be determined in the measurement procedure-with precision despite the presence of speckle noise. A fringe sharpening procedure is shown to considerably enhance the fringe period measurement procedure. The range of distances to be measured using the proposed technique can be adjusted via the manipulation of either the amount of expander lens translation or the amount of shear. Coupled with ease of usage, rugged
457
Optical distance sensing
25
0 0
50
100
150
200
250
300
350
400
450
500
0
50
100
150
200
250
300
350
400
450
500
350
400
450
500
r” : d C
30
20
10
0 Phi
Location
60
50
40 r” ;
30
E 20
10
0
50
100
150
200 Plxml
Fig. 4.
250
300
Locatlon
Intensity distribution of a cross-section in the x direction of fringe patterns processed using: (a) two; (b) four; and (c) six frames of speckle data.
T. W. Ng
458
400 g
5 : 300
2
c
.
I
500
(c)
-234
0
Frmqumoy 5. Fourier spectra of cross-sectional intensity distribution in the x direction of fringe-patterns processed using: (a) two; (b) four; and (c) six frames of speckle data. The zero-frequency is clipped to enhance the detail of the remaining spectra.
fig.
Optical distance sensing
design and the possibility for complete ises to be an attractive tool for distance
459
automation, measurement.
the technique
prom-
ACKNOWLEDGEMENT
The author is indebted to the reviewer for contributing many comments which have helped tremendously towards the improvement in presentation of this paper. REFERENCES
1. Strand, Engng,
T. C., Optical
three-dimensional
sensing for machine
vision. Opt.
24 (1985) 33-40.
2. Takezaki, J. & Hung, Y. Y., Direct measurement of flexural strains in plates by shearography. J. Appl. Mech., 53 (1986) 125-9. 3. Hung, Y. Y. & Liang, C. Y., Image-shearing camera for direct measurement of surface strains. Appl. Opt., 18 (1979) 1046-51. 4. Chau, F. S. & Ng, T. W., A real-time digital shearing speckle interferometer. Meas. Sci. Technol., 3 (1992) 381-3. 5. Ganesan, A. R., Sharma, D. K. & Kothiyal, M. P., Universal digital speckle shearing interferometer. Appl. Opt., 27 (1988) 4731-4. 6. Ng, T. W. & Chau, F. S., A digital shearing speckle interferometry technique for modal analysis. Appl. Acoustics, 42 (1994) 175-85. 7. Ganesan, A. R., Joenathan, C. & Sirohi, R. S., Sharpening of fringes in digital speckle pattern interferometry. Appl. Opt., 27 (1988) 2099-100.
APPENDIX
The optical path difference for the ray travelling from the light source &(x1, yl, zi) and S2(xz, yz, z2) to the camera 0(x,, y,, z,) via an arbitrary point P on the test target, as shown in Fig. Al, is given by
SlP = [(Xl - g2 + (y1 - y)’ + (Zl - z)2]“2 ZR 1
S2P
=
[(x2
[
1+x~+Y2+z2-2xx1-2Yyl-2zz1 2R: -x)’
+
1
(AlI
(y2 - y)2 + (z2 - Z)2]lD
x2+y2++
2xX2
2R;
-
2yy2 - 2222
I
(A21
T. W. Ng
460
Fig.Al. Optical path diagram of the digital speckle shearing interferometry technique for distance measurement. where Rf=x:+yf+z: and R$= xz + ys + z$. The equations are obtained by neglecting high-order terms in the series expansion. Therefore, the optical path length change dl from S, and S, to the point in the object is d&x, y, z) = (S,P + OP) - (&P + OP)
=R,-RI+-
; [gj(x2+y2+z2) 2
For a neighbouring as
643)
1
point P’(x + 6x, y, z), the optical path change d’ is expressed
d,(x + 6x, y, z) = (SIP + OP) - (S,P + OP)
[(x + 8x)’ +y2 + z2]
6-W
Neglecting small terms in the equation, the relative optical path change Sd is expressed as Sd = dz(x + 6x, y, z) - d,(x, y, z) R2 =
-
-
RI
[ R2R,
1
x8x
If RI = R2 >> 1, eqn (A5) can be rewritten as
(fw where SR is the amount of translation between the expander and test target.
of the expander,
and R is the distance
ELSEVIER
Optical Distance Sensing Using Digital Speckle Shearing Interferometry T. W. Ng Department
(Received
of Mechanical & Production Engineering, National University 10, Kent Ridge Crescent, Singapore 0511 15 September
of Singapore,
1994; revised version received and accepted 13 March 1995)
ABSTRACT A novel distance sensing technique using digital speckle shearing interferometry is outlined in this paper. The primary incentive for the development of this technique is of one that can be: (i) incorporated into any TV imaging system; (ii) used as a focusing mechanism; (iii) suited to measure from diffuse targets; and (iv) relatively insensitive towards the environment. Results obtained reveal the feasibility of further research and development of this technique for practical applications. Copyright 0 1996 Elsevier Science Ltd.
1 INTRODUCTION Distance is one of the fundamental measurements in metrology. Although it is possible to measure spatial distance in 3D space, inevitably it is the horizontal equivalent that is required. The advent of optical measurement techniques has rendered physical methods like taping obsolete for all base-line measurements. Currently, there is a wide range of optical distance measuring techniques. These techniques can be grouped into four different classes and are conveniently named the time-of-flight, interferometric, geometric and diffraction techniques.’ Time-of-flight techniques operate on the simple principle of measuring the time delay between the propagation and receipt of pulsed or sinusoidial optical waves which are travelling with known velocity. Interferometric techniques involve splitting a coherent signal beam into two components, whereby one component is sent to the target and back whereas the other is used as the reference. The shift in phase is 449
T. W. Ng
used to sense the distance between the target and detector in this class of technique. Geometric techniques, in essence, function on the principle of triangulation, whereby a pair of similar triangles, one between the target and the measurement fulcrum, and the other between the image and measurement fulcrum, are established to sense distance. Diffraction techniques basically involve measuring the contrast properties of coherent illumination and relating them to determine the target distance. A general comparison of the effectiveness of individual techniques in each class will reveal various levels of strengths and weaknesses. For example, interferometric techniques generally give higher accuracy measurements as compared with the other class of techniques. Nevertheless, it is also more prone to noise arising from environmental vibration. Very often, the distance sensing technique that is adopted is dependent on the needs of the measurement scheme itself. The primary incentives for the development of the distance measurement technique outlined in this work is of one that can be (i) incorporated into any TV imaging system; (ii) used as a focusing mechanism; (iii) suited to measure from diffuse targets; and (iv) relatively insensitive towards the environment. These requirements restrict the direct use of distance measurement techniques that have been previously reported. The novel distance measuring technique outlined in this paper is based on digital speckle shearing interferometry (DSSI). Speckle shearing interferometry is a laser interferometric technique that has been used widely in the determination of flexural* and in-plane3 strains of diffuse object surfaces subjected to deformation. The optical configuration of speckle shearing interferometry renders it relatively insensitive towards environmental noise. Coupled with the incorporation of video camera recording and digital processing schemes,- which yield the fringe pattern almost instantaneously, DSSI has proliferated as a tool for non-destructive inspection. That the output signal from DSSI is digital in nature, facilitates post-processing. In this paper, image-processing techniques are incorporated to enhance the distance measurement procedure. A Fourier transform technique is shown to accurately measure the fringe period, which is an important parameter in the distance measurement procedure, despite the presence of speckle noise. A secondary fringe-sharpening technique is demonstrated to improve the accuracy of fringe-period determination.
2 TECHNIQUE
DESCRIPTION
The setup used to measure distance via DSSI is illustrated in Fig. 1. The target, which comprises any flat diffuse object, is illuminated by a laser
Optical distance sensing
451 Laser
Ta
Video
Image
Fig. 1.
ProcesIor
Personal
Digital speckle shearing interferometry
Camera
Computer
set-up for optical distance measurement.
light source expanded using a lens. The first step involves focusing the imaging system. In DSSI, intensity fringe-patterns are obtained when the light source is translated between two recordings and the two speckle images subsequently differenced. When the imaging system is out-offocus, fringes with low to zero contrast are obtained whereas high contrast fringes are obtained when the system is in-focus. By doing a successive number of double recordings and monitoring the fringe contrast, it is possible to bring the imaging system to the condition of best focus. After the imaging system is in-focus, measurement of the fringe distance in relation to expander source displacement will reveal the desired distance measurement.
3 FRINGE FORMING PRINCIPLES AND THEIR RELATIONSHIP TO DISTANCE MEASUREMENT Since speckle images are recorded before and after the expander lens is translated along the direction of illumination by an amount SR, the intensities obtained using the interferometric system are described by z1= a’, + a’b+ 244
cos (4)
Z2= a’, + a’b+ 24~~ cos (4 + A)
(la) (lb)
T. W. Ng
4.52
where a,, and ab are the amplitudes of the interfering wavefronts, 4 the random phase angle, and A’ the phase change due to the expander lens translation. Differencing the two speckle images yield a third speckle image described by
(2) When the light source is shifted from S, to S,, the relative phase shift A between the two sheared wavefronts induced by the relative optical path change Sd (see the Appendix) is described by A=ySd=
29@R)(&)X AR2
(3)
where A is the wavelength of the coherent light, R the distance between the light source and test target, and SX the magnitude of shearing. The fringe pattern is hence related to the phase change by A=(2n
+ l)n=
27r(SR)(6x)x hR2
where n is the fringe order. If the fringe period is denoted can be rewritten using
as x’, eqn (4)
Ii2 I From eqn (4), it is obvious that the fringe pattern basically comprises equally-spaced bands orientated normal to the direction of shear. Based on eqn (5), R can be measured provided that SR, 6x, A, and x’ are known,
4 FOURIER
TRANSFORM METHOD OF FRINGE DETERMINATION
SPACING
An important parameter to be determined in the distance measurement technique is the fringe period x’. The amount of speckle noise in the fringe pattern often does not allow this parameter to be accurately determined. The incorporation of image processing schemes is therefore necessary in order to extract this parameter precisely. Since the fringe pattern obtained comprises equally-spaced bands orientated normal to
Optical distance sensing
453
Amplitude
I\
* -ve
Fig. 2.
w
+ve
Frequency
Separated
Fourier spectra of a fringe pattern.
the x direction, the spatial intensity function d(x, y), described in eqn (2), may be rewritten as 44 y) = 4x7 y) + &, Y > cos [27cf,xl
(6)
where a(_~,y) describes the background variation, a(~, y) is related to the local contrast of the pattern, and fo the frequency of the equally-spaced fringes. Expressing eqn (6) in a more convenient form, we obtain i(-&y) = a(% y ) + 0(x, y ) + 6*(x, Y)
(7)
whereby 44 Y > = (MT and * denotes a complex respect to x gives
conjugate.
Y > exp
W7G4
Fourier
(8)
transforming
eqn (6) with
JUY)=J4f,Y)+~W%Y)+~*(f+f,,Y)
(9)
where the capital letters denote the Fourier spectra and f is the spatial frequency in the x direction. Since the spatial variations of a and 6 are slow compared with fo, the Fourier spectra is as shown in Fig. 2. The frequency values corresponding to either % or %‘*, which appear as peaks in the spectra apart from the DC term, depict f0 in the speckle image. The fringe period is basically determined by taking the inverse of fo.
5 FRINGE
SHARPENING
OF FRINGE
PATTERNS
Fringe sharpening has been previously demonstrated to enhance the visibility of speckle fringe patterns. ’ In this paper, fringe sharpening is incorporated to enhance the fringe period determination procedure using the Fourier transform technique mentioned in the previous section. This
T. W. Ng
454
feature is important in cases where the reflectance of illumination from the target surface is low; leading to fringes with low contrast. From eqn (3), it is obvious that A is directly proportional to the expander lens translation SR. If SR is incremented in equal progressive steps, the intensity values described in eqns (la) and (lb) can be generalized using z, = af + a’b+ 2a,ab cos (4 + (m - l)A)
(10)
where m = 1,2, 3, . . . is the number of data frames used. By analysing the binomial coefficients, it is shown that operations on three, four, five and six frames, respectively, produce fringe patterns given by I(Z1+ 1,)/2 - 121= (4u,ab cos (4 - A) sin’ t ( 11 l(Z1- Z,)/3 - Z2+ Z31= ((~)a,a,
cos (4 -
36/2) sin3 ($) /
/(I1 + Z,)/6 - 2(Zz+ Z,)/3 + Z3[= (y)a,a,, cos (4 - 2A) sin4 $ ( )I
(11) (12) (13)
](I1- Z,)/lO - (Z2- ~,)/2 + Z3- &I = /(~)a,a, cos (4 - SA/2) sin5 (g) I (14)
It is obvious that the fringe function is directly proportional to sin (A/2) raised to the power of (m - 1). As the number of data frames increase, the visibility of the fringe patterns improves, since the bright fringes become spatially narrower. Since the information to derive the fringe period is found in the Fourier spectra, it is necessary to study the accompanying effect in this domain. According to the multiplication property of Fourier transforms, g(W@) = G(f) @H(f) where Cl9denotes the convolution
(15)
of two functions which is defined by
G(f)eH(f)-l+=g(t)h(t-2)dz --m
(16)
It is easy to see that when more data frames are used, the spectrum described in Fig. 2 is convolved with itself since g(t) equals h(t) and G(f) equals H(f). The convolution operation of a function with itself produces a resultant function that is more defined at the peaks. The fringe sharpening process therefore also sharply defines the peaks of % and %* in the Fourier spectra; thereby allowing the frequency value of f0 to be more easily determined.
Optical distance sensing
6 EXPERIMENTAL
4.55
DEMONSTRATION
An experimental demonstration using the proposed technique is performed on a flat diffuse test target located 56.5 mm (measured using a tape) from the expander lens. In the test, A = 6.32 X lOA mm and Sx = 7.67 mm. Six speckle images are obtained for 6R = O-O-5 mm at equal steps of 0.1 mm. Fringe patterns are obtained using the operations described in eqns (2) and (ll)-(16). A cross-sectional intensity distribution is taken in the x direction from each fringe pattern. These distributions are Fourier transformed in order to determine the fringe period. To reduce the effect of spectral leakage at the ends, a Hanning window is multiplied over each intensity distribution before it is Fourier-transformed.
7 RESULTS
AND DISCUSSION
Figure 3(a)-(e) gives digital speckle shearing interferometry fringe patterns obtained by processing two, three, four, five and six frames of speckle pattern data, respectively. It can be seen that as increased frames of data are used, the fringe patterns progressively become more visible due to the narrowing of the bright fringes. Figure 4(a)-(c) gives the cross-sectional intensity distributions in the x direction of the fringe
Fig. 3.
Digital speckle shearing interferometry fringe pattern obtained by processing: (a) two; (b) three; (c) four; (d) five; and (e) six frames of speckle pattern data.
456
T. W. Ng
patterns processed using two, four, six frames of speckle data, respectively. It is obvious that when more frames of data are used, the fringes becomes more clearly defined despite the presence of speckle noise. However, even with the improved definition of the intensity distribution using six frames of speckle data, as shown in Fig. 4(c), it is difficult to determine the fringe period with certainty due to the effect of speckle noise. Figure 5(a)-(c) provides the respective Fourier spectra of the intensity distributions given in Fig. 4(a)-(c). With increasing frames of data, it can be seen that peaks of Ce and q* in eqn (9) increase in magnitude accordingly. With six frames of data, as shown in Fig. 5(c), these peaks are very pronounced and there is little trouble in determining fo from the spectrum. The fringe sharpening technique therefore enables improved measurements of the fringe period in order to determine distance. The fringe period is found to be 26*7mm using the proposed DSSI technique of distance measurement. Incorporation of this value and values of 6R, A and SX already determined in eqn (5) yields a distance value of 56.9 mm, which is barely 0.71% off the distance of 56.5 mm measured using tape. As with all physical measurement techniques, it is necessary to study the various influencing factors involved with the proposed technique in order to draw solid conclusions on the limits of its practical application. The list of possible contributing factors which may affect performance include beam divergence, speckle noise, electronic noise and decorrelation between each recorded frame. In this work, the focus is to demonstrate the working principles behind this technique as well as to present a set of measurement results to validate the technique. The results obtained, however, do reveal the feasibility of pursuing further research and development work on the technique. 8 CONCLUSION In this paper, a novel technique adapted from digital speckle shearing interferometry, is demonstrated to be able to sense distance optically. By performing a Fourier transform on the speckle fringe pattern, it is possible to extract the fringe period-which is an importrant parameter to be determined in the measurement procedure-with precision despite the presence of speckle noise. A fringe sharpening procedure is shown to considerably enhance the fringe period measurement procedure. The range of distances to be measured using the proposed technique can be adjusted via the manipulation of either the amount of expander lens translation or the amount of shear. Coupled with ease of usage, rugged
457
Optical distance sensing
25
0 0
50
100
150
200
250
300
350
400
450
500
0
50
100
150
200
250
300
350
400
450
500
350
400
450
500
r” : d C
30
20
10
0 Phi
Location
60
50
40 r” ;
30
E 20
10
0
50
100
150
200 Plxml
Fig. 4.
250
300
Locatlon
Intensity distribution of a cross-section in the x direction of fringe patterns processed using: (a) two; (b) four; and (c) six frames of speckle data.
T. W. Ng
458
400 g
5 : 300
2
c
.
I
500
(c)
-234
0
Frmqumoy 5. Fourier spectra of cross-sectional intensity distribution in the x direction of fringe-patterns processed using: (a) two; (b) four; and (c) six frames of speckle data. The zero-frequency is clipped to enhance the detail of the remaining spectra.
fig.
Optical distance sensing
design and the possibility for complete ises to be an attractive tool for distance
459
automation, measurement.
the technique
prom-
ACKNOWLEDGEMENT
The author is indebted to the reviewer for contributing many comments which have helped tremendously towards the improvement in presentation of this paper. REFERENCES
1. Strand, Engng,
T. C., Optical
three-dimensional
sensing for machine
vision. Opt.
24 (1985) 33-40.
2. Takezaki, J. & Hung, Y. Y., Direct measurement of flexural strains in plates by shearography. J. Appl. Mech., 53 (1986) 125-9. 3. Hung, Y. Y. & Liang, C. Y., Image-shearing camera for direct measurement of surface strains. Appl. Opt., 18 (1979) 1046-51. 4. Chau, F. S. & Ng, T. W., A real-time digital shearing speckle interferometer. Meas. Sci. Technol., 3 (1992) 381-3. 5. Ganesan, A. R., Sharma, D. K. & Kothiyal, M. P., Universal digital speckle shearing interferometer. Appl. Opt., 27 (1988) 4731-4. 6. Ng, T. W. & Chau, F. S., A digital shearing speckle interferometry technique for modal analysis. Appl. Acoustics, 42 (1994) 175-85. 7. Ganesan, A. R., Joenathan, C. & Sirohi, R. S., Sharpening of fringes in digital speckle pattern interferometry. Appl. Opt., 27 (1988) 2099-100.
APPENDIX
The optical path difference for the ray travelling from the light source &(x1, yl, zi) and S2(xz, yz, z2) to the camera 0(x,, y,, z,) via an arbitrary point P on the test target, as shown in Fig. Al, is given by
SlP = [(Xl - g2 + (y1 - y)’ + (Zl - z)2]“2 ZR 1
S2P
=
[(x2
[
1+x~+Y2+z2-2xx1-2Yyl-2zz1 2R: -x)’
+
1
(AlI
(y2 - y)2 + (z2 - Z)2]lD
x2+y2++
2xX2
2R;
-
2yy2 - 2222
I
(A21
T. W. Ng
460
Fig.Al. Optical path diagram of the digital speckle shearing interferometry technique for distance measurement. where Rf=x:+yf+z: and R$= xz + ys + z$. The equations are obtained by neglecting high-order terms in the series expansion. Therefore, the optical path length change dl from S, and S, to the point in the object is d&x, y, z) = (S,P + OP) - (&P + OP)
=R,-RI+-
; [gj(x2+y2+z2) 2
For a neighbouring as
643)
1
point P’(x + 6x, y, z), the optical path change d’ is expressed
d,(x + 6x, y, z) = (SIP + OP) - (S,P + OP)
[(x + 8x)’ +y2 + z2]
6-W
Neglecting small terms in the equation, the relative optical path change Sd is expressed as Sd = dz(x + 6x, y, z) - d,(x, y, z) R2 =
-
-
RI
[ R2R,
1
x8x
If RI = R2 >> 1, eqn (A5) can be rewritten as
(fw where SR is the amount of translation between the expander and test target.
of the expander,
and R is the distance