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Abstract

Aiming at the problem of camera calibration with multiple parameters, this paper proposes the fusion optimization algorithm of improved differential evolution and particle swarm in the calibration of the camera. Adaptive judgment factors is Introduced to control the improvement of differential evolution during each iteration (IDE) algorithm and particle swarm optimization (PSO) algorithm solve the multiple parameters of traditional camera calibration algorithm. The proposed algorithm can ensure the diversity and effectiveness of individual evolution of the population. Experimental results show that the algorithm has excellent global search capabilities and local optimizations ability. It can accurately calibrate the camera. The convergence precision, speed and robustness performance significantly is superior to PSO (Particle swarm optimization algorithm), DE (differential evolution algorithm), GA (Genetic algorithm) and Zhang’s method. It improves the precision and speed of the proposed calibration method. The root mean square error of the calibration algorithm proposed in this paper is only 0.182, the calibration error result is smaller than other several algorithms. The reprojection error of our method is 0.05938 (Ex/pixel) and 0.02988 (Ey/pixel). It is smaller than GA, PSO, DE, and Zhang’s method. So, the algorithm performance is excellent.

Keywords

Camera calibration differential evolution particle swarm

Introduction

Camera parameter calibration is a necessary process and key technology to achieve accurate target pose acquisition based on vision measurement.¹ Camera calibration needs to determine the internal orientation elements and external elements, and the reliability of calibration results directly affects the stability and accuracy of target position and attitude information measurement.^2,3 Therefore, high precision calibration is an important guarantee to realize accurate vision measurement technology. The early direct linear transformation method⁴ established the geometric linear model of camera imaging, and the imaging model can be obtained directly through the linear equation. On this basis, Tsai⁵ proposed a two-step calibration method introducing lens distortion coefficient. Zhang’s calibration model was used to automatically calculate parameters, and a kind of automatic camera calibration algorithm was proposed.⁶ The calibration method can reduce the complexity of the calibration process to some extent, but the calibration accuracy is not improved. The above three classical calibration methods lay the foundation for the development of current calibration technology.

Research on self-calibration method^7–9 and camera calibration method based on active vision^10,11 have also matured successively. Some calibration algorithms^12,13 models with intelligent ideas have also begun to appear.

In recent years, scholars have applied the population optimization algorithm with intelligent optimization function to the camera parameter calibration. Particle swarm optimization and its improved algorithm are used to optimize the camera parameters to obtain accurate camera parameters. Zhou et al.¹⁴ proposed a virtual three-dimensional calibration method of camera internal parameters based on mutation mechanism particle swarm optimization algorithm. Through the idea of phased optimization, the external parameters and some internal parameters of the camera imaging model were established for initial value estimation. Deng et al.¹⁵ adopted hybrid particle swarm optimization algorithm to effectively calibrate camera parameters and it has good optimization ability, avoiding local convergence to a certain extent.

At the same time, the differential evolution (DE) algorithm is also applied to the camera parameter calibration.^16–19 Zhang and Zhong²⁰ adopted the camera calibration method combining step-by-step method and improved differential evolution algorithm to solve the problem of multiple local extreme points of the objective function to be optimized. De la Fraga and Schütze²¹adopted an improved differential evolution algorithm to solve the optimization problem of highly nonlinear camera calibration, overcoming the disadvantages of poor convergence accuracy or non-convergence in conventional calibration process. However, the convergence rate of PSO algorithm is fast in the initial stage of solving the optimization problem, and in the later stage, because all the particles are close to the optimal particle, the diversity of the whole population is lost, and the particles are easy to fall into the local optimal. IDE algorithm has the ability to maintain population diversity and explore local search, but it has no mechanism to store previous processes and use global information about the search space, so it can easily lead to the waste of computing power.

Based on the above analysis, it can be found that the future focus will be on the development of intelligent calibration algorithms, so as to adapt the camera calibration parameters to the diversified practical application requirements. So this paper combine the advantages of PSO and IDE, the adaptive judgment factor is introduced to control the particle swarm optimization in the process of each iteration and proportion of improved differential evolution algorithm. Consider using PSO algorithm or IDE algorithm to update individuals according to probability rules, and guide the evolution process of individuals in PSO operation by using the individuals obtained from IDE operation through information exchange mechanism. In this way, the greater the crossover probability, the more information the new individual inherits from the mutant individual, and the richer the population diversity, thus ensuring the global solution accuracy and efficiency of the algorithm.

Camera calibration

Camera calibration is a process of establishing equations based on the corresponding relationship between the 3D physical coordinates of the known feature points and the image pixel coordinates, and obtaining the internal and external parameters of the camera model through solving and optimizing equations. The commonly used pinhole projection model is shown in Figure 1.

Figure 1.

Pinhole imaging model of camera.

$O_{w} - X_{w} Y_{w} Z_{w}$ is the world coordinate system. $O - xy$ is two dimensional image physical coordinate system. $O_{c} - X_{c} Y_{c} Z_{c}$ is the camera coordinate system.

Under the ideal pinhole imaging model, a point in the world coordinate system can be converted to the pixel coordinate system of two-dimensional image by the following formula (1),

s (\begin{matrix} u \\ v \\ 1 \end{matrix}) = (\begin{matrix} \begin{matrix} α & γ & μ_{0} & 0 \\ 0 & β & ν_{0} & 0 \\ 0 & 0 & 1 & 0 \end{matrix} \end{matrix}) (\begin{matrix} R & t \\ 0 & 1 \end{matrix}) (\begin{matrix} X_{w} \\ Y_{w} \\ Z_{w} \\ 1 \end{matrix})

(1)

In the formula, R is the rotation matrix and t is the translation vector between the coordinate systems $O_{w} - X_{w} Y_{w} Z_{w}$ and $O_{c} - X_{c} Y_{c} Z_{c}$ . S is the scale factor, $μ$ and $ν$ are the pixel coordinates of the two-dimensional image plane of the point P, $α$ and $β$ are respectively the equivalent focal length in the direction of the axis, $μ_{0}$ and $ν_{0}$ are the coordinates of the intersection point (main point) of the optical axis and the image plane. The parameter $γ$ is the distortion coefficient of the Angle between $X_{c}$ and $Y_{c}$ pixel coordinate axis.

Algorithm design

Particle swarm optimization algorithm

PSO algorithm based on population evolution has been widely used to solve multi-dimensional optimization problems. Each particle in the population is constructed by a D-dimensional vector. Let’s call it $X_{i} = [x_{i 1}, x_{i 2}, \dots, x_{iD}]$ , where $i = 1, 2, \dots, N_{p}$ , $N_{p}$ represents the size of the population, which will directly affect the search ability of the PSO algorithm and the amount of computing to search the optimal value.

The particle swarm optimization algorithm is expressed as follows:

v_{id}^{k + 1} = v_{id}^{k} + c_{1} r_{1} (p_{id}^{k} - x_{id}^{k}) + c_{2} r_{2} (p_{gd}^{k} - x_{id}^{k})

(2)

x_{id}^{k + 1} = x_{id}^{k} + v_{id}^{k + 1}

(3)

Where, $x_{id}^{k}$ represents the position of the ith particle at time k. $v_{k}^{i}$ represents the velocity of the ith particle at time k ( $i = 1, \dots, N$ , N is the number of particles; $d = 1, \dots, D$ , D is the particle dimension) .

C1 and C2 are acceleration coefficients, namely learning factors. R1 and R2 are random numbers in [0,1]; $p_{i}^{k}$ is the location of the history optimal of the ith particle; $p_{g}^{k}$ is population history optimal position.

Particle trajectory analysis shows that if each particle converges to its local attractor $a_{i}^{k}$ , then the particle swarm optimization algorithm converges. Among them, $a_{i}^{k}$ is expressed in equation (4).

a_{id}^{k} = ϕ p_{id}^{k} + (1 - ϕ) p_{gd}^{k} ϕ = \frac{c_{1} r_{1}}{c_{1} r_{1} + c_{2} r_{2}}

(4)

Improved differential evolution algorithm

Improved differential evolution (IDE) algorithm can find a balance between local search and global search in order to overcome the shortcomings of traditional differential evolution algorithms that are easy to fall into local optimal solutions,

Let the number of the population $i_{0} = [1 : 2 N_{p}]$ and the dimension $j_{0} = [1 : D]$ of the individual. Denoted as $i = [1, \dots, τ, i^{*}, \dots N_{p}]$ , where $i, i_{0}, j_{0} \in Z^{+}$ ; Then the population is initialized as follows:

P_{(i_{0}, j_{0})} ~ U (lo w_{(j_{0})}, u p_{(i_{0})} | (2 N_{p}, D) \leftarrow size (P)

(5)

In formula (5), $N_{p}$ is the number of variable individuals in the population in the operating mode; D is the dimensionality of the individuals. U is the initial parameter range, $lo w_{(j_{0})}$ is the lower limit of the parameter and $u p_{(i_{0})}$ is upper limit of the parameter. The minimum fitness value of the population $p_{(i_{0})}$ can be calculated by the minimum objective function $η$ in (6):

fit P_{(i_{0})} = η (P_{(i_{0})})

(6)

Improved differential evolution method must first randomly select $N_{p}$ from the number $2 N_{p}$ of individuals to form a sub-population $P_{sub}$ , and the number and dimensions of the individuals in the sub-population are $(N_{p}, D) \leftarrow size (p_{sub})$ . The random selection and generation of subpopulation $P_{sub}$ are carried out by formula (7).

P_{sub} = P_{(k)} | {ζ = j_{(1 : N_{p})}} | j = randperm (1 : 2 N_{p})

(7)

Where, $randperm (\cdot)$ represents the re-random ordering of individual element labels.

The objective function value of the randomly selected subpopulation $P_{sub}$ can be calculated by the following formula

fit P_{sub} = fit P_{(ζ)}

(8)

The weighted operation of the population produces a temporary population during each generation of evolution:

\begin{matrix} Temp P_{index = 1 : N} = [\begin{matrix} Temp P_{1} \\ \dots \\ Temp P_{N} \end{matrix}] \\ Temp P_{(index)} = \sum (ω ⊙ P_{(l)}) \end{matrix}

(9)

In formula (9), $l$ is the individual label with index number $randperm (1 : 2 N_{p})$ after sorting by $(N_{p} + 1 : 2 N_{p})$ ; Operator ⊙ represents the Hadamard product between matrices. $index = 1 : N_{p}, index \in Z^{+}$ .

In addition, $ω^{*} = ζ_{(N)}^{3} | [N_{p}, 1] = size (ω^{*})$ , then assign $ω^{*}$ by formula $ω^{*} : = \frac{ω^{*}}{\sum ω^{*}}$ , and finally the weight coefficient matrix can be expressed as $ω = ω^{*} \times [1]_{[1, D]}$ .

During the mutation process, the binary mapping matrix $M$ is initialized to $M_{(1 : N_{p}, 1 : D)} = 0$ ; in each iteration cycle, the matrix $M$ is updated by formula (10)

M_{(index, 1)} : = 1

(10)

Where, $J = V (1 : [H \times D]) | V = randperm (j_{0})$ , $[]$ Indicates rounding up; Let $α, β ~ U (0, 1)$ , $k_{(\cdot)} ~ U (0, 1)$ , $k_{(\cdot)} \neq 0$ , Then H is selected by formula (11)

If $α < β$ then $H = k_{(1)}^{3}$ else

H = 1 - k_{(1)}^{3}

(11)

Where, the operator $(\cdot) = size (k_{(1)}^{3})$ , to ensure the same dimension. The scaling factor $F$ is generated using equation (12):

\begin{matrix} {if α < β then F = [λ_{(0)}^{3}]' [1, D] = size (F) \\ else F = (λ_{(N_{p})}^{3} \times Δ) [N_{p}, D] = size (F) \end{matrix}

(12)

Where $Δ = [1]_{(1, D)}$ ; $λ_{(\cdot)} ~ N (0, 1)$ , That is, it obeys the standard normal distribution.

In the improved differential evolution algorithm, the trial vector $ϕ$ is calculated by formula (13):

\begin{matrix} ϕ = P_{sub} + F ⊙ M ⊙ (TempP - P_{sub (m)}) \\ | m = randperm (i) | m \neq [1 : N_{p}] \end{matrix}

(13)

According to the greedy selection rule, $P_{sub}$ and $fit P_{sub}$ are updated within the upper and lower bounds through $ϕ$ and $fit ϕ$ . And complete the update by using the rules in equation (14):

\begin{matrix} if (fit ϕ_{(i^{*})} < fit P_{sub (i^{*})}) then [P_{sub (i^{*})}, fit P_{sub (i^{*})}] \\ : = [[ϕ_{(i^{*})}, fit ϕ_{(i^{*})}] | i^{*} \in i \end{matrix}

(14)

The update of $P_{sub}$ and $fit P_{sub}$ is used to update $P_{(l)}$ and $fit P_{(l)}$ . The update rule is as follows:

[P_{(l)}, fit P_{(l)}] = [P_{sub}, fit P_{sub}]

(15)

The final global optimal solution:

[g min, gbest] = [fit P_{(τ)}, P_{(τ)}] | fit P_{(τ)} = min (fitP)

(16)

Where, $τ \in i .$

It can be seen that the algorithm expands the scope of population search space and avoids the problems such as gradual singleness and falling into prematurity in the process of new population evolution.

Camera calibration algorithm

The evolution criterion of the differential evolution algorithm is based on adaptive information, and does not require additional conditions such as function derivation and continuity. In addition, differential evolution has inherent parallelism and is suitable for large-scale parallel sub-processing. But the differential evolution algorithm does not use the prior knowledge of the individual, that is, there is no mechanism to store the previous process and use the global information about the search space. The particle swarm optimization algorithm makes decisions based on the experience of itself and other particles, so it can effectively make up for the difference.

This section integrates IDE and PSO algorithms, and introduces adaptive judgment factors to control the proportion of particle swarm optimization and differential evolution algorithms used in each iteration process, and considers the use of PSO algorithm or IDE algorithm to update individuals according to the law of probability. It is ensured that the greater the crossover probability, the more information the new individual inherits from the mutated individual, and the richer the population diversity. The judgment factor is used to select the update method of the individual, and its calculation method is:

η_{G} = η_{0} \frac{2}{π} arccot \frac{G_{max} - G + 1}{G^{2}}

(17)

Where $G_{max}$ is the total number of iterations, G is the current evolutionary algebra and $G \in [1, G_{max}]$ , $η_{0}$ is a constant.

Then when the ith individual generates a new individual, a random number of $s_{i} ~ [0, 1]$ is first generated. If $s_{i} < η_{0}$ , the IDE algorithm is used to update the individual, otherwise the particle swarm optimization algorithm is used to update the individual.

According to the camera model analysis, the hybrid algorithm of IDE and PSO is proposed to obtain the camera calibration parameters. The obtained image 2D coordinates are compared with the actual measured image coordinates to verify the feasibility and superiority of the algorithm.

Suppose there are n planar template images, each with m calibration points, each of the same size, in the same noisy environment. The objective function is established as follows:

\begin{array}{l} f_{o b j} = \min \sum_{i = 1}^{n} \sum_{j = 1}^{m} \\ | | {\hat{p}}_{i j} - p (M_{A}, k_{1}, k_{2}, k_{3}, p_{1}, p_{2}, R_{i}, T_{i}, P_{j} | | \end{array}

(18)

Where ${\hat{p}}_{ij}$ is the pixel coordinate of the jth calibration point in the ith image. $R_{i}$ and $T_{i}$ are rotation and translation matrices corresponding to the ith image.

Camera calibration experiment results analysis and performance comparison

Camera calibration test

Two CF1000C cameras are used, with an image resolution of 3664 × 2748 pixels. The experimental platform is the Windows 7 operating platform, the CPU clock speed is 4 GHz, and the running memory is 6 GHz. In the differential particle swarm optimization model, set the maximum number of iterations of 1000, the number of consecutive optimizations is 100, the number of particles is 100, and the acceleration factor is $c_{1} = c_{2} = 2$ . The calibration plate used in this experiment is shown in Figure 2.

Figure 2.

Calibration target.

The calibration plate is fabricated by photographically, and a total of white circles, four relatively large white circles, each adjacent white circle spacing of 25 mm, used for camera calibration to match the corresponding spatial points and image points. The target production error is 1 mm, and the accuracy of this target can meet the requirements of the camera calibration on the calibration plate.

By changing the position angle of the calibration target, 10 sets of images of different positions and angles are taken as the detection image. The camera calibration calculation solution flow as shown.

Algorithm flow:

Step 1 Feature point center extraction

Step 2 Initialize population and individual parameters randomly

Step 3 Determine the minimum objective function. $f_{obj} = min \sum_{i = 1}^{n} \sum_{j = 1}^{m} ‖ {\hat{p}}_{ij} - p (M_{A}, k_{1}, k_{2}, k_{3}, p_{1}, p_{2}, R_{i}, T_{i}, P_{j} ‖$

Step 4 Initialize the individual optimal position and the global optimal position

Step 5 Determine whether the iteration termination condition is met

Step 6 If not, calculate the adaptive judgment factor $η_{G} = η_{0} \frac{2}{π} arccot \frac{G_{max} - G + 1}{G^{2}}$ ; If yes, output the optimal solution.

Step 7 Judge s < g? if yes, then improved algorithm to update individuals; If not, pso algorithm updates individuals

Step 8 Individual selection and update by calculating the objective function, then go to step 4

It can be seen from Table 1 that the algorithm starts from the maximum number of iterations of 600, and the iteration basically reaches convergence. Starting from the number of iterations of 800, the iteration basically begins to stabilize, and the required parameter values are the camera internal parameter values.

Table 1.

Experimental results of solving camera internal parameters based on differential particle swarm optimization.

Parameter	Number of iterations
Parameter	200	400	600	800	1000
$f_{x}$ (pixels)	1078.65897	1079.13895	1080.98213	1081.95963	1082.01238
$f_{y}$ (pixels)	1065.85362	1066.52963	1068.74852	1068.995284	1069.74963
$c_{x}$ (pixels)	263.12856	265.12762	264.22641	264.68247	265.09521
$c_{y}$ (pixels)	203.49632	203.69741	203.93674	204.10963	204.39421
$k_{1}$ (pixels)	0.00756	0.00582	0.00563	0.00395	0.00398
$k_{2}$ (pixels)	0.061685	0.056125	0.054397	0.052753	0.051761
$k_{3}$ (pixels)	0.099632	0.0853614	0.0837425	0.078365	0.0779351
$p_{1}$ (pixels)	−0.045761	−0.0449631	−0.0438634	−0.0430749	−0.0429987
$p_{2}$ (pixels)	−0.0397562	−0.0396322	−0.0395123	−0.0393756	−0.0391938
$fitness$	0.022963	0.0057531	$2.532 \times 10^{- 5}$	$3.246 \times 10^{- 5}$	$3.259 \times 10^{- 5}$

According to the transformation relationship of the random calibration model, the corresponding 2D image coordinates $(u, v)$ of the calibrated 2D camera parameters and 3D spatial coordinates $(X_{w}, Y_{w}, Z_{w})$ are derived and compared with the actual image coordinates $(\bar{u}, \bar{v})$ obtained through image processing, which can effectively verify the accuracy and calibration reliability of the calibration method. This experiment is based on 20 groups of randomly selected actual calibration image data, as shown in Table 2. It can be seen from Table 2 that the image coordinates derived from any 20 groups of standard detection images are close to the actual image coordinates, and the difference between the derived values and the actual detection values is small, all of which are less than 0.10 standard detection pixels. This shows that the calibration method proposed in this paper has good performance.

Table 2.

Reliability verification results.

$X_{w}$ (mm)	$Y_{w}$ (mm)	$Z_{w}$ (mm)	$\bar{u}$ (pixels)	$u$ (pixels)	$\| u - \bar{u} \|$ (pixels)	$\bar{v}$ (pixels)	$v$ (pixels)	$\| v - \bar{v} \|$ (pixels)
−5.5	121	180	298.453681	298.345158	0.108523	583.748213	583.658737	0.089476
−19.5	156	180	158.723652	158.625295	0.098357	438.932542	438.828586	0.103956
14.5	115	180	352.954612	352.864975	0.089637	368.459762	368.361433	0.098329
16.2	131	180	510.965232	510.870598	0.094632	638.397621	638.297858	0.089763
17.5	160	180	429.941257	429.849772	0.0914853	267.977612	267.888248	0.089364
−4.5	110	180	216.438523	210.336167	0.102356	726.395962	726.298226	0.097736
−25.5	120	180	198.641562	198.560026	0.081536	392.676834	392.583367	0.093467
32.5	105	180	625.492831	625.407555	0.085276	536.349793	536.259726	0.090067
6.5	135	180	561.943721	561.864335	0.079385	259.366281	259.273406	0.092875
3.5	99	180	360.943152	360.8453	0.097852	195.356213	195.26719	0.089023

Performance comparison of algorithms

In order to evaluate the iterative performance and average error of the calibration results of the proposed algorithm, the performance was compared with that of GA, PSO, DE, Zhang’s method and Improved Differential Evolution and Particle Swarm Optimization (IDE-PSO) respectively. As shown in Figure 3, improved algorithm in this paper converges quickly and has a small average error. Its iterative performance and average error are better than GA, PSO, DE, Zhang’s method. See also in Table 3.

Figure 3.

Iteration and average error.

Table 3.

Comparison of camera calibration parameters for five algorithms.

Parameter	GA	PSO	DE	Zhang’s method	Our methods
$f_{x}$ (pixels)	1159.58423	1155.34652	1153.22832	1152.98236	1151.68721
$f_{y}$ (pixels)	1038.29541	1036.57342	1034.58126	1034.01985	1030.29467
$c_{x}$ (pixels)	229.94621	228.59423	228.29842	227.92510	226.83543
$c_{y}$ (pixels)	221.25974	220.59462	220.29824	219.16782	218.57642
$k_{1}$ (pixels)	0.008521	0.008235	0.008123	0.008106	0.008038
$k_{2}$ (pixels)	0.059852	0.055621	0.053051	0.052869	0.051672
$k_{3}$ (pixels)	0.122942	0.119837	0.105342	0.098236	0.095673
$p_{1}$ (pixels)	−0.075361	−0.061891	−0.050823	−0.04826	−0.039562
$p_{2}$ (pixels)	−0.04623	−0.041687	−0.038462	−0.037268	−0.036492

In order to further verify the performance of the experimental results, the MSE (Maximum Mean Square Error) and RMS (Root Mean Square Error) of the entire test data are used as evaluation indicators, and the GA, PSO, DE , Zhang’s method are also compared, as shown in Table 4. The experimental results show that the algorithm in this paper converges prematurely and can improve the algorithm performance more effectively. The root mean square error of the calibration algorithm proposed in this paper is only 0.182, the calibration error result is smaller than other several algorithms, and the algorithm performance is excellent.

Table 4.

Comparison of calibration results of five algorithms (mm).

Methods	MSE	RMS
GA	0.569	0.852
PSO	0.312	0.286
DE	0.255	0.265
Zhang’s method	0.358	0.339
Our methods	0.086	0.182

It can be seen from Table 5, Our method’s calibration time less than other method.

Table 5.

Calibration times comparison of five algorithms.

Methods	Times (s)
GA	8.56
PSO	7.28
DE	5.96
Zhang’s method	4.58
Our methods	3.66

Comparison of reprojection errors

The reprojection errors of the proposed algorithm are compared with those of GA, PSO, and DE algorithms, and the results are shown in Table 5. It can be seen from Table 6 that E_X and Ey represent the reprojection error in the X and Y directions respectively. The reprojection error of our method is smaller than GA, PSO, DE, Zhang’s method.

Table 6.

Comparison of reprojection errors.

Reprojection error	GA	PSO	DE	Zhang’s method	Our method
E_x/pixel	0.06582	0.06369	0.06295	0.06216	0.05938
E_y/pixel	0.03286	0.03196	0.03085	0.03028	0.02988

Conclusions

In this paper, improved differential evolutionary particle swarm hybrid algorithm can solve the multiple parameters of traditional camera calibration algorithm, and is applied to camera calibration experiment, which can verify the practical feasibility and effectiveness of the new camera calibration algorithm. Experiments show that the camera calibration method proposed in this paper has better calibration performance compared with GA, PSO, DE, Zhang’s method and the calibration result data error iteration performance obtained by the new algorithm is better. Compared with GA, PSO, DE, and Zhang’s method, the image coordinates derived by the new algorithm are close to the actual image coordinates, and the difference values are all less than 0.10 pixels. This indicates that the calibration method proposed in this paper has good performance and has certain theoretical significance and research value. The root mean square error of the calibration algorithm proposed in this paper is only 0.182, the calibration error result is smaller than other several algorithms. The reprojection error of our method is 0.05938 (Ex/pixel) and 0.02988 (Ey/pixel). It is far smaller than GA, PSO, DE, and Zhang’s method, so the algorithm performance is excellent.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This article is supported by scientific research project of Jiangxi Department of Education.

ORCID iD

Wei Fu

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Camera calibration based on improved differential evolution particle swarm

Abstract

Keywords

Introduction

Camera calibration

Algorithm design

Particle swarm optimization algorithm

Improved differential evolution algorithm

Camera calibration algorithm

Camera calibration experiment results analysis and performance comparison

Camera calibration test

Performance comparison of algorithms

Comparison of reprojection errors

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References