**2.2.3 The sub-pixel fitting precision**

Third, it is well known that estimation error of the curve fitting may become smaller when the order of a fitting polynomial increases. In this experiment, the comparison of the subpixel shift estimation error among different fitting polynomials including quadratic curve, cubic curve, quartic curve and spline curve, was conducted. Here, we used the same five fitting points, where the middle point is the peak and the other four points are on its both sides evenly. The main results are shown as Fig.5 to Fig.8, from which the following results can be obtained,


Applications of Computer Vision in Micro/Nano Observation 535

ZSAD-estimation error with different fitting methods

Original Quadratic Cubic Quartic Spline

20x20 25x25 30x30 35x35 40x40 45x45 50x50 55x55 60x60 65x65 70x70 -0.5

Different model size

Based on the theory and experiment results in 2.2, if we want to attain a matching result with high precision and low computational task using the grid block, the following parameters should be chosen: the reduced searching region, ZSAD object function, 50 50

In order to validate the precision of our improved BAM practically, we used the PI nano platform to control the motion of the standard grid and calculated it with our method. Because the platform can output the nano motion, the shift of the grid is known. The KH-7700 microscope was used to capture the images of the grid, and the shift of each step was 50nm. Since the horizontal pixel is 57.47nm, the practical calculation result should be 0.89 pixels. Fig.9 is the result of the integral shift and Fig.10 is the shift of sub-pixel where the vertical axis denotes the motion, with unit of pixel, and the horizontal axis denotes the shift steps; the line with "\*" is the true movement and the line with "o" is the calculation

From Fig.9-10, we can see that the precision of our improved sub-pixel motion measurement method is very high, the integral pixel measurement is exactly equal to the true value and the sub-pixel measurement result is close to the true value. Therefore, the method can be

Then, the sub-pixel block matching method of displacement measurement based on computer vision was used to measure the driving characteristic curve of a piezoelectric actuator practically. The result was shown in Fig.11, where the vertical axis denotes the motion, with unit of nm, and the horizontal axis denotes the driving voltage, with unit of V. Fig.11(a) is the driving curve when the voltage increases to 200V and then decreases to 0V smoothly; Fig.11(b) is the driving curve when the voltage increases to 150V and then decreases to 0V; Fig.11(c) is the driving curve when the changing routine of the voltage is 0V-200V-0V,0V-150V-0V,0V-100V-0V,0V-50V-0V; Fig.11(d) is the driving curve when the changing routine of the voltage is 0V-200V-0V,0V-160V-0V,0V-120V-0V,0V-80V-0V,0V-

0 0.5 1 1.5 2 2.5 3

pixels block size and the quartic sub-pixel fitting function.

**2.3 The driving characteristic of a piezoelectric actuator** 

used to measure the practical shifts in micro/nano manipulation.

Estimation error(pixel)

Fig. 7. Estimation error with ZSAD

movement.

40V-0V.

3.5 x 10-3

Fig. 4. Estimation errors with NCC

Fig. 5. Estimation error with SSD

Fig. 6. Estimation error with LSAD

#### Fig. 7. Estimation error with ZSAD

534 Mechanical Engineering

NCC-estimation error with different fitting methods

20x20 25x25 30x30 35x35 40x40 45x45 50x50 55x55 60x60 65x65 70x70 -0.025

Different model size

SSD-Estimation error with different fitting methods

20x20 25x25 30x30 35x35 40x40 45x45 50x50 55x55 60x60 65x65 70x70 -2

Different model size

LSAD-estimation error with different fitting methods

20x20 25x25 30x30 35x35 40x40 45x45 50x50 55x55 60x60 65x65 70x70 -0.025

Different model size

Original Quadratic Cubic Quartic Spline

Original Quadratic Cubic Quartic Spline

Original Quadratic Cubic Quartic Spline


<sup>6</sup> x 10-3



Estimation error(pixel)

Fig. 6. Estimation error with LSAD



0

Estimation error(pixel)

Estimation error(pixel)

Fig. 4. Estimation errors with NCC

Fig. 5. Estimation error with SSD

Based on the theory and experiment results in 2.2, if we want to attain a matching result with high precision and low computational task using the grid block, the following parameters should be chosen: the reduced searching region, ZSAD object function, 50 50 pixels block size and the quartic sub-pixel fitting function.

### **2.3 The driving characteristic of a piezoelectric actuator**

In order to validate the precision of our improved BAM practically, we used the PI nano platform to control the motion of the standard grid and calculated it with our method. Because the platform can output the nano motion, the shift of the grid is known. The KH-7700 microscope was used to capture the images of the grid, and the shift of each step was 50nm. Since the horizontal pixel is 57.47nm, the practical calculation result should be 0.89 pixels. Fig.9 is the result of the integral shift and Fig.10 is the shift of sub-pixel where the vertical axis denotes the motion, with unit of pixel, and the horizontal axis denotes the shift steps; the line with "\*" is the true movement and the line with "o" is the calculation movement.

From Fig.9-10, we can see that the precision of our improved sub-pixel motion measurement method is very high, the integral pixel measurement is exactly equal to the true value and the sub-pixel measurement result is close to the true value. Therefore, the method can be used to measure the practical shifts in micro/nano manipulation.

Then, the sub-pixel block matching method of displacement measurement based on computer vision was used to measure the driving characteristic curve of a piezoelectric actuator practically. The result was shown in Fig.11, where the vertical axis denotes the motion, with unit of nm, and the horizontal axis denotes the driving voltage, with unit of V. Fig.11(a) is the driving curve when the voltage increases to 200V and then decreases to 0V smoothly; Fig.11(b) is the driving curve when the voltage increases to 150V and then decreases to 0V; Fig.11(c) is the driving curve when the changing routine of the voltage is 0V-200V-0V,0V-150V-0V,0V-100V-0V,0V-50V-0V; Fig.11(d) is the driving curve when the changing routine of the voltage is 0V-200V-0V,0V-160V-0V,0V-120V-0V,0V-80V-0V,0V-40V-0V.

Applications of Computer Vision in Micro/Nano Observation 537

(a) 0V-200V-0V

(b) 0V-150V-0V

(c) 0V-200V-0V,0V-150V-0V,0V-100V-0V,0V-50V-0V

2000

4000

6000

8000

10000

12000

14000

Movement(nm)

Movement (nm)

2000

4000

6000

8000

10000

12000 14000

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>160</sup> <sup>180</sup> <sup>200</sup> <sup>0</sup>

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>0</sup>

Driving voltage(v)

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>160</sup> <sup>180</sup> <sup>200</sup> <sup>0</sup>

Driving voltage(v)

Driving volage(v)

Movement(nm)

The measurement results of the piezoelectric actuator driving characteristic are consistent with the physics analysis. Furthermore, the proposed method, which is simple in manipulation and credible in measurement results, satisfies the requirement of the micro/nano measurement with high precision.

Fig. 8. The integral pixel measurement result

Fig. 9. The sub- pixel measurement result

The measurement results of the piezoelectric actuator driving characteristic are consistent with the physics analysis. Furthermore, the proposed method, which is simple in manipulation and credible in measurement results, satisfies the requirement of the

Intergral pixel measurement result

True movment Measurement result

True movement Measurement result 0 2 4 6 8 10 12 14 16 18 20

PI movement steps(n)

Sub-pixel measurement result

0 2 4 6 8 10 12 14 16 18 20

PI Movement steps(n)

micro/nano measurement with high precision.

0


Fig. 9. The sub- pixel measurement result



位位

*s/pixelShift value (pixel)*






Fig. 8. The integral pixel measurement result

0.2 0.4 0.6 0.8 1

1.2 1.4 1.6 1.8 2

Shift value(pixel)

(b) 0V-150V-0V

Driving volage(v)

(c) 0V-200V-0V,0V-150V-0V,0V-100V-0V,0V-50V-0V

Applications of Computer Vision in Micro/Nano Observation 539

only one microscope with unchanged camera parameters, so the reconstruction process is very simple. The experiments and error analysis results show that it can reconstruct shape

In the defocus imaging model, a defocused image can be theoretically considered as the summation of some defocused points, and this process can be denoted by the following

where *E*(*x*, *y*) and *I*(*x*, *y*) are the defocused image and the focused image, respectively, *h*(*x*, *y*)

When the point spread function is approximated by a shift-invariant Gauss function, the

( , ,) ( , ,) [, ) (, )

*t*

*u*

If the depth map is an equifocal plane, *a* is constant. Otherwise, *a* is shift-variant, and the

( , , ) ( ( , ) ( , , )) ( , )

where " "denotes the gradient operator and " "is the divergence operator,

*uxyt axy uxyt t*

<sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup>

1

2 2

1

 

2 

2 2 1

2 2

Suppose there are two images *E*1(*x*, *y*) and *E*2(*x*, *y*) for two different focus setting, also,

(that is, *E*1(*x*, *y*) is more defocused than *E*2(*x*, *y*)), then *E*2 (*x*, *y*) can be written as:

) ( )( ) ( , ) exp( (,)

2 2

*xu yv x y r u v dudv*

( )( ) exp( ) (,)

  2 2

*E*

*xu yv u v dudv*

imaging model in Eq.(17) can be formulated in terms of the isotropic heat equation:

*uxyt a uxyt a t*

0

0

.

*uxy rxy*

(,,) (,)

where *a* is the diffusion coefficient, *<sup>u</sup>*

(,,) (,)

*uxy Exy*

*Exy Ixy hxy* (,) (,) (,) (17)

0 0

(18)

0

is related to the diffusion coefficient *a* via:

2*ta* (20)

(21)

(19)

," "denotes the Laplacian operator,

on micr/nano scale.

**3.1 The imaging model for defocus**

convolution function normally:

is the point spread function.

2 2

diffusion equation becomes:

, *T*

1 2 

*xy xy* 

It is also easy to verify that the variance

*E*

2 2 *uxyt uxyt* ( , ,) ( , ,) *<sup>u</sup> x y*

.

(d) 0V-200V-0V,0V-160V-0V,0V-120V-0V,0V-80V-0V, 0V-40V-0V

Fig. 10. The driving characteristic of a piezoelectric actuator
