are used for the case of the in-plane rotation (rota- ... and the normal rotation (rotation about a normal to ...... an aluminous strip was mounted to a rotary stage ...
Optik
00, No. 0 (0000) 1{0
1
c Wissenschaftliche Verlagsgesellschaft mbH, Stuttgart
Measurement of an object rotation using the theory of speckle pattern decorrelation M. Hrabovsky, Z. Baca, P. Horvath
Joint Laboratory of Optics - Palacky University and Institute of Physics of Academy of Sciences
Measurement of an object rotation using the theory of speckle pattern decorrelation. The
equations are derived for the determination of the components of the small deformation tensor in the free space and in the image eld. These general equations are used for the case of the in-plane rotation (rotation about an axis lying in the object surface plane) and the normal rotation (rotation about a normal to the object surface plane). The analysis of sensitivity and accuracy is done in Chapter 5. Some experimental congurations and results are also presented.
.
fy P2
a1
Pf
f1,2
I
L1
Received Monat 00, 0000. 17.listopadu 50, 77207 Olomouc,Czech Republic
∆g g
Pg2
L2 fz
Of
Og
fx
gz
gx Lo
1 Introduction Speckle is the peculiar appearance which is generated, if an object with a rough surface is illuminated by a coherent beam. The speckle eect is a resultant expression of interference of elementary coherent beams of radiation 1], 2]. The real and imaginary parts of the complex amplitude of light at any point of the plane under consideration have zero mean value and the visibility of the speckle pattern is maximal. A theoretical study of speckle localization and its visibility was made by I. Yamaguchi 3], 4] and in 1981 5] he made the rst study of speckle displacement and decorrelation in the diraction and image elds for small object deformation. When the investigated object is deformed (zone of so-called small deformation), then the tensor of the object elementary surface deformation can be determined from the mutual shift of the speckle elds that characterize two states of the object deformation. I. Yamaguchi and some other authors made the study of the object translation 6], 7], object deformation 8], 9], 10] and object rotation 11], 12] by means of the electronic speckle correlation. Material testing by the laser speckle strain gauge was made by I. Yamaguchi and K. Kobayashi 13], Yu Lung Lo used ber Bragg-grating sensors for the simultaneous measurement of axial strain and temperature on surfaces of structures, T. Yoshimura, E. Miyazaki and K. Nakanishi 14] monitored surface roughness by means of double scattered image speckle, e.t.c.
gy
Pg1 Ic
Is
P1 a2
S Ls
Fig. 1. Coordinate systems for derivation of speckle displacement in the free-space geometry The authors of this paper made a study of the speckle correlation in the free space and in the image
eld 15], 16] and its use in mechanics for measurement of translation and deformation 17], 18].
2 Correlation of intensity uctuations of two speckle patterns 2.1 Free space According to 5], 15] we can write the following relation for the correlation of uctuations of two intensities
hI1 I2 i = jhU1 U2 ij2 :
(1)
Let us introduce the arrangement according to Fig.1. We illuminate an object in the plane (fx fy ) with a spherical wave originating from the point S and detect a speckle pattern in the plane (gx gy ). Ls is the distance from the source S to the object point Pf , Lo is the distance from the detection plane (gx gy ) to the object plane (fx fy ). In this case we can write the relation according to 14], 15] for the modulus
jhU1 U2ij2 jhU1 (g) U2 (g + g)ij2 =
2
Autor: Titel
= jK2 j2
2 I1 (f )I2 (f )e Liko A(f ); g]f df (2)
R R p
Object plane
where K2 is a constant, U1 (g) is the complex amplitude of light in the point Pg1 U2 (g + g) is the complex amplitude of light in the point Pg2 , I1 I2 are the intensities of light in the points P1 P2 , k = 2 , f g are the position vectors in the planes (fx fy ), (gx gy ) and
fy
; 2 ; 1 + l 2 ; 1 ; Ax = ;ax LLo lsx x s ;ay LLo lsx lsy + lx ly ; s L o ;az L lsx lsz + lx lz ; s ;Lo xx (lsx + lx ) + xy (lsy + ly )] ; ;Lo z (lsy + ly ) ; 2y (lsz + lz )] (4)
fz A’x gx
’
fx
Lo
Lp
g’x L’p
Lc
Fig. 2. Coordinate systems for derivation of speckle displacement in the image eld
Ay = ;ax LLo lsx lsy + lx ly ; s ; L o ;ay L lsy2 ; 1 + ly2 ; 1 ; s ;az LLo lsy lsz + ly lz ; s ;Lo yy (lsy + ly ) + xy (lsx + lx)] ; ;Lo ;z (lsx + lx) + 2x (lsz + lz )] (5)
g’y
gy
Ax
A(f ) = ;Lo fr Ic(f )a(f )]g0 : (See Fig: 1) (3) According to 5], 14], 15] the equation (3) represents the relationship between the small deformation tensor and the correlation of uctuations of two intensities in the free space and the vector A(f ) gives the position of the maximum of the correlation function. Let us now transcribe this equation into components A(f ) = (Ax Ay ):
Detection plane
Lens
where Is = (lsx lsy lsz ) I = (lx ly lz ) a = (ax ay az ), @ ls = 1 ;l2 ; 1 lsx lsy lsx lsz , sx
@x Ls @ ls = 1 ;lsx lsy l2 ; 1 lsy lsz , sy @y Ls @l 1 ; 2 @x = L lx ; 1 lx ly lx lz , @l 1 ; 2 @y = L lx ly ly ; 1 ly lz , ; xx = @a@xx 0 , yy = @a@yy 0 , h ; i xy = yx = 21 @a@yx 0 + @a@xx 0 ,
z , y = ; 1 @az , z = 1 @ay ; @ax . x = 21 @a @y 2 @x 2 @x @y (ax ay az ), (x y z ), (xx xy yy ) are the translation, rotation and deformation components of the small deformation tensor, (lsx lsy lsz ) are the components of the unit vector of the illumination direction, (lx ly lz ) are the components of the unit vector of the observation direction. The components of the small deformation tensor can be evaluated in the free space using the relations (4),(5).
2.2 Image eld In this part we would like to show in what manner the relations (4),(5) can be transcribed for the case when a thin lens is placed between the object plane and the detection plane Fig.2. Let us consider again the object placed in the plane (fx fy ). The speckle pattern, arising by a diusive reection from its surface, propagates in the free space into the plane (gx gy ), being imaged by means of a thin converging lens on the plane (gx gy ). It follows from the above mentioned section that for the calculation of small deformation tensor components with utilization of a speckle pattern displacement in the plane (gx gy ) we can use the relations (4),(5). However, we need to calculate Lo Ax Ay . In the case of a thin lens we have Lo = Lo = Lc ; fL;pLfp Ax = fA;xLfp = 0
0
0
0
0
Ax A = Ay f = Ay , where L is the distance bec m y m f ;Lp 0
0
0
0
tween the object and the lens, f is the focal lenght of the lens, Ax Ay are the measured displacements of the speckle pattern in the plane (gx gy ) Lp is the dis0
0
0
Autor: Titel
tance between the detection plane and the lens, and m = ; Lpf;f is the magni cation of the lens. These relations are valid if we assume the conditions of the ideal imaging. By substitution into (4), (5) we obtain
3
0
"
#
Ax = ;a Lo ;l2 ; 1 + l2 ; 1 ; x L sx x m s " # L o ;ay L lsx lsy + lxly ; s " # L o ;az L lsx lsz + lxlz ; s ;Lo xx (lsx + lx ) + xy (lsy + ly )] ; ;Lo z (lsy + ly ) ; 2y (lsz + lz )] (6) 0
0
0
y
OBJECT PLANE P
θs
θo
x
z
Ls Y
0
0
0
"
#
Ay Lo l l + l l ; = ; a x m Ls sx sy x y " # ; L ;ay Lo lsy2 ; 1 + ly2 ; 1 ; s # " L o ;az L lsy lsz + ly lz ; s ;Lo yy (lsy + ly ) + xy (lsx + lx)] ; ;Lo ;z (lsx + lx) + 2x (lsz + lz )] : (7) 0
0
0
0
0
0
The (6), (7) are relations, from which the individual components of the small deformation tensor can be evaluated for the case of the image eld.
3 Object rotation in free space From the practical point of view it is suitable to establish the arrangement according to Fig.3. The object is located in the plane (x y) and it is illuminated by coherent light beam radiated from the point source S. The linear CCD detector sensing a sample of the speckle pattern is located in the plane (X Y ). Points O, P, S lie in the plane (x z ). It follows from Fig. 3 that the components of the unit vector Is of the direction of illumination and the unit vector I of the direction of observation have the following form I = (lx ly lz ) = (sin o 0 cos o) Is = (lx ly lz ) = (sin o 0 cos o ). Futhermore, if we take into account the rotation of the plane (x y) compared with (X Y ) we can transcribe the equations (4), (5) into the form 2 s Ax = ax LLo cos + cos o ; cos o s s sin s + sin ; ;az Lo Lcoscos o s o
S-SOURCE
Lo
X
O
OBSERVATION PLANE
Fig. 3. Coordinate systems for rotation determination in the free space
sin s + tan + ;Lo xx cos o
o
cos s + 1 +Lo 2y cos o
(8)
Ay = ay LLo + 1 ; Lo xy (sin s + sin o )] ; s ;Lo 2x (cos s + cos o )] + +Loz (sin s + sin o ) : (9)
From the equations (8), (9) it is evident that two types of rotation can be measured: rotation about an axis in the object surface plane - in-plane rotation (components x y ), rotation about a normal to the object surface plane - normal rotation (component z ). For simplicity, let us consider that the object is only rotated. The translation (ax ay az ) and deformation (xx yy xy ) components of the small deformation tensor are equal to zero. In the case of object in-plane rotation (for example about y) the equation (8) can be written as
cos s + 1 (10) Ax = Lo !y cos o where !y = 2y is the rotation angle of the object about the y-axis measured in radians, Lo is the distance from the object plane to the observation plane, s is the angle of illumination direction, o is the angle of observation direction, and Ax is the x-component of the position vector of the correlation function maximum. If the object is rotated about the normal (about the z -axis) we can write
Ay = Lo !z (sin s + sin o )
(11)
4
Autor: Titel
where !z = z is the rotation angle of the object about the z -axis in radians, Ay is the y-component of the position of the correlation function maximum and Lo s o have the same meaning as above.
4 Object rotation in image eld In case that in Fig. 3 a thin lens is placed between the object plane and the observation plane the equations (8), (9) have the form "
#
Ax = a Lo cos2 s + cos ; x L cos o m s o " # L cos sin s s o ;az L cos + sin o ; s o
sin s + tan + ;Lo xx cos o o
cos s + 1 +Lo 2y cos 0
0
0
o
(12)
!
Ay Lo m = ay Ls + 1 ; Lo xy (sin s + sin o )] ; ;Lo 2x (cos s + cos o )] + +Loz (sin s + sin o ) (13) 0
0
0
0 0
where Lo = Lc ; fL;pLfp m = ; Lpf;f (see Fig. 2). Similarly the relations (10),(11) can be transcribed as follows 0
0
0
0
!
fL ; LAx;f f = Lc + L ;p f !y p p
cos s + 1 cos o 0
0
0
0
0
0
0
(14)
!
; LAy;f f = Lc + LfL;p f !z p p (sin s + sin o ) :
From the theoretical relations (10), (11) it is evident that the sensitivity and the accuracy of the arrangement in Fig. 3 depend crucially on the geometrical con guration of the experiment and furthermore on the accuracy of Ax Ay shift determination. By means of the arrangement in Fig. 3 it is possible to measure rotation angles within the range of 10;2 ; 102 minutes. 5.1.1 Rotation about the y-axis (in-plane rotation)
0
0
5.1 Free space
0
(15)
5 Sensitivity and Accuracy The analysis of sensitivity and accuracy is done in this chapter for the rotation angles !y and !z both in the free space and in the image eld.
Let us determine the boundary condition !ymax 300 and require the sensitivity of magnitude !y measurement grater than 0.1' (i.e. !ymin < 0:10). Then the arrangement in Fig. 3 seems to be realizable within intervals demonstrated in Fig. 4, i.e. o 2 (0 60)o, Ax 2 (1 480)pixel, and for constant values Lo = 0:4 m, s = 0o. For example, for the observation angle o = 15o, and for the shift Ax = 1 pixel, the sensitivity of the magnitude !y measurement is !ymin = 0:060. From the Fig. 4 it is obvious that the sensitivity of the magnitude !y grows with the increasing o (for example, the sensitivity !ymin = 0:050 corresponds to the angle o = 35o). From the analysis we also ascertain that the sensitivity grows with the increasing Lo . However, the magnitude Lo is more inuential on the sensitivity of the measurement than the magnitude o . On the contrary the sensitivity decreases with the increasing s . To keep the maximum sensitivity and for easier practical realization, it is suitable to determine the angle of the illumination direction s = 0o. Furthermore, the accuracy analysis of the selected experimental arrangement follows from the theory of errors. At common operating accuracies of determination of magnitudes Lo, o and s , the arrangement in Fig. 4 corresponds with Fig. 5. On the basis of performed accuracy analysis we can say, that the relative mean square error r !y decreases with the increasing measured magnitude !y and grows slightly with the increasing angles o and s . The eect of the change of the magnitude Lo on the measurement accuracy can be omitted if Lo 2 (0:2 0:6)m. But, in general, the magnitude r !y decreases with the increasing Lo . For the presented example of the arrangement with o = 15o, Lo = 0:4 m and s = 0o, the magnitude r !y 5 %0 for !y > 0:30 , r !y 2 (5 10) % for !y 20 (0:12 0:30) and r !y > 10 % for !y 2 (0:06 0:12) . From the Fig. 5 it is evident, that for practical measurement of rotation angles within the limits 5' - 30' the relative mean square error r !y is less than 5%.
5
Autor: Titel
ω \
ω ]
θο
[
δ U ω \
θο
θR
Fig. 4. Sensitivity !y for experimentalo arrangement in Fig. 3 for Lo = 0:4 m and s = 0 .
[
Fig. 5. Relative mean square error r !y for experimental arrangement in Fig. 3 for Lo = 0:4 m and s = 0o . 5.1.2 Rotation about the z -axis (normal rotation) If we consider the same boundary condition !zmax 300 and the arrangement as in the chapter 5.1.1, i.e. the CCD detector distance Lo = 0:4 m and the illumination angle s = 0o, we ascertain that the sensitivity of the measurement of the magnitude !z in the case of the object rotation about the normal (z -axis) is approximately by order lower than in the case of the object rotation about the y-axis (compare Fig.4 with Fig.6). For example, for the observation angle o = 15o, and for the shift Ay = 1 pixel, the sensitivity is !zmin = 0:50 and for the angle o = 35o the sensitivity is !zmin = 0:20. From the analysis of
\
Fig. 6. Sensitivity !z for experimentalo arrangement in Fig. 3 for Lo = 0:4 m and s = 0 .
the sensitivity it follows, that the sensitivity grows again with the increasing Lo and o . On the contrary, the increase of the distance Lo allows the angle o to decrease (approximately linearly) while keeping the sensitivity of the !y determination on the same level. This fact is very useful for the experiment. The sensitivity of the measurement would grow (nearly up to the level of the case mentioned in the chapter 5.1.1) with the increasing angle of the illumination direction s . However, in this speci c case we chose s = 0o to simplify the experimental calculations. Next we will analyse the accuracy of the selected experimental arrangement. The arrangement in Fig. 6 corresponds with Fig. 7. It is obvious that the relative mean square error r !z of the determination of the magnitude !z decreases with the increasing measured magnitude !y and signi cantly with the increasing angle o . For example, for o = 15o, Lo = 0:4 m and s = 0o , the magnitude r !z 100 % for !z > 3:50 , r !z 2 (10 15) % for !z 2 (1 3:5) and for o = 35o, the magnitude r !z 5 % for !z > 1:80, r !z 2 (5 10) % for !z 2 (0:6 1:8)0 and r !z 2 (10 15) % for !z 2 (0:4 0:6)0 . In the Fig. 7 we can see, that for practical measurement of rotation angles within the limits 5'- 30' the relative mean square error r !z is less than 5% for o > 30o and within interval (5, 10)% for o 2 (15 30)o. From the performed accuracy analysis we further accertain a strong inuence of the illumination angle s on the magnitude r !z ( r !z decreases with the increasing s ). On the other hand, the inuence of the distance Lo on the measurement accuraccy can be omitted if Lo 2 (0:2 0:6)m.
6
Autor: Titel
δ U ω ]
\
θR
Fig. 7. Relative mean square error r !z for experimental arrangement in Fig. 3 for Lo = 0:4 m and s = 0o . 5.2 Image eld From the theoretical relations (14) and (15) in the case of the image eld it is evident that the sensitivity and the accuracy while determining the object rotation depend (except the parameters mentoined in the chapter 5.1) also on the properties of the used optical system (the thin lens). Therefore in this speci c case, i.e. the ideal imaging, they depend only on the lens magni cation m. We can considerably increase the sensitivity of the measurement by placing the thin lens between the object plane and the observation plane. From the sensitivity analysis we will then ascertain that the measurement sensitivity in the image eld is approximately jmj times greater than the measurement sensitivity in the free space (compare Fig. 4 with Fig. 8). This conclusion is valid both for the object rotation about the y-axis and the object rotation about the normal (z -axis). The trends derived for the sensitivity and the accuracy in the free space are kept also in the image eld. The accuracy of the arrangement, that is shown in Fig. 8 can be seen in Fig. 9 as an example.
6 Object in-plane rotation To verify validity of the equation (10), we designed the arrangement in Fig. 10. The He-Ne laser ( = 632:8 nm) was used with the output power 40 mW as the light source. Its beam was directed by means of a plane mirror (Z1) upright to the object, that was represented by an aluminous cuboid with a rough surface. A linear CCD detector (2048 pixels, pixel size 14x14
ω \
θο
[
Fig. 8. Sensitivity !y for experimental arrange-o ment with thin lens in Fig. 3 for Lc = 0:4 m, s = 0 and jmj = 4.
δ U ω \
θο
[
Fig. 9. Relative mean square error r !y for experimental arrangement with thin lens in Fig. 3 for Lc = 0:4 m, s = 0o and jmj = 4. m) was placed at the distance Lo from the object and at the angle o . The linear CCD detector regis-
tered a speckle pattern sample. It was connected to a PC, which interpreted the measured readings. The value of the parameter Ls was high to eliminate the inuence of translation components on measurement results. Some results are presented in tables 1, 2, 3. Table 1 shows records for the experiment con guration when the distance from the object surface to the linear CCD detector is Lo = 0:22 m, the angle of illumination direction is s = 0o , the angle of observation direction is o = 16o. The object was rotated through the angle of 30 minutes with the accuracy 1 . B12 is 0
Autor: Titel
7
Table 2. Rotation through the angle of -20 minutes Mirror Z1 z
Linear CCD detector Lo
x
θo
He-Ne Laser
Rotary stage with an object
Fig. 10. Experimental arrangement for an object rotation about the y-axis Table 1. Rotation through the angle of 30 minutes Ax B12 !y (i)2 pixel % min: i = !y ; !yi 281.3 26 30.16 0.1764 280.4 25 30.07 0.1089 279.5 17 29.97 0.0529 277.5 21 29.75 0.0001 281.5 31 30.18 0.1936 281.4 26 30.17 0.1849 277.6 20 29.76 0.0004 268.5 22 28.79 0.9025 280.3 28 30.05 0.0961 265.5 25 28.47 1.6129 Average = 29.74 0.19 o Configuration : Lo = 0:22m s = 0 o = 16o M. number n 1 2 3 4 5 6 7 8 9 10
the maximum value of the correlation function in percents, (i)2 represents the square deviation. Similarly in tables 2, 3. We can see that measured results are in a good accord with the theory. During the experiment realization we discovered that the rotation by greater angles (greater than 30 minutes) caused decline of the correlation and the calculated position of the correlation function maximum didn't correspond with the speckle pattern displacement. Futhermore, with the cumulative angle o and the cumulative mea-
Ax B12 !y (i)2 pixel % min: i = !y ; !yi -343.0 64 -20.23 0.0529 -340.0 54 -20.05 0.0025 -338.5 77 -19.96 0.0016 -341.8 90 -20.16 0.0256 -338.7 48 -19.97 0.0009 -339.4 44 -20.02 0.0004 -338.5 33 -19.96 0.0016 -336.5 32 -19.84 0.0256 -335.6 33 -19.79 0.0441 -340.0 36 -20.05 0.0025 Average = -20.00 0.04 o Configuration : Lo = 0:4m s = 0 o = 16o M. number n 1 2 3 4 5 6 7 8 9 10
Table 3. Rotation through the angle of 5 minutes Ax B12 !y (i)2 pixel % min: i = !y ; !yi 49.7 66 5.05 0 50.4 45 5.12 0.0049 46.7 43 4.74 0.0961 48.3 43 4.90 0.0225 48.4 50 4.91 0.0196 53.2 55 5.40 0.1225 49.7 28 5.05 0 49.5 25 5.03 0.0004 49.7 26 5.05 0 51.3 57 5.21 0.0256 Average = 5.05 0.06 o Configuration : Lo = 0:22m s = 0 o = 30o M. number n 1 2 3 4 5 6 7 8 9 10
sured angle the maximum correlation declined more quickly.
7 Object normal rotation To verify validity of the equation (11) we used the con guration in Fig. 11. The object represented by an aluminous strip was mounted to a rotary stage
8
Autor: Titel
Table 5. Rotation through the angle of -10 minutes Mirror Z1
z Mirror Z2
Microcomputer
Linear CCD detector
θo Lo
x Argon Laser
Rotary stage with an object
Fig. 11. Experimental arrangement for an object rotation about the z-normal and iluminated by an argon laser beam (514 nm). The output power was 20 mW. The laser beam was directed by means of plane mirrors (Z1,Z2) upright to the object. The rotary stage could be rotated about the z -axis with the accuracy of 0.25 minutes. The linear CCD detector (2048 pixels) placed paralelly to the y-axis at the distance Lo from the object and at the angle o was operated by a microcomputer. It was connected by means of a serial cable to the PC, which again interpreted the measured readings. Some re-
Table 4. Rotation through the angle of 30 minutes Ay B12 !z (i)2 pixel % min: i = !y ; !yi 126 48 30.94 0.0004 127.4 47 31.29 0.1089 126.1 48 30.97 0.0001 127.3 48 31.26 0.0900 126.6 48 31.09 0.0169 125.3 48 30.77 0.0361 125.5 47 30.82 0.0196 125.4 48 30.79 0.0289 125.1 48 30.72 0.0576 125.9 47 30.92 0.0016 Average = 30.96 0.06 Configuration : Lo = 0:413m s = 0o o = 28:3o M. number n 1 2 3 4 5 6 7 8 9 10
sults are presented in tables 4, 5, 6 and we can see that they are again in a good accord with the theory.
Ay B12 !z (i)2 pixel % min: i = !y ; !yi -25.7 66 -10.13 0.2601 -25.7 65 -10.13 0.2601 -23.5 67 -9.26 0.1296 -26.4 65 -10.4 0.6084 -24.2 66 -9.54 0.0064 -24.7 64 -9.73 0.0121 -23.2 67 -9.14 0.2304 -23.2 67 -9.14 0.2304 -24.2 66 -9.54 0.0064 -23.4 57 -9.22 0.1600 Average = -9.62 0.15 o Configuration : Lo = 0:41m s = 0 o = 17:3o M. number n 1 2 3 4 5 6 7 8 9 10
Table 6. Rotation through the angle of -5 minutes Ay B12 !z (i)2 pixel % min: i = !y ; !yi -12.3 73 -4.96 0.0529 -14.3 75 -5.76 0.3249 -10.8 77 -4.35 0.7056 -13.5 75 -5.44 0.0625 -13.8 75 -5.56 0.1369 -12.6 75 -5.08 0.0121 -12.8 80 -5.16 0.0009 -13.8 76 -5.56 0.1369 -12.6 80 -5.08 0.0121 -12.3 74 -4.96 0.0529 Average = -5.19 0.13 o Configuration : Lo = 0:223m s = 0 o = 32:4o M. number n 1 2 3 4 5 6 7 8 9 10
8 Conclusion In this paper we deal with the problem of measurement of an object rotation using the theory of the speckle pattern decorrelation. From the general equations (4) and (5) expressing the relationship between the small deformation tensor and the maximum of the correlation function in the free space and from the
Autor: Titel
equations (6) and (7) which are valid in the image
eld we derived the theoretical relations for in-plane and normal rotation measurement in the free space (10), (14) and in the image eld (11), (15). The analysis of the sensitivity and the accuracy show, that this method is convenient for the measurement of rotation angles within the range of 10;2 ; 102 minutes. The sensitivity and the accuracy depend crucially on the geometrical arrangement of the experiment and in the case of the image eld they depend also on the properties of the used optical system. Furthemore, we ascertained that the measurement sensitivity in the image
eld is aproximately jmj times greater than the measurement sensitivity in the free space. The accuracy of the measurement conforms to the common operating measurement (r ! < 10 %) and grows with the increasing value of the measured angle. The verifying experiments and their results con rmed validity of the theory and a good accord with the performed sensitivity and accuracy analysis.
References
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16] Z. Baca, M. Hrabovsky, P. Horvath: Theory of speckle displacement and decorrelation in free-space geometry. SPIE Proc. 3820 (1999) 64{69. 17] M. Hrabovsky: Theory of speckle displacement and decorrelation: application in mechanics. SPIE Proc. 3479 (1998) 345{354. 18] M. Hrabovsky, Z. Baca, P. Horvath: Applications of speckle displacement and decorrelation in free-space geometry: sensitivity and accuracy of deformation and displacement measurement in mechanics. SPIE Proc. 3820 (1999) 355{362.
