This paper introduces the basic principle of stripe reflection method and proposes an improved algorithm on the traditional Southwell gradient iterative integration algorithm. The algorithm adds a coefficient value with an attenuation factor to the compensation height value and the value of the attenuation factor is changed by the determination of the compensation height threshold. Through computer simulation, the fitting error of the reconstructed surface show that the RMS of the new method is one order of magnitude better than the traditional algorithm and the PV value of the high frequency part is about 1/15 of the traditional algorithm. It is proved that the improved algorithm can effectively improve the convergence and noise resistance of the iterative algorithm.
The fringe reflection method [
The measurement model is shown in
The intensity of each point on the deformed fringe pattern after reflection by the mirror to be tested can usually be expressed as:
I n ( x , y ) = a ( x , y ) + b ( x , y ) cos [ φ ( x , y ) + σ n ] (1)
where I n ( x , y ) is the light intensity of each point of the stripe diagram, a ( x , y ) is the background light intensity, b ( x , y ) is amplitude modulation, and φ ( x , y ) is the phase modulation to be solved, and σ n is the additional phase, which is determined when the fringes are generated. The computer generated grating can accurately control the magnitude of the single phase shift. The calculation of the phase modulation φ ( x , y ) is shown in Equation (2)
φ ( x , y ) = − tan − 1 [ ∑ n − 1 N I n ( x , y ) sin σ n ∑ n − 1 N I n ( x , y ) cos σ n ] , n = 1 , 2 , ... , N (2)
The arctangent function obtains the truncated phase of [−π, π). In order to determine the one-to-one correspondence between the pixel of the screen and the pixel of the mirror, it is necessary to expand the phase to form a continuous phase. The method of phase unwrapping includes the spatial phase expansion method [
includes branch cutting method, quality map guiding method, minimum norm method; the latter mainly includes binary encoding and multi-frequency heterodyne method.
Measuring the surface of the object by the stripe reflection system, we can obtain two gradient distributions in the mutually perpendicular direction. The gradient refers to the partial derivatives g x ( x , y ) and g y ( x , y ) obtained by the height z ( x , y ) of the surface in the directions of x and y. The calculation is as shown in Equation (3) and Equation (4):
g x ( x , y ) = ∂ z ( x , y ) ∂ x , (3)
g y ( x , y ) = ∂ z ( x , y ) ∂ y , (4)
In order to obtain the specular surface 3D height information, we need to integrate the derivative:
z ( x , y ) = ∫ g x d x + ∫ g y d y , (5)
In the actual testing process, the gradient field is not a conservative field due to the influence of system error and image noise. Southwell model based on regional wave front reconstruction method can not only effectively suppress noise but also deal with complex connected regions. Therefore, it has become the mainstream algorithm of gradient integral.
The Southwell grid model is shown in
The gradient matrix is the same size as the height matrix. The relationship between the gradient data and the height data is only related to the same row or the same column. The above relationship is expressed as
{ 1 2 ( g i , j + 1 x + g i , j x ) = 1 h ( z i , j + 1 − z i , j ) i = 1 , 2 , ... , M ; j = 1 , 2 , ... , N − 1 1 2 ( g i + 1 , j y + g i , j y ) = 1 h ( z i + 1 , j − z i , j ) i = 1 , 2 , ... , M − 1 ; j = 1 , 2 , ... , N (6)
Z represents a height matrix, the size is MN × 1; D represents a coefficient matrix, which is a sparse matrix size [(M − 1)N + M(N − 1)] × MN; G represents a gradient matrix, and the size [(M − 1)N + M(N − 1)] × 1. Equation (6) can be rewritten into:
D Z = G , (7)
Generally, the surface matrix is relatively large. In this case, the D matrix is a singular matrix. The augmented matrix D f needs to be replaced by the matrix
D, and the least norm solution of Z can be obtained by using the least square method:
Z = ( D f T D f ) − 1 D f T G . (8)
The rate of convergence and the ability to suppress noise in the late iteration are two conditions for evaluating the merits of an iterative algorithm. Especially for large-caliber mirrors, the gradient data needs to be reprocessed for each iteration. The higher the convergence efficiency, the fewer the number of iterations. As the number of iterations increases, the ability to suppress gradient noise in the iteration ensures the accuracy of the final fitted shape.
The traditional iterative algorithm is showed as follows [
1) Using the traditional integration method to obtain the original surface height z 0 ( x , y ) as the initial value of the iteration;
2) Calculate the derivatives q ( x , y ) and p ( x , y ) of the current height z n ( x , y ) in the x and y directions, and make a difference with the original gradient to obtain gradient residuals d g x ( x , y ) and d g y ( x , y ) ;
3) Using traditional integration method to integrate d g x ( x , y ) and d g y ( x , y ) to obtain the compensation height value z c n ( x , y ) ;
4) The compensated height values z n ( x , y ) = z n − 1 ( x , y ) + z c n ( x , y ) / n k , nk = 3, 4.0909, 4.9476, 5.6768 (k = 1, 2, 3, 4);
5) Repeat the iterative process of 2-4 until the iterations of adjacent two iterations are less than the threshold, max | z n − z n − 1 | < threshold.
This section presents an improved algorithm that includes two improvements. First, use the two variables of attenuation factor t and iteration number n to control the contribution value of the compensation height to participate in the iteration. Second, the threshold is set for the nth compensation height during iteration. The formula of the new iterative algorithm can be written as:
z n ( x , y ) = z n − 1 ( x , y ) + ω ⋅ z c n ( x , y ) , (9)
ω = 1 ( n − 1 ) t , (10)
The height of the nth compensation is related to the height threshold:
RMS ( z c n ) < threshold n ⋅ m , (11)
The value of m is generally between [
The simulated face is divided into two parts: z 1 is the standard paraboloid:
z 1 = x 2 50 + y 2 50 , (12)
z 2 is set to a high-order shape in a certain area:
z 2 = 1 0.5 cos 2 ( x 6 + y 4 ) + ( 0.25 ( 0.5 x − 2 ) 4 + 2 ) 2 + 0.125 ( ( 0.5 y ) 5 + 2 ) 2 , (13)
The size of z 1 and z 2 is 501 × 501 pixels, and the pixel size is 0.04 mm. In order to get closer to the actual testing conditions, we combined the gradient data of surface shape with the gaussian noise whose SNR = 100:1. Using the traditional Southwell integral z 0 as the initial shape, after four iterations we compare the high frequency partial fit results on the green lines in
In order to compare the capability of the three iterative methods to iterative errors in the high-frequency part, we gather the high-frequency errors fitted by the three methods into the same figure, as shown in
In order to verify the effect of controlling the attenuation factor by setting compensation high threshold in the later iteration, we will perform 10 iterations without changing the initial attenuation coefficient t and controlling the attenuation coefficient, and compare the rms results of the last four fittings. The result is shown in
This paper introduces the basic principle of the stripe reflection method, and proposes an improved algorithm based on traditional Southwell iterative
integration. The algorithm uses the attenuation coefficient to control the compensation value and the attenuation coefficient t is controlled by the compensation height threshold. Through computer simulation, it is verified that the algorithm has better convergence rate in the early iteration period and better fitting
| RMS(mm) | |||
|---|---|---|---|
| Southwell integration method | Traditional iterative integration | Improved iterative integration | |
| Initial value | 1.2472 × 10 − 5 | 1.2472 × 10 − 5 | 1.2472 × 10 − 5 |
| 1st iteration | - | 6.2299 × 10 − 6 | 1.9712 × 10 − 6 |
| 2nd iteration | - | 4.7936 × 10 − 6 | 7.4142 × 10 − 7 |
| 3rd iteration | - | 3.8844 × 10 − 6 | 3.6974 × 10 − 7 |
| 4th iteration | - | 3.2445 × 10 − 6 | 2.2140 × 10 − 7 |
| Change the value of t | 5.5637 × 10 − 8 | 4.8215 × 10 − 8 | 4.2415 × 10 − 8 | 3.7773 × 10 − 8 |
|---|---|---|---|---|
| Keep tunchanged | 8.2965 × 10 − 8 | 7.2555 × 10 − 8 | 6.4581 × 10 − 8 | 5.8273 × 10 − 8 |
of high frequency detail. In the latter part of the iteration, by changing the size of t, it can effectively prevent too many noise points from participating in the iteration and leading to poor surface performance. The simulation results show that the improved algorithm effectively suppresses the influence of noise and has better fitting accuracy than the traditional algorithm at low SNR.
Supported by the NSFC (NO.61378055), Advantages of disciplines in colleges and universities in Jiangsu Province construction grant program.
The authors declare no conflicts of interest regarding the publication of this paper.
Gao, X. and Guo, P.J. (2019) An Improved Algorithm for 3D Reconstruction Integration Based on Stripe Reflection Method. Optics and Photonics Journal, 9, 11-19. https://doi.org/10.4236/opj.2019.98B002