Cu₂ZnSn(S,Se)₄ thin film solar cells fabricated with benign solvents

Abstract

Cu₂ZnSn(S,Se)₄ (CZTSSe) is considered as the promising absorbing layer materials for solar cells due to its earth-abundant constituents and excellent semiconductor properties. Through solution-processing, such as various printing methods, the fabrication of high performance CZTSSe solar cell could be applied to mass production with extremely low manufacturing cost and high yield speed. To better fulfill this goal, environmental-friendly inks/solutions are optimum for further reducing the capital investment on instrument, personnel and environmental safety. In this review, we summarized the recent development of CZTSSe thin films solar cells fabricated with benign solvents, such as water and ethanol. The disperse system can be classified to the true solution (consisting of molecules) and the colloidal suspension (consisting of nanoparticles).Three strategies for stabilization (i.e., physical method, chemical capping and self-stabilization) are proposed to prepare homogeneous and stable colloidal nanoinks. The one-pot self-stabilization method stands as an optimum route for preparing benign inks for its low impurity involvement and simple procedure. As-prepared CZTSSe inks would be deposited onto substrates to form thin films through spin-coating, spraying, electrodeposition or successive ionic layer adsorption and reaction (SILAR) method, followed by annealing in a chalcogen (S- or Se-containing) atmosphere to fabricate absorber. The efficiency of CZTSSe solar cell fabricated with benign solvents can also be enhanced by constituent adjustments, doping, surface treatments and blocking layers modifications, etc., and the deeper research will promise it a comparable performance to the non-benign CZTSSe systems.

Keywords

Cu₂ZnSn(S / Se)₄ (CZTSSe) / solar cell / benign solvents / metal chalcogenide complexes (MCCs) / solution processing

1 Introduction

Precise calibration for a camera is fundamental in computer vision and vision metric. A camera is often modeled as an ideal pin-hole one, which images on the focus plane without distortion. However, lens distortion will make the image distorted from ideal one.

There are several distortion models depending on the type of lens. In this paper, radial, decentering and prism distortion models proposed by Brown [ 1, 2] were to be considered.

To identify each distortion model, several methods have been proposed. The first method uses the known object 3D world coordinates [ 3], with which the result could be inaccurate when both camera intrinsic and extrinsic parameters are estimated at the same time. The second method uses point correspondences in different views [ 4]. This method is based on fundament matrix without the intrinsic and extrinsic explicitly. However, it is difficult to find correct point correspondences in multi-views. The third method, which estimates the distortion without camera parameters, is based on projective invariants [ 5- 7]. Nowadays, more and more researches depend on this method. This method estimates the distortion without camera parameters. The most widely used projective invariant is straight line, which remains straight from different view if there is no lens distortion based on pin-hole camera. This method needs to know the scene in advance, as it is important to find out the straight line or other invariant objects in distortion image. The forth method is also based on point correspondences between image points and world points [ 8], and it uses planar invariant instead. Planar calibration pattern points and non-distortion image points are mapped by homography matrix. This method calculates the distortion model parameters to make the correspondence most fit.

There are two different ways to compute the distortion models, backward mapping and forward mapping. Zhang [ 4] used forward mapping in both model definition and computing, while Hartley and Kang [ 9] used backward mapping.

In this paper, a new camera calibration method is presented, we used only one chessboard pattern image to solve the distortion models, which includes three main types of lens distortion–radial, decentering and prism distortions. It has four steps. First, image feature points are detected by chessboard point detecting algorithm. Second, the non-distortion feature point coordinate is calculated from detected feature point on distortion image. Third, nonlinear optimization is done to get the distortion model. Last, iterate former the three steps until converge. This method is based on the idea that the distortion near the optic axis is tiny, and we can use iterative procedure to reduce influence of this distortion. Therefore, non-distortion feature points can be calculated using the near center feature points on distortion image.

The paper is organized as follows. Section 2 describes the camera lens distortion model. Section 3 proposes the method of distortion model calculating. In Section 4, several experimental results are reported on both real and synthetic data. The paper ends with some concluding remarks.

2 Lens distortion model

Lens distortion makes the actual image different from ideal pin-hole image. Three forms of lens distortion, namely radial distortion, decentering distortion and thin prism distortion are taken into account in this paper [ 8]. The distortion can be formed as follow:

(1) $$\left[\begin{array}{l}x\\ y\end{array}\right]=\left[\begin{array}{l}{x}^{*}\\ y^{*}\end{array}\right]+\left[\begin{array}{l}{\mathrm{\Delta }}_{x^{*}}\\ {\mathrm{\Delta }}_{y^{*}}\end{array}\right].$$

${[x,y]}^T$ is the ideal image without distortion, ${[x^{*},y^{*}]}^T$ is the actual image with distortion, ${[{\mathrm{\Delta }}_{x^{*}},{\mathrm{\Delta }}_{y^{*}}]}^T$ is the amount of distortion. Different forms of lens distortion result in the different part of distortion factor.

2.1 Radial distortion

Radial distortion is caused by inconsistent magnification in the field of view. This distortion is the significant part of lens distortion. Many researches only dealt with radial distortion [ 5, 7, 10].

The amount of radial distortion is circularly symmetric about optical center as shown in Fig. 1, which means there is no distortion at the optical center. In general, the farther the image point is away from the optical axis, the greater the distortion is. Radial distortion can be expressed as follows:

(2) $$\{\begin{array}{l}{\mathrm{\Delta }}_{rx}(x^{*},y^{*})=k_{\mathrm{1}}{x}^{*}({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})+k_{\mathrm{2}}{x}^{*}{({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})}^{\mathrm{2}}\\ {\mathrm{\Delta }}_{ry}(x^{*},y^{*})=k_{\mathrm{1}}{y}^{*}({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})+k_{\mathrm{2}}{y}^{*}{({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})}^{\mathrm{2}}\end{array},$$

where k₁, k₂ are distortion factors of radial distortion model.

Fig.1 (a) Barrel distortion image; (b) pincushion distortion image

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2.2 Decentering distortion

Decentering distortion is due to non-strict collineation of the optical centers of lens elements, as shown in Fig. 2. The decentering distortion model is formed as Eq. (3).

(3) $$\{\begin{array}{l}{\mathrm{\Delta }}_{dx}(x^{*},y^{*})=p_{\mathrm{1}}(\mathrm{3}{(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})+\mathrm{2}{p}_{\mathrm{2}}{x}^{*}{y}^{*}\\ {\mathrm{\Delta }}_{dy}(x^{*},y^{*})=p_{\mathrm{2}}({(x^{*})}^{\mathrm{2}}+\mathrm{3}{(y^{*})}^{\mathrm{2}})+\mathrm{2}{p}_{\mathrm{1}}{x}^{*}{y}^{*}\end{array},$$

where p₁, p₂ are distortion factors of decentering distortion model.

Fig.2 Image with decentering distortion

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2.3 Thin prism distortion

Thin prism distortion arises from the imperfection in lens design and manufacturing as well as camera, as shown in Fig. 3.The thin prism distortion model is described in Eq. (4).

(4) $$\{\begin{array}{l}{\mathrm{\Delta }}_{px}(x^{*},y^{*})=s_{\mathrm{1}}({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})\\ {\mathrm{\Delta }}_{py}(x^{*},y^{*})=s_{\mathrm{2}}({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})\end{array},$$

where s₁, s₂ are distortion factors of prism distortion model.

Fig.3 Image with thin prism distortion

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In this paper, three forms of distortion are considered. Then, the total amount of distortion is the sum of three distortions.

(5) $$\{\begin{array}{l}{\mathrm{\Delta }}_{x^{*}}(x^{*},y^{*})={\mathrm{\Delta }}_{rx}(x^{*},y^{*})+{\mathrm{\Delta }}_{dx}(x^{*},y^{*})+{\mathrm{\Delta }}_{px}(x^{*},y^{*})\\ {\mathrm{\Delta }}_{y^{*}}(x^{*},y^{*})={\mathrm{\Delta }}_{ry}(x^{*},y^{*})+{\mathrm{\Delta }}_{dy}(x^{*},y^{*})+{\mathrm{\Delta }}_{py}(x^{*},y^{*})\end{array}.$$

3 Distortion correct with one chessboard pattern image

The proposed method consists following steps: 1) preprocess chessboard image and pick up the coordinate of corner points; 2) reconstruct non-distortion corner coordinate from distortion image; 3) initialize distortion model parameters using Levenberg-Marquardt nonlinear algorithm (LM); 4) use iterative procedure to minimize distortion residual; 5) use forward mapping method to correct image.

3.1 Corner points detection on distortion image

The performance of distortion correction is affected by the accuracy of the corner point coordinate. In this paper, a new corners detecting method of chessboard pattern image is proposed. This method has high corner detection rate.

The proposed method detects corner points according to the dissimilar distribution of pixels gray-scale between corner region and non-corner region. It defines three operators to calculate the dissimilarity distribution on different directions as follows. S is centrosymmetric operator. V is vertical symmetry operator. H is horizontal symmetry operator.

(6) $$\{\begin{array}{l}s(x_{\mathrm{0}},y_{\mathrm{0}})={\displaystyle \sum _{(x,y)\in neighborhood(x_{\mathrm{0}},y_{\mathrm{0}})}\left|p_{(x,y)}-p_{(\mathrm{2}{x}_{\mathrm{0}}-x,\mathrm{2}{y}_{\mathrm{0}}-y)}\right|}/card(neighborhood(x_{\mathrm{0}},y_{\mathrm{0}}))\\ v(x_{\mathrm{0}},y_{\mathrm{0}})={\displaystyle \sum _{(x,y)\in neighborhood(x_{\mathrm{0}},y_{\mathrm{0}})}\left|p_{(x,y)}-p_{(x,\mathrm{2}{y}_{\mathrm{0}}-y)}\right|}/card(neighborhood(x_{\mathrm{0}},y_{\mathrm{0}}))\\ h(x_{\mathrm{0}},y_{\mathrm{0}})={\displaystyle \sum _{(x,y)\in neighborhood(x_{\mathrm{0}},y_{\mathrm{0}})}\left|p_{(x,y)}-p_{(\mathrm{2}{x}_{\mathrm{0}}-x,y)}\right|}/card(neighborhood(x_{\mathrm{0}},y_{\mathrm{0}}))\end{array},$$

where $neighborhood(x_{\mathrm{0}},y_{\mathrm{0}})$ is 8 × 8rectangle region with point $(x_{\mathrm{0}},y_{\mathrm{0}})$ located at the center, card() returns the pixel number of the region.

Define M as pixels response, whose value is associated with three operators. The response value of corner point is calculated by the following steps:

1)Compute the value of pixels by the three operators;

2)If all of the three operator results are between the pre-defined ranges, the final response value is the result by operator S. Otherwise, M is set to zero.

(7) $$M=\{s|s\in [s_{\min },s_{\max }],v\in [v_{\min },v_{\max }],h\in [h_{\min },h_{\max }]\}\cup \{\mathrm{0}|s\notin [s_{\min },s_{\max }]orv\notin [v_{\min },v_{\max }]orh\notin [h_{\min },h_{\max }]\},$$

Median filtering algorithm is used to remove isolated detective point which is fault detection caused by noise. The corner coordinate can be computed by calculating the connected region centroid of response M, described as follows:

(8) $$centroid={\displaystyle \frac{{\displaystyle \sum _{connected}m*position}}{{\displaystyle \sum _{connected}m}}},$$

where $m\in M$ , connected region is detected by 8-adjacent connection, position_m is the coordinate of the point with value m.

In this paper, distortion of the four corner points around the center is tiny for distortion image. Non-distortion points are reconstructed by the geometry feature (considering the same size of each block) of chessboard, which exploits these four corner points, as shown in Fig. 4, defined in Eq. (9).

(9) $$\{\begin{array}{l}{x}_{(i,j)}=x_{(\mathrm{0},\mathrm{0})}+j*(x_{(\mathrm{0},\mathrm{1})}-x_{(\mathrm{0},\mathrm{0})})+i*(x_{(\mathrm{1},\mathrm{0})}-x_{(\mathrm{0},\mathrm{0})})\hfill \\ y_{(i,j)}=y_{(\mathrm{0},\mathrm{0})}+j*(y_{(\mathrm{0},\mathrm{1})}-y_{(\mathrm{0},\mathrm{0})})+i*(y_{(\mathrm{1},\mathrm{0})}-y_{(\mathrm{0},\mathrm{0})})\hfill \end{array},$$

where (i,j) is the points array order, i and j could be negative integer, center four points are (0,0), (0,1), (1,0), (1,1).

To minimize the influence of the distortion of four center points, this paper introduces iterative process. The distortion of the four points would reduce after image correction, and then the corrected image is treated as an input distortion image for next iteration to reconstruct non-distortion points, iterate this procedure until four points unchanged after correction.

Fig.4 (a) Distortion image; (b) the dots are detected corners in image (a) and the circles are ideal points location.

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3.2 Distortion model calculate

Three forms of distortion indicate that the optical center region is always non-distortion.

As long as the distortion and non-distortion points are achieved, the total amount of distortion would be computed by Eq. (1).

Expansion Eq. (5) is the whole model formed by three distortions.

(10) $$\{\begin{array}{l}{\mathrm{\Delta }}_x(x^{*},y^{*})=k_{\mathrm{1}}{x}^{*}({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})+k_{\mathrm{2}}{x}^{*}{({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})}^{\mathrm{2}}+[p_{\mathrm{1}}(\mathrm{3}{(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})+\mathrm{2}{p}_{\mathrm{2}}{x}^{*}{y}^{*}]+s_{\mathrm{1}}({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})\\ {\mathrm{\Delta }}_y(x^{*},y^{*})=k_{\mathrm{1}}{y}^{*}({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})+k_{\mathrm{2}}{y}^{*}{({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})}^{\mathrm{2}}+[p_{\mathrm{2}}({(x^{*})}^{\mathrm{2}}+\mathrm{3}{(y^{*})}^{\mathrm{2}})+\mathrm{2}{p}_{\mathrm{1}}{x}^{*}{y}^{*}]+s_{\mathrm{2}}({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})\end{array}.$$

In this paper, the optical center is set to the image center. Different center only affects the distortion model values, which would not influence the result of distortion correction. Then, the linear equation is showed as follows

(11) $$X-X^{\mathit{*}}=AP,$$

where

$$\{\begin{array}{l}A=\left[\begin{array}{l}{x}^{*}({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}}),x^{*}{({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})}^{\mathrm{2}},\mathrm{3}{(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}},\mathrm{2}{x}^{*}{y}^{*},{(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}},\mathrm{0}\\ y^{*}({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}}),y^{*}{({(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}})}^{\mathrm{2}},\mathrm{2}{x}^{*}{y}^{*},{(x^{*})}^{\mathrm{2}}+\mathrm{3}{(y^{*})}^{\mathrm{2}},\mathrm{0},{(x^{*})}^{\mathrm{2}}+{(y^{*})}^{\mathrm{2}}\end{array}\right]\\ P=\mathit{}{\left[\begin{array}{llllll}{k}_{\mathrm{1}}& k_{\mathrm{2}}& p_{\mathrm{1}}& p_{\mathrm{2}}& s_{\mathrm{1}}& s_{\mathrm{2}}\end{array}\right]}^T\\ X=\mathit{}{[x\hspace{1em}y]}^T\\ X^{\mathit{*}}={[x^{*}\hspace{1em}{y}^{*}]}^T\end{array}.$$

P is the required model parameters, X is the non-distortion position, X* is the distortion position.

The model P can be calculated by LM nonlinear optimization [ 11]. The objective function is defined as follows:

(12) $${\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}[{\Vert X_{\mathrm{0}}-X(P,X_{\mathrm{0}}^{*})\Vert }^{\mathrm{2}},\cdots ,{\Vert X_j-X(P,X_j^{*})\Vert }^{\mathrm{2}},\cdots ,{\Vert X_m-X(P,X_m^{*})\Vert }^{\mathrm{2}}]=\mathrm{0},$$

where $X_j$ is non-distortion pixel coordinate, $X(P,X_j^{*})$ is the pixel coordinate transferred by Eq. (11) related to distortion image.

The Jacobian matrix is used for LM algorithm. From Eq. (12), the Jacobian matrix can be easy to get.

Then, the procedure for calculating distortion parameters is:

1)use corner detection to obtain point coordinate;

2)calculate the non-distortion point location depending on near center point;

3)using LM method to optimize distortion model parameters;

4)calculate the distortion residual, repeat steps 1), 2) and 3) until distortion residual convergence;

The parameter of convergence is the best solution of proposed method.

3.3 Distortion image correction

Image is corrected by the distortion parameters obtained in last steps. This procedure contains two steps.

1)Coordinate transfer: transfer the pixel coordinate from distortion image to non-distortion image (forward-mapping, as shown in Fig. 5) or from non-distortion image to distortion image (backward-mapping). This paper uses forward-mapping method to rectify image.

Fig.5 Forward-mapping from distortion image to correction image

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2)Gray-scale interpolation: in this paper, bilinear interpolation is used to reconstruct the correction image.

4 Experiments

4.1 Corner points detection algorithm

The performance of the proposed corner detective algorithm is tested by comparing with the chessboard corner detection algorithm in OpenCV. It is shown in Fig. 6 that the proposed method achieves higher detection rate and is more accurate than the OpenCV method.

Fig.6 White rectangles are the detected corners: (a) the proposed method; (b) the OpenCV method

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4.2 Synthetic data experiment

The performance of the proposed method is tested in several experiments. First, use the proposed method to estimate parameters of artificial distortion images. We distorted the origin image (Fig. 7) by adding all the three forms of distortion, and then corrected them with the proposed method.

Fig.7 Origin image without distortion

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Fig.8 (a) and (b) are images with different distortion parameters; (c) and (d) are corrected image by proposed method, which parameters are shown in Table 1

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Tab.1 Compares the distortion between man-made parameters and the results of proposed method

origin distortion image	Figure 8(a)	Figure 8 (b)
distortion model	$k_{\mathrm{1}}=\mathrm{3.0}e-\mathrm{6},k_{\mathrm{2}}=-\mathrm{5.8}e-\mathrm{13}$ $p_{\mathrm{1}}=-\mathrm{2.4}e-\mathrm{5},p_{\mathrm{2}}=\mathrm{2.86}e-\mathrm{5}$ $s_{\mathrm{1}}=\mathrm{6.5}e-\mathrm{6},s_{\mathrm{2}}=-\mathrm{7.2}e-\mathrm{6}$	$k_{\mathrm{1}}=-\mathrm{2.3}e-\mathrm{6},k_{\mathrm{2}}=\mathrm{4.3}e-\mathrm{13}$ $p_{\mathrm{1}}=\mathrm{5.0}e-\mathrm{5},p_{\mathrm{2}}=-\mathrm{3.4}e-\mathrm{5}$ $s_{\mathrm{1}}=\mathrm{7.8}e-\mathrm{6},s_{\mathrm{2}}=\mathrm{5.6}e-\mathrm{6}$
correct image by proposed method	Figure 8(c)	Figure 8(d)
correct model by proposed method	$k_{\mathrm{1}}=\mathrm{3.67}e-\mathrm{6},k_{\mathrm{2}}=-\mathrm{3.47}e-\mathrm{12}$ $p_{\mathrm{1}}=-\mathrm{1.99}e-\mathrm{5},p_{\mathrm{2}}=\mathrm{2.36}e-\mathrm{5}$ $s_{\mathrm{1}}=-\mathrm{1.26}e-\mathrm{6},s_{\mathrm{2}}=\mathrm{2.87}e-\mathrm{7}$	$k_{\mathrm{1}}=-\mathrm{2.67}e-\mathrm{6},k_{\mathrm{2}}=\mathrm{1.37}e-\mathrm{13}$ $p_{\mathrm{1}}=\mathrm{4.90}e-\mathrm{5},p_{\mathrm{2}}=-\mathrm{2.79}e-\mathrm{5}$ $s_{\mathrm{1}}=\mathrm{1.48}e-\mathrm{5},s_{\mathrm{2}}=-\mathrm{6.87}e-\mathrm{6}$

This method has a good estimation about parameters $k_{\mathrm{1}},p_{\mathrm{1}},p_{\mathrm{2}}$ , which are the main factors of distortion. The correction images are very similar to the origin image without distortion, shown in Fig. 8.

Synthetic data are used for further analysis for the performance of the proposed method. The synthetic data are setup as Gao and Yin did inRef [ 8], shown in Fig. 9(a), defining the optical axis as z-axis, the horizon direction on the image plane as x-axis, and vertical direction on the image plane as y-axis.

Fig.9 Performance of the proposed method with the synthetic data, dots are distorted points and circles are non-distorted points. (a) The distorted image; (b) the corrected image by the proposed method

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Fig. 10 shows the three methods’ performance versus noises. In this experiment, Gaussian noise with zero mean and σ standard deviation is added to distorted points. We test the performance with noise standard deviation σ from 0.1 to 1.5 pixels. Define ERROR as root of mean square error in Eq. (13), which is to describe the correction accuracy. For each noise level, we used 100 images to evaluate the method’s performance. Brown’s method does not correct prism distortion. It indicates that the method without considering the entire distortion model could not make a good image correction.

(13) $$ERROR={\displaystyle \frac{{\displaystyle \sum _i\sqrt{{(x_d-x_r)}^{\mathrm{2}}+{(y_d-y_r)}^{\mathrm{2}}}}}{n}},$$

where n is the number of samples, (x_d,y_d) is test data and (x_r,y_r) is the real data.

Fig.10 Compare ERROR of the proposed method with Gao’s and Brown’s method versus different noises levels

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To test the robust of the proposed method, the image data are rotated around z-axis or x-axis. The results are shown in Figs. 11 and 12. Figure 11 shows the ERROR results when the chessboard is rotated around z-axis. It shows that the correction accuracy is similar between different rotation angles. In Fig. 12, though the rotation around x-axis is even up to 7.5°, the ERROR is similar with Gao’s method.

Fig.11 ERROR of different rotations around the z-axis of image versus noises levels

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Fig.12 ERROR versus different rotation around the x-axis of image

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The proposed method is very fast to converge. In these experiments, the iteration times are nearly 2 to 4, and the total time consuming to solve distortion model is almost 10 ms (image size is 752 × 480, using PC with CPU of i3 2.5 GHz).

4.3 Real data experiment

Real distortion images are taken by the Pentax CCTV lens (F1.2) with the focal lengths of 4 and 8 mm, which are captured by smart camera based on TMS320DM6437 DSP. Figure 7 is the object for taking photos. Figure 13 shows the pictures with different lens.

Fig.13 (a) Pictures taken with 4 mm lens, image size is 752 ×480; (b) pictures is taken with 8 mm lens, image size is 640 ×480

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Use the proposed method to correct the distortion images in Fig. 13, and results are shown in Fig. 14. It shows that proposed method makes the collinear point in straight pattern in image. Table 2 presents the solutions in detail. We evaluate distortion residual of the distance between point and line. Since the points on each columns and rows are actually collinear, the distortion residual is the average distance between points and line. First, least square method is used to estimate the line function. Second, the absolute distance from points to the line is computed. The average absolute distance of all the point indicates the error of distortion, named as root mean square error (RMSE). Maximum distortion rate is calculated by the maximum distance from point to line dividing the image diagonal length.

Fig.14 (a) Corrected image taken with 4 mm lens; (b) corrected image taken with 8 mm lens

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Tab.2 Experiment results with proposed method

	len 1 focal f = 4 mm	len 2 focal f = 8 mm
correct model	$k_{\mathrm{1}}=\mathrm{8.86}e-\mathrm{7},k_{\mathrm{2}}=\mathrm{3.97}e-\mathrm{12}$ $p_{\mathrm{1}}=\mathrm{1.23}e-\mathrm{5},p_{\mathrm{2}}=\mathrm{5.64}e-\mathrm{5}$ $s_{\mathrm{1}}=\mathrm{1.43}e-\mathrm{5},s_{\mathrm{2}}=-\mathrm{2.23}e-\mathrm{5}$	$k_{\mathrm{1}}=\mathrm{7.72}e-\mathrm{7},k_{\mathrm{2}}=\mathrm{1.98}e-\mathrm{14}$ $p_{\mathrm{1}}=\mathrm{1.05}e-\mathrm{8},p_{\mathrm{2}}=\mathrm{2.89}e-\mathrm{8}$ $s_{\mathrm{1}}=\mathrm{1.37}e-\mathrm{8},s_{\mathrm{2}}=\mathrm{2.15}e-\mathrm{8}$
RMSE	0.34 pixel	0.25 pixel
maximum distortion rate	0.13%	0.10%

Table 2 indicates that proposed method is effective, which makes the image with little distortion and even the maximum distortion rather small.

5 Conclusions

This paper proposed a new method for lens distortion correction. Only one chessboard pattern image was used to rectify the three distortion models, which makes the implement convenient. To minimize the influence by the tiny distortion of the center four points, iterative procedure was introduced to optimize the distortion model and reevaluate the non-distorted points. Both synthetic data and real data have been adopted to test by the proposed method. Synthetic data experiment showed that the proposed method had good performance versus noises, and it was more accurate for lens distortion correction by considering the whole distortion models. Besides, chessboard rotation experiment was conducted to show the robustness of this method. The result of real data experiment also indicated that the proposed method was effective.

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Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 61274055 and 61322401), China Postdoctoral Science Foundation (No. 2013M542015) and Fundamental Research Funds for the Central Universities, Huazhong University of Science and Technology (No. CXY12M008). Jie Zhong acknowledges the support of Foundation for Scientific Research of Wuhan University of Technology.

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg