**4.2 Effect of number of windowed holograms on accuracy and computation processing time**

The effect on accuracy by changing the number *n* of windowed holograms into 1, 4, 16, 64, 256, 1024, 4096, 14400, 25600 and 57600 is examined. Some reconstructed images obtained from each one of the windowed holograms are shown in Figure 9. When the number of windows becomes larger, that is, the window size becomes smaller, the speckle size becomes larger. It is considered that the spatial resolution would be worse when the speckle size becomes larger. Figure 10 shows some of the results. The displacement distributions along the centerline of the beam are shown in Figure 11. The theoretical displacement distribution for a cantilever is obtained by fitting a cubic curve with the minimum error by means of the least square method to each obtained distribution from the fixed point to the loading point. The standard deviations of the errors from the theoretical cubic curves are shown in Table 2 and Figure 11. The standard deviation changes according to the number *n*. Though the standard deviation when *n* =1 is 22 nm, it is 2.4 nm when *n* =16. The value becomes the minimum value of 680 pm when *n* =1024. It shows the proposed method

Three-Dimensional Displacement and Strain Measurements

Fig. 13. Relationship between number of windows and processing time

**4.3 Effect of window size on standard deviation of errors and spatial resolution**  As shown in Figures 9(e) and 10(e), large speckles cannot provide accurate results. In this section, the relationship between window size and spatial resolution is examined. The phase-difference distributions along the vertical white line A (*x* = 632 pixels, *y* = 260–380 pixels) in Figure 10(a) are shown in Figure 14. The theoretical phase difference along the line A on the cantilever is π/2. The theoretical phase difference along the line A on the fixed back plate is 0. Although the theoretical phase-difference distribution has a discontinuity at the edge of the cantilever (y = 318), the analysed phase-difference distributions are continuous slopes. Figure 14 also shows the result obtained by a conventional averaging method using values at the peripheral 11 × 11 pixels. However, the spatial resolution obtained from a small window may be bad. That is, bad spatial resolution changes a step function along line A into a continuous function. In order to define the spatial resolution quantitatively in this study, it is defined as the distance (in pixels) between the positions at

the 25% and 75% values of the step of the phase difference as shown in Figure 15.

Fig. 14. Phase difference distribution along line A in Figure 10

The results of the relationships between the number of windows, the standard deviation of error and the spatial resolution are shown in Table 2 and Figure 16. When the window size becomes

by Windowed Phase-Shifting Digital Holographic Interferometry 39

provides very high-resolution measurement by selecting appropriate number *n*. The calculation time is, however, almost proportional to the number *n* as shown in Table 2 and Figure 13. In practice, 16, 64 or 256 windowed holograms are recommended by considering the balance of the computation time and accuracy.

Fig. 11. Displacement distributions along centerline of cantilever of Figure 4


Table 2. Relationship among number of windows, standard deviation of error and processing time

Fig. 12. Relationship between number of windows and standard deviation

provides very high-resolution measurement by selecting appropriate number *n*. The calculation time is, however, almost proportional to the number *n* as shown in Table 2 and Figure 13. In practice, 16, 64 or 256 windowed holograms are recommended by considering

(a) *n* = 1 (c) *n* = 64 (e) *n* = 57600

1 960 21.93 0.435 3 0.36 4 480 4.87 0.097 9 0.44 16 240 2.35 0.047 29 1.5 64 120 1.61 0.032 110 2.8 256 60 0.89 0.018 430 6.66 1024 30 0.68 0.013 1704 16.24 4096 15 0.94 0.019 6791 27.68 14400 8 2.29 0.045 23751 - 25600 6 2.8 0.056 42237 - 57600 4 15.7 0.312 95430 -

11x11 -- 1.69 0.034 4 6.77

Table 2. Relationship among number of windows, standard deviation of error and

Fig. 12. Relationship between number of windows and standard deviation

Standard deviation Processing

[pixels] [nm] [rad] [pixel]

time[s] Spatial

resolution

Fig. 11. Displacement distributions along centerline of cantilever of Figure 4

the balance of the computation time and accuracy.

Window size

Number of window *n*

Smoothing

processing time

Fig. 13. Relationship between number of windows and processing time

### **4.3 Effect of window size on standard deviation of errors and spatial resolution**

As shown in Figures 9(e) and 10(e), large speckles cannot provide accurate results. In this section, the relationship between window size and spatial resolution is examined. The phase-difference distributions along the vertical white line A (*x* = 632 pixels, *y* = 260–380 pixels) in Figure 10(a) are shown in Figure 14. The theoretical phase difference along the line A on the cantilever is π/2. The theoretical phase difference along the line A on the fixed back plate is 0. Although the theoretical phase-difference distribution has a discontinuity at the edge of the cantilever (y = 318), the analysed phase-difference distributions are continuous slopes. Figure 14 also shows the result obtained by a conventional averaging method using values at the peripheral 11 × 11 pixels. However, the spatial resolution obtained from a small window may be bad. That is, bad spatial resolution changes a step function along line A into a continuous function. In order to define the spatial resolution quantitatively in this study, it is defined as the distance (in pixels) between the positions at the 25% and 75% values of the step of the phase difference as shown in Figure 15.

Fig. 14. Phase difference distribution along line A in Figure 10

The results of the relationships between the number of windows, the standard deviation of error and the spatial resolution are shown in Table 2 and Figure 16. When the window size becomes

Three-Dimensional Displacement and Strain Measurements

speckles cannot provide accurate results.

Number of windows *n*

Window size *w*

resolution

by Windowed Phase-Shifting Digital Holographic Interferometry 41

and 30 are shown in Table 3 and Figure 17. The spatial resolution becomes worse when the window size becomes smaller. The standard deviation becomes also better when the window size becomes smaller. However, the standard deviation is the best when the window size is 60. The standard deviation when the window size is 30 is worse than that when the window size is 60. Although the case *w* = 60 and *n* = 256 uses all the data of the hologram, the case *w* = 30 and *n* = 256 does not use all the data of the hologram. Large

Fig. 17. Relationship between standard deviation and spatial resolution in Table 3

Standard deviation [nm]

Table 3. Relationship among Window size, standard deviation of error and spatial

In this windowed PSDHI study, the effect of the number of windows or window size on accuracy, spatial resolution and computation time were examined. When the number of windows increases, accuracy becomes better at first because of speckle noise reduction and then it becomes worse because of larger speckle size. The best accuracy is of the subnanometer scale when the number of windows is 1024 in our experiment. However, the accuracy is better than that when using the conventional averaging method with 11 × 11 pixels. The spatial resolution becomes worse when the number of windows becomes larger, and correspondingly the window size becomes smaller. If the number of windows is constant, the spatial resolution is better when the window size becomes larger and the accuracy is better when all the data are used and the window size is smaller. The computation time increases according to the number of windows. In practice, it is useful when the number of

480 256 3.81 0.93 493 240 256 1.68 1.81 457 120 256 1.07 3.55 439 60 256 0.89 6.66 430 30 256 1.06 12.16 428

Spatial resolution [pixels]

Processing time [s]

smaller, the standard deviation of errors becomes better and the spatial resolution becomes worse. The computation time becomes worse. By comparing the spatial resolution (6.77 pixels) of the conventional averaging method using 11 × 11 pixels and the spatial resolution (2.80 pixels) when the number of windows is 64, which have almost the same standard deviation of error, the windowed PSDHI is better than the conventional averaging method.

### Fig. 15. Definition of special resolution

Although the standard deviation of error when *n* = 1024 is the best in Table 2, the data when *n* = 1024 in Figure 14 show a bad slope function far from the step function. Although the result from the hologram with a small window has good accuracy in an area with almost constant values of intensity near the centerline, it is not very accurate in the area with nonconstant values near the edge because of large speckle size.

Fig. 16. Standard deviation and spatial resolution against number of windows in Table 2

### **4.4 Effect of larger window size on standard deviation of error and spatial resolution**

In the previous section, the effect of window size on the standard deviation of error and the spatial resolution were examined. However, the numbers of windows are different. In this section, the effect of window size on the standard deviation of error and the spatial resolution are examined when the numbers of the windows are the same (*n* = 256). The standard deviation and the spatial resolution when the window sizes are 480, 240, 120, 60

smaller, the standard deviation of errors becomes better and the spatial resolution becomes worse. The computation time becomes worse. By comparing the spatial resolution (6.77 pixels) of the conventional averaging method using 11 × 11 pixels and the spatial resolution (2.80 pixels) when the number of windows is 64, which have almost the same standard deviation of error, the

Although the standard deviation of error when *n* = 1024 is the best in Table 2, the data when *n* = 1024 in Figure 14 show a bad slope function far from the step function. Although the result from the hologram with a small window has good accuracy in an area with almost constant values of intensity near the centerline, it is not very accurate in the area with non-

Fig. 16. Standard deviation and spatial resolution against number of windows in Table 2

**4.4 Effect of larger window size on standard deviation of error and spatial resolution**  In the previous section, the effect of window size on the standard deviation of error and the spatial resolution were examined. However, the numbers of windows are different. In this section, the effect of window size on the standard deviation of error and the spatial resolution are examined when the numbers of the windows are the same (*n* = 256). The standard deviation and the spatial resolution when the window sizes are 480, 240, 120, 60

windowed PSDHI is better than the conventional averaging method.

Fig. 15. Definition of special resolution

constant values near the edge because of large speckle size.

and 30 are shown in Table 3 and Figure 17. The spatial resolution becomes worse when the window size becomes smaller. The standard deviation becomes also better when the window size becomes smaller. However, the standard deviation is the best when the window size is 60. The standard deviation when the window size is 30 is worse than that when the window size is 60. Although the case *w* = 60 and *n* = 256 uses all the data of the hologram, the case *w* = 30 and *n* = 256 does not use all the data of the hologram. Large speckles cannot provide accurate results.

Fig. 17. Relationship between standard deviation and spatial resolution in Table 3


Table 3. Relationship among Window size, standard deviation of error and spatial resolution

In this windowed PSDHI study, the effect of the number of windows or window size on accuracy, spatial resolution and computation time were examined. When the number of windows increases, accuracy becomes better at first because of speckle noise reduction and then it becomes worse because of larger speckle size. The best accuracy is of the subnanometer scale when the number of windows is 1024 in our experiment. However, the accuracy is better than that when using the conventional averaging method with 11 × 11 pixels. The spatial resolution becomes worse when the number of windows becomes larger, and correspondingly the window size becomes smaller. If the number of windows is constant, the spatial resolution is better when the window size becomes larger and the accuracy is better when all the data are used and the window size is smaller. The computation time increases according to the number of windows. In practice, it is useful when the number of

Three-Dimensional Displacement and Strain Measurements

angle at Point P

(8);

 and 

sensitivity vector matrix *S* as follows;

*eee*

by Windowed Phase-Shifting Digital Holographic Interferometry 43

*d x*

Fig. 18. Relationship between displacement of object, observation direction and incident

111 1 2 222 3 333

 

where the suffixes 1, 2 and 3 show the corresponding illumination directions.

*xyz x x y z y x y z z*

*eee d eee d eee d*

111 222 333

*eee Se e e eee* 

Each component of the sensitivity vector matrix *S* is obtained by the geometric parameters of the optical system. When each component of the matrix *S* of Equation (7) is specified, the displacement components *dx dy* and *dz* can be obtained from the phase-difference

for each incident light, respectively, using the inverse matrix *S*-1 of the

*x y z x y z x y z*

When an object is illuminated from three different directions, the number of parameters of the sensitivity vector components increases, and Equation (7) can be extended as Equation

*d z*

*xyz d ed ed ed*

*y xx yy zz*

(7)

(8)

(9)

windows is 16, 64 or 256, as in our experiment by considering the balance of the computation time and accuracy.
