**3.3 Experiment results**

544 Mechanical Engineering

As a global algorithm, we construct the following optimization problem to calculate the

arg min (,, ) (,)

2 2

argmin (,, ) (,)

(,) (,)

where the additional term imposes a smoothness constraint on the depth map. In practice,

 <sup>2</sup> 2

34 37

, *v*,*s*0;two defocus images *E*1,*E*2;a threshold

2 2 *F s u x y t E x y dxdy* () ( , , ) ( , )

> arg min ( ) . . .( ), .( ) *s s Fs s t Eq Eq*

Eq. (39) is a dynamic optimization which can be solved by the gradient flow, the algorithm can be divided into the following steps (the detailed process can be seen in literature (Favaro

2. Initialize the depth map with a plan *s*, to be simple, we can suppose that the initial

,

So if the initial depth is known, maybe it is just a general value, the dynamic depth, as well

 *s ks*

*k*

 *s xy s xy*

*s u x y t E x y dxdy*

However, the optimization process above is ill posed(Favaro et al 2008), that is, the minimum may not exist, and even if it exists, it may not be stable with respect to data noise.

2 2

, 0 *k* which are all very small, because this term has no practical influence on

2

2

2

2

(38)

(37)

(39

 ;and

, stop; or

*s u <sup>x</sup> <sup>y</sup> t Ex <sup>y</sup> dxdy* (36)

solutions of the diffusion equations.

we use 0 

et al 2008)):

6. '( ) *<sup>s</sup> F s t*

the cost energy denoted as:

1. Give camera parameters *f*, *D*,

plane is an equifocal plane;

;

3. Compute Eq.(28)and attain the relative blurring;

compute the following equation with step

as the expected shape, can be reconstructed.

optimization step

(40)

2

2

(,)

Thus the solution process is equal to the following:

7. Compute Eq.(26), update the depth map, and return to step(3).

4. Compute Eq.(27) and attain the solution *uxy t* (,, ) of diffusion equations ; 5. Compute Eq.(38) with the solution of step(4). If the cost energy is below

*s xy*

(,)

A common way to regularize the problem is to add a Tikhonov Penalty:

*s xy*

In order to validate the new algorithm, we used it to reconstruct the shapes of a nano standard grid which is 500nm high, two AFM cantilevers. We used the microscope of HIROX -7700 shown as Fig.17, and magnify the grid into 7000 times. The rest parameters are as the following: *f*=0.357mm, *s*0 =3.4mm, *F*-number =2, *D*= *f*/2.

Fig. 17. HIROX-7700

In order to investigate the influence of different region size on the algorithm, we tested the grid with three kinds of region size and two kinds of AFM cantilever. As for the grid, through comparing to the true grid, the error maps in each experiment are constructed and the mean square error of the proposed method was calculated to test the precision. When testing the AFM cantilevers, we used PI nano platform to test the reconstruction precision.
