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基于结构先验与协同优化的城市场景分段平面重建

王伟 任国恒 陈立勇 张效尉

罗毅平, 周笔锋. 时滞扩散性复杂网络同步保性能控制. 自动化学报, 2015, 41(1): 147-156. doi: 10.16383/j.aas.2015.c140202
引用本文: 王伟, 任国恒, 陈立勇, 张效尉. 基于结构先验与协同优化的城市场景分段平面重建. 自动化学报, 2019, 45(11): 2187-2198. doi: 10.16383/j.aas.2017.c170458
LUO Yi-Ping, ZHOU Bi-Feng. Guaranteed Cost Synchronization Control of Diffusible Complex Network Systems with Time Delay. ACTA AUTOMATICA SINICA, 2015, 41(1): 147-156. doi: 10.16383/j.aas.2015.c140202
Citation: WANG Wei, REN Guo-Heng, CHEN Li-Yong, ZHANG Xiao-Wei. Piecewise Planar Urban Scene Reconstruction Using Structure Priors and Cooperative Optimization. ACTA AUTOMATICA SINICA, 2019, 45(11): 2187-2198. doi: 10.16383/j.aas.2017.c170458

基于结构先验与协同优化的城市场景分段平面重建

doi: 10.16383/j.aas.2017.c170458
基金项目: 

模式识别国家重点实验室开放课题基金 201700004

河南省高校重点科研项目 16A5201 05

河南省高校重点科研项目 17A520019

河南省自然科学基金 162300410347

河南省高校重点科研项目 17A520018

河南省科技攻关项目 172102310727

周口师范学院校本项目 zknuC2015103

周口师范学院校本项目 zknuB2201604

详细信息
    作者简介:

    任国恒   周口师范学院讲师.2011年获得西安工业大学硕士学位.主要研究方向为大数据分析.E-mail:rengguoheng@126.com

    陈立勇  周口师范学院讲师.2010年获辽宁科技大学硕士学位.主要研究方向为大数据分析.E-mail:chenliyongup@163.com

    张效尉  周口师范学院讲师.2009年获得郑州轻工业学院硕士学位.主要研究方向为大数据分析.E-mail:xwzhang286@163.com

    通讯作者:

    王伟  周口师范学院副教授.2015年获得中国科学院自动化研究所博士学位.主要研究方向为计算机视觉与模式识别.本文通信作者.E-mail:wangwei@zknu.cn

Piecewise Planar Urban Scene Reconstruction Using Structure Priors and Cooperative Optimization

Funds: 

Open Projects Program of National Laboratory of Pattern Recognition 201700004

College Key Research Project of Henan Province 16A5201 05

College Key Research Project of Henan Province 17A520019

Natural Science Foundation of Henan Province 162300410347

College Key Research Project of Henan Province 17A520018

Key Technology Research and Development Project of Henan Province 172102310727

School-based Project of Zhoukou Normal University zknuC2015103

School-based Project of Zhoukou Normal University zknuB2201604

More Information
    Author Bio:

      Lecturer at Zhou- kou Normal University. He received his master degree from Xi'an Technological University in 2011. His main research interest is big data analysis

      Lecturer at Zhou- kou Normal University. He received his master degree from University of Science and Technology Liaoning. His main research interest is big data analysis

      Lecturer at Zhoukou Normal University. He received his master degree from Zhengzhou University of Light Industry in 2009. His main research interest is big data analysis

    Corresponding author: WANG Wei   Associate professor at Zhoukou Normal University. He received his Ph. D. degree from the Institute of Automation, Chinese Academy of Sciences in 2015. His research interest covers computer vision and pattern recognition. Corresponding author of this paper
  • 摘要: 在基于图像的城市场景三维重建中,场景分段平面重建算法可以克服场景中的弱纹理、光照变化等因素的影响而快速恢复场景完整的近似结构.然而,在初始空间点较为稀疏、候选平面集不完备、图像过分割质量较低等问题存在时,可靠性往往较低.为了解决此问题,本文根据城市场景的结构特征构造了一种新颖的融合场景结构先验、空间点可见性与颜色相似性的平面可靠性度量,然后采用图像区域与相应平面协同优化的方式对场景结构进行了推断.实验结果表明,本文算法利用稀疏空间点即可有效重建出完整的场景结构,整体上具有较高的精度与效率.
    Recommended by Associate Editor WANG Liang
  • With the increasingly complicated engineering problems during the past few years, many researchers devote themselves to researching new intelligent optimization algorithms. In 2011, a new heuristic optimization algorithm named fruit fly optimization algorithm (FOA) is proposed by Pan [1] who is inspired by the feeding behaviors of drosophila. FOA is easy to be understood, and it can deal with the optimization problems with fast speed and high accuracy, while, the results are influenced a lot by the initial solutions [2]. Based on the phototropic growth characteristics of plants, a new global optimization algorithm called plant growth simulation algorithm is proposed by Li et al., which is a kind of bionic random algorithm and suitable for large-scale, multi-modal and nonlinear integer programming [3], however, for its complex calculation theory, the algorithm is not widely applied in industry and scientific research. Artificial bee colony algorithm [4] is a new application of swarm intelligence, which simulates the social behaviors of bees, whose defects are slow convergence speed and easy to trap into local optimum [5].

    Mirror is a common necessity, which plays an important role in daily life. Inspired by the optical function of mirror, a new algorithm called specular reflection algorithm (SRA) is raised by this paper. SRA, similar to genetic algorithm [6]-[8], particle swarm optimization [9]-[11], simulated annealing algorithm [12], [13], differential evolution algorithm [14], [15], etc, can be widely used in science and engineering. The SRA has many outstanding advantages, such as simple principle, easy programming, high precision and fast calculation speed, and its unique non-population searching mode distinguishes itself from original swarm algorithm. Furthermore, the global searching ability is significantly improved by the specific acceptance criterion of the new solution. In order to verify above mentioned features of SRA, a great deal of comparative experiments are adopted in this paper. At last, the reliability based design and robust design are combined with the SRA, in order to evaluate the ability of SRA in reliability based robust optimization design.

    Mirror is a life necessity and a product of human civilization, which can change the direction of propagation of light. There are various kinds of mirrors, such as magnifying glass, microscope, etc. With the help of mirror, a great deal of stuff can be observed, even if they are out of the range of visibility. For example, the submarine soldier is able to catch sight of the object above the water by periscope. This reflection property of mirror is simulated by the SRA.

    Object, suspected target, eyes and mirror are the four basic elements of specular reflection system.

    Object is the objective function of optimization. Getting its exact coordinate is the purpose of the SRA. It is not involved in the optimization procedure for the location of the object is unpredictable.

    Suspected target is the coordinate of the object observed by eyes, which is approximate to the optimal solution. There is an error between the suspected target and object, because the coordinate of the object observed by eyes is not accurate. The suspected target is located around the object, and it is the element nearest to the object.

    Mirror can change the direction of propagation of light. The vision of eyes can be broaden by mirror. All the things that can reflect light (glass, water, etc.) are taken as mirror.

    Eyes are the subject of the SRA, which can acquire the approximate coordinate of the object. And it is the element farthest from the object.

    $ \begin{align}\label{eq1} &\min f(X), \ X = (x^1, x^2, \ldots, x^N), \quad X \in \mathbb{R}^N \notag\\ & {\rm s.t.}\ \ g_j (x) = 0, \ \ j = 1, 2, \ldots, m \notag\\ &\qquad h_k (x) \le 0, \ \ k = 1, 2, \ldots, l. \end{align} $

    (1)

    Taking the constrained optimization problem showed in (1) as an example, the definition of SRA will be drawn as following:

    Set the specular reflection system as a $ 4\times N$ dimensional Euclidean space, where $N$ is the number of design variables. The elements in the system are defined as $X_i$ , $x_i^N)$ , $i = (0, 1, 2, 3)$ , and , $X_{\rm Suspect} = X_1$ , $X_{\rm Mirror}$ $=$ $X_2$ , . Where $x_i^n$ $(n=1, 2, \ldots, N)$ is the position of the $i$ th variable in the $N$ dimensional space. The four elements of SRA can be defined as $f(X_i)$ , and the relationship among the four elements is $f(X_0)\leq f(X_1) \leq$ $f(X_2)$ $\leq$ $f(X_3)$ .

    Searching the new coordinate: the coordinates of $X_{\rm New1}$ and $X_{\rm New2}$ can be acquired by (2), and the new coordinate of $X_{\rm New}$ can be got by (2).

    $ \begin{align} \begin{cases} X_{\rm New1}^n = x_1^n + \xi (2{\rm rand} - 1)(x_1^n - x_3^n ) \\[2mm] X_{\rm New2}^n = x_1^n + \xi (2{\rm rand} - 1)(2x_1^n - x_2^n - x_3^n ) \end{cases} \end{align} $

    (2)

    where $\xi$ is coefficient, which is determined by (11).

    $ \begin{align} \label{eq3} \begin{cases} X_{\rm New} = X_{\rm New1}, f(X_{\rm New1} ) \leq f(X_{\rm New2} ) \\[2mm] X_{\rm New} = X_{\rm New2}, f(X_{\rm New1} ) \ge f(X_{\rm New2} ). \end{cases} \end{align} $

    (3)

    Updating the specular reflection system: Once the coordinate of $X_{\rm New}$ is acquired, the eyes will change its place to continue searching for the "object", the four elements of the system are $X_0 X_1 X_2$ and $X_{\rm New}$ under the current situation. The specular reflection system will be adjusted by the modification of the four elements, the system will be changed by the rules shown in Fig. 1.

    图 1  Coordinate update of the specular reflection system.
    Fig. 1  Coordinate update of the specular reflection system.

    The optimization steps of the SRA are shown as follows:

    Step 1: Define the initial value $X_i$ , $i = 0, 1, 2, 3$ , and the maximum iteration number $Iter_{\max}$ .

    Step 2: If the precision or the maximum iteration number reaches the design requirements, the coordinate of $X_{\rm Object}$ will be output which is the optimum solution. Otherwise, execute the next step continually.

    Step 3: Search the coordinate of $X_{\rm New}$ by (2) and (3), the new iteration process will begin, then go back to Step 2 and Continue to calculate.

    In conclusion, the optimization flow chart of the SRA is given by Fig. 2.

    图 2  Optimization flow chart of the SRA.
    Fig. 2  Optimization flow chart of the SRA.

    Theorem 1: The constraint optimization problem presented in (1) can converge to the global extremum with 100 % probability by the SRA.

    Proof: Provided that $X_{\rm Object} = \min f(X)$ , $X\in D$ which is the global optimal solution, where $f(X_{\rm Object})$ is the optimal value of objective function, $D$ is the feasible region and $D=\{X|g_j (X_{\rm Object}) = 0$ , $j = 1, 2, \ldots, m$ ; , $k$ $=$ $1$ , $2$ , $\ldots$ , $l$ ; and $D\in \mathbb{R}^N$ .

    First, get the feasible initial solutions $X_{\rm Suspect}^0$ , $X_{\rm Mirror}^0$ and $X_{\rm Eyes}^0$ randomly among the searching space, where $X_{\rm Suspect}^0$ , $X_{\rm Mirror}^0$ , , and the corresponding values of objective function $f (X_{\rm Suspect}^0)$ , $f(X_{\rm Mirror}^0)$ and $f(X_{\rm Eyes}^0)$ can be worked out, where $f(X_{\rm Mirror}^0)$ $\leq$ $f(X_{\rm Eyes}^0)$ .

    Second, the new solutions $f(X_{\rm Suspect}^k)$ , and $f(X_{\rm Eyes}^k)$ can be acquired according to the new specular reflection system, where are the randomly produced solutions which are uniformly distributed in , $X_{\rm Suspect}^k$ is the solution of the $k$ th ( iteration, $X_{\min}^k$ and $X_{\max}^k$ are the boundaries of design variable in the current iteration, and the maximum iteration number $Iter_{\max}$ should be big enough. Therefore, under the uniform distribution, the probability of generating the feasible solutions is:

    $ \begin{align} p^k =&\ \int\nolimits_{X_{\rm Object} - \varepsilon }^{X_{\rm Object} + \varepsilon } \frac{1}{X_{\max }^k - X_{\min }^k }dX = \frac{2\varepsilon }{X_{\max }^k - X_{\min }^k } \nonumber\\[2mm] \ge&\ \frac{2\varepsilon }{X_{\max } - X_{\min } } > 0 \end{align} $

    (4)

    where $\varepsilon$ is a real number which is sufficiently small; $X_{\rm max}$ and $X_{\rm min}$ are the extreme values of the 4 $\times N$ dimensional Euclidean space.

    The probability that the feasible solution $X_{\rm Suspect}^0$ is optimal is $P^1$ , and the probability that $X_{\rm Suspect}^0$ is not optimal is $Q^1$ , both $P^1$ and $Q^1$ are expressed as follows:

    $ \begin{align} \begin{cases} P^1 = P\{X_{\rm Suspect}^0 \subseteq [X_{\rm Object}-\varepsilon, X_{\rm Object} + \varepsilon]\} \\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\mbox{ }P^1 = P \\ Q^1 = P\{X_{\rm Suspect}^0 \not\subset [X_{\rm Object}-\varepsilon, X_{\rm Object} + \varepsilon]\} \\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\mbox{ }Q^1 = P \end{cases} \end{align} $

    (5)

    where $X_{\rm Suspect}^0$ is the feasible solution gotten for the first time.

    The probability that the feasible solution gotten for the second time still failing to be the optimal value is:

    $ \begin{align} Q^2=Q^1(1-P)=(1-P)^2. \end{align} $

    (6)

    So, the probability that the solution is optimal is:

    $ \begin{align} P^2=1-(1-P)^2. \end{align} $

    (7)

    After $n$ times iteration, the probability of getting the optimum solution can be acquired by the following inference.

    $ \begin{align}\label{2} P^n& = 1 - (1 - P)^n = 1 - \prod _{i = 1}^n \left( {1 - \frac{2\varepsilon }{X_{\max }^i - X_{\min }^i }} \right) \nonumber\\[1mm] &\ge 1 - \left( {1 - \frac{2\varepsilon }{X_{\max } - X{ }_{\min }}} \right) ^n. \end{align} $

    (8)

    Calculate the extreme value of (8):

    $ \begin{align} \lim _{n \to \infty } P^n& = \lim\limits_{n \to \infty } \left[{1- \prod _{i = 1}^n \left( {1-\frac{2\varepsilon }{X_{\max }^i- X_{\min }^i }} \right)} \right] \nonumber\\ &\ge \lim _{n \to \infty } \left[{1-\left( {1- \frac{2\varepsilon }{X_{\max }-X_{\min } }} \right)^n} \right] = 1. \end{align} $

    (9)

    With the iterations going on, it is more and more likely to achieve the optimum solution. When $n\rightarrow \infty$ , , it indicates that the searching process of SRA can converge to the global extreme with 100 % probability.

    The control parameter is closely related to the space complexity of optimized target, which has an effect on the capability of algorithm. The control parameters of classical optimization algorithm are gotten by experience or experiment, such as the learning parameter $c_1 = c_2$ = 2 by PSO [16], [17], and the crossover probability and mutation probability of GA [18]. It is impossible that the control parameter acquired by experience is suitable for all optimization problems. The SRA only has the control parameter $\xi$ , whose value will have a prominent effect on SRA. In this section, a classical test function is used to confirm the most appropriate value of $\xi$ , and the results are listed in Table Ⅰ.

    表 Ⅰ  JUDGEMENT OF $\xi$
    Table Ⅰ  JUDGEMENT OF $\xi$
    Value of $\xi$ N =2 N =10 N=20 N=50 N = 100
    Optimal solution (10-6) Iteration times Optimal solution (10-6) Iteration times Optimal solution (10-6) Optimal solution (103) Optimal solution (10-6) Optimal solution (103) Optimal solution (10-6) Optimal solution (104)
    0.4 4.7776 402.70 7.1895 1103 7.2883 2.0369 8.9324 6.1015 9.5844 1.4945
    0.5 3.3845 341.04 6.4267 940.12 7.3111 1.8001 8.4771 5.3149 9.6383 1.2586
    0.6 3.9884 737.76 5.4844 936.46 7.2327 1.6802 9.0155 4.8292 9.3691 1.1344
    0.7 3.5625 515.18 6.9587 810.24 7.2858 1.5971 8.5419 4.4544 9.5971 1.0679
    0.8 4.2770 509.46 6.7379 747.90 7.5046 1.4992 8.8811 4.2697 9.3384 1.0741
    0.9 4.0589 259.08 6.3850 732.90 7.4304 1.4562 8.3421 4.2036 9.4009 1.0976
    1.0 4.9287 193.26 5.9257 694.18 6.8977 1.3677 8.3414 4.3603 9.5947 1.1404
    1.1 4.6702 142.60 6.1496 674.28 8.0852 1.2946 9.4538 4.2854 9.4944 1.1889
    1.2 4.6250 142.42 5.8875 626.54 7.7654 1.3608 8.6969 4.4775 9.6771 1.2434
    1.3 5.1501 139.08 6.5208 654.72 7.2172 1.4050 8.9588 4.5342 9.5792 1.3215
    1.4 5.4409 131.02 5.6072 695.40 6.9556 1.4699 8.9053 4.6930 9.6898 1.3675
    1.5 4.7099 103.72 5.7050 675.02 7.6612 1.4740 9.0472 4.8329 9.5134 1.4173
    1.6 4.7625 93.82 5.8038 713.20 6.4546 1470 8.9756 4.8634 9.7748 1.4768
    1.7 4.9327 91.94 4.9871 783.90 5.8034 1.6036 9.1825 5.0851 9.6612 1.4985
    1.8 5.9076 87.32 5.4104 856.30 7.1143 1.6917 8.7372 5.3202 9.4446 1.5536
    1.9 4.9402 82.44 5.5724 832.12 6.4092 1.8641 8.7754 5.5962 9.6617 1.6423
    2.0 4.7168 89.08 4.8307 998.300 5.7508 2.0544 8.1780 6.3700 9.4975 2.5117
    下载: 导出CSV 
    | 显示表格

    $ \begin{align} f (x_1, x_2, \ldots, x_N)=\sum\limits_{j=1}^N j\times x_j^2. \end{align} $

    (10)

    The test function is illustrated by (10), and its three-dimension diagram is shown in Fig. 3. The global minimum value in theory of this function is 0 $(0, 0, \ldots, 0)$ and the constraint condition is $-5.12\leq x_j\leq 5.12$ , $j=1, 2, \ldots, N$ . In consideration of $N = (2, 10, 20, 50, 100, 500)$ and $\xi = (0.4$ , $0.5$ , $\ldots$ , $2.0)$ , do the calculation 50 times using every possible combination of $N$ and $\xi$ , then put the average results in Table Ⅰ. Assume that the convergence condition is $Iter_{\max}$ $=$ $10^5$ or $f (x_1, x_2, \ldots, x_N )\leq 10^{-5}$ .

    图 3  Three-dimensional surface of test function.
    Fig. 3  Three-dimensional surface of test function.

    As shown in Table Ⅰ, all the results fall in between 10 $^{-5}$ and 10 $^{-6}$ , the optimization efficiency which is influenced by $\xi$ cannot be evaluated by the optimal solutions, therefore, iteration times is the only factor to be considered.

    According to the Table Ⅰ, the conclusions can be drawn as follows: when $N = 2$ and $\xi = 1.9$ , the efficiency of the optimization is highest, the corresponding iteration is 82.44; When $N = 10$ , , $N = 50$ and $N = 100$ , the best $\xi$ and its corresponding iteration times are 1.3 and 654.72, 1.1 and $1.2946\times 10^3$ , 0.9 and $4.2036\times 10^3$ , 0.7 and $1.0679$ $\times$ $10^4$ , respectively. In addition, the value of $\xi$ will be reduced gradually with the increasing of $N$ , and the relationship between $\xi$ and $N$ (as shown in (11)) can be speculated by the method of data fitting.

    $ \begin{align} \xi=\frac{2.15}{N}+0.84. \end{align} $

    (11)

    To verify the global optimization ability of SRA, four numerical test functions in [10] are used, each test function is listed in Table Ⅱ in detail. The total iteration time is set as 2000. The SRA will be executed 50 times, and the average values are listed in Table Ⅲ, other results are references from [10], Figs. 4-7 show the iteration curves of the objective functions of each test function respectively.

    表 Ⅱ  NUMERICAL CALCULATION FUNCTION
    Table Ⅱ  NUMERICAL CALCULATION FUNCTION
    Name Expression Interval of convergence Global extreme Dimension
    Sphere $f_1 = \sum\limits_{i = 1}^n {x_i^2 }$ $x_i\in [-50,50]$ 0 (0, 0, $\ldots$ , 0) $n$ = 30 100
    Griewank $f_2 = 1 + \sum\limits_{i = 1}^n {\left( {\frac{x_i^2 }{4000}} \right) -\prod\limits_{i = 1}^n {\cos \left( {\frac{x_i }{\sqrt i }} \right)} }$ $x_i\in [-600,600]$ 0 (0, 0, $\ldots$ , 0) $n$ = 30 100
    Rosenbrock $f_3 = \sum\limits_{i = 1}^{n - 1} {[100(x_{i + 1}-x_i^2 )^2 + (x_i-1)^2]}$ $x_i\in [-100,100]$ 0 (1, 1, $\ldots$ , 1) $n$ = 30 100
    Restrigin $f_4 = \sum\limits_{i = 1}^n {[10 + x_i^2-10\cos (2\pi x_i )]}$ $x_i\in [-5.0, 5.0]$ 0 (0, 0, $\ldots$ , 0) $n$ = 30 100
    下载: 导出CSV 
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    表 Ⅲ  CALCULATION RESULTS OF TEST FUNCTION
    Table Ⅲ  CALCULATION RESULTS OF TEST FUNCTION
    Name PSO
    (n = 30) [10]
    Kalman swarm
    (n = 30) [10]
    Chaos ant colony optimization
    (n = 30)[10]
    Chaos PSO
    (n = 30) [10]
    New chaos PSO
    (n = 30) [10]
    SRA
    (n = 30)
    SRA
    (n = 100)
    Sphere 3.7004×102 4.723 3.815×10-1 2.4736×10-3 2.0729×10-9 1.1080×10-24 2.3160×10-12
    Griewank 2.61×107 3.28×103 23.414 6.8481×10-2 9.9051×10-11 4.6629×10-15 2.7978×10-14
    Rosenbrock 13.865 9.96×10-1 4.669×10-1 1.0404×10-2 2.9068×10-4 9.8730×10-7 6.1173×10-5
    Restrigin 1.0655×102 53.293 22.6361 9.5258×10-1 4.3741×10-4 3.9373×10-21 8.7727×10-7
    下载: 导出CSV 
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    图 4  Iteration curve of sphere.
    Fig. 4  Iteration curve of sphere.
    图 5  Iteration curve of griewank.
    Fig. 5  Iteration curve of griewank.
    图 6  Iteration curve of rosenbrock.
    Fig. 6  Iteration curve of rosenbrock.
    图 7  Iteration curve of restrigin.
    Fig. 7  Iteration curve of restrigin.

    The results in Table Ⅳ indicate that: when $n$ = 30, the results of the four test functions calculated by SRA are , $4.6629 \times 10^{-15}$ , $9.8730 \times 10^{-7}$ and $3.9373$ $\times$ $10^{-21}$ respectively, which are , $2.12 \times 10 ^4$ , $2.90$ $\times$ $10^2$ , times higher than the results gotten by new chaos PSO algorithm which possesses the highest accuracy in [10]; When , the results of the four test functions calculated by SRA are $2.3160\times 10^{-12}$ , $2.7978$ $\times$ $10^{-14}$ , $6.1173\times10^{-5}$ , $8.7727\times 10^{-7}$ respectively, and the computational accuracy are still $8.95\times 10^2$ , , $4.75$ , $4.99$ $\times$ $10^2$ times higher than the results calculated by new chaos PSO algorithm. All in all, the SRA is an efficient optimization algorithm.

    表 Ⅳ  CALCULATION RESULTS
    Table Ⅳ  CALCULATION RESULTS
    Design method Design variables (mm) Objective function (mm2) Reliability Sensitivity of reliability/(10-3)
    x1 x2 x3 x4 x5 A Rv $\frac{\partial R_v}{\partial S}$ $\frac{\partial R_v}{\partial E}$ $\frac{\partial R_v}{\partial \rho}$
    SRA Optimization 6 6 205 635 257 10 704 0.5071 14.3985 0.0017 9.15×10-9 0.0011
    Reliability Optimization 6 6 258 632 310 11 304 0.9968 13.0816 0.0015 8.77×10-9 0.0010
    Robust Reliability Optimization 6 6 324 595 376 11 652 0.9813 12.6270 0.0015 9.23×10-9 0.0010
    PSO Optimization 10 6 185 567 619 11 544 0.5314 13.0714 0.0015 9.8×10-9 0.0010
    Reliability Optimization 7 7 222 605 276 12 334 0.9806 12.9119 0.0015 9.73×10-9 0.0011
    Robust Reliability Optimization 9 6 302 534 354 12 780 0.9810 11.5262 0.0013 1.01×10-8 0.0010
    FOA Optimization 9 6 190 581 633 11 328 0.5132 13.3500 0.0016 9.64×10-9 0.0010
    Reliability Optimization 6 6 491 532 543 13 068 0.9802 11.3697 0.0013 1.01×10-8 9.95×10-4
    Robust Reliability Optimization 8 11 237 536 299 16 576 1.0 11.6479 0.0013 1.27×10-9 0.0013
    Note: The index of reliability R0 = 0.98 is deflned.
    下载: 导出CSV 
    | 显示表格

    According to the reliability design theory, the reliability can be calculated by (12):

    $ \begin{align} R=\int_{g(X)} f_x(X) dX \end{align} $

    (12)

    where $f_x (X)$ is the joint probability density of basic random variables $X=(X_1, X_2, \ldots, X_n)^T$ , which shows the state of the components.

    $ \begin{align} \begin{cases} g(X)\leq 0, &{\rm failure}\\[2mm] g(X)>0, &{\rm safe.} \end{cases} \end{align} $

    (13)

    The basic random variables $X_i$ ( $i = 1, 2, \ldots, n$ ) are independent of each other and follow certain distribution. The reliability index $\beta$ and the reliability $R=\Phi(\cdot)$ can be calculated by Monte Carlo method [19], where $\Phi(\cdot)$ is the standard normal distribution function.

    Robust design is a modern design technique that can improve the efficiency and quality and reduce the cost of products [20], [21]. The robust design of mechanical products can make the products insensitive to the changes of design parameters. The product which is designed by robust design method has the characteristic of stability. Even if there is an error in the designed parameters, the product still has excellent performance. Reliability is a kind of design method to eliminate the weaknesses, failure modes and guard against malfunction. The reliability robust optimization design is a new method by combining the robust design and reliability design, which possess all the merits of the two methods. The products designed by the reliability robust optimization design method are reliable and have robustness.

    $ \begin{align} &\min f(X)=\omega_1 f_1(X)+\omega_2 f_2(X)\notag\\ & {\rm s.t.} \ \ R\geq R_0\notag\\ &\qquad p_i(X)\geq 0, \ i=1, 2, \ldots, l\notag\\ &\qquad q_j(X)\geq 0, \ j=1, 2, \ldots, m \end{align} $

    (14)

    where $f_1(X)$ and $f_2(X)$ are the objective functions of the Reliability Robust Optimization design, $f_1(X)=R$ and $f_2(X)$ is the design criterion related to robust design which can be acquired by (15); $R$ is the reliability; $R_0$ is the constraint condition of reliability; $p_i$ and $q_j$ are equality and inequality constraints of the robust reliability optimization design respectively.

    $ \begin{align} f_2 (X) = \sqrt {\sum\limits_{i = 1}^n \left( {\frac{\partial R}{\partial X_i }} \right)^2} \end{align} $

    (15)

    where $\omega_1$ and $\omega_2$ are weighting coefficients, which are related to the importance of $f_1(X)$ and $f_2(X)$ , both of them are calculated by (16), and $\omega_1+\omega_2 = 1$ .

    $ \begin{align} \begin{cases} \omega _1 = \dfrac{f_2 (X^{1\ast }) - f_2 (X^{2\ast })}{[f_1 (X^{2\ast })- f_1 (X^{1\ast })] + [f_2 (X^{1\ast })-f_2 (X^{2\ast })]} \\[4mm] \omega _2 = \dfrac{f_1 (X^{2\ast }) - f_1 (X^{1\ast })}{[f_1 (X^{2\ast })- f_1 (X^{1\ast })] + [f_2 (X^{1\ast })-f_2 (X^{2\ast })]} \end{cases} \end{align} $

    (16)

    where $X^{1*}$ and $X^{2*}$ are the best values when $\min f(X)$ $=$ $f_1 (X)$ and $\min f(X)=f_2 (X)$ respectively.

    The bridge crane is taken as an example to verify the capability of the SRA in solving the engineering problems. The SRA is adopted to design the structure with optimized design, reliability optimization design and robust reliability optimization design, and the results are listed in Table Ⅲ together with the results calculated by PSO and FOA, which are used for analysing the performance of the SRA.

    The mechanical model of the bridge crane is shown in Fig. 8, the uniform load $q$ and the concentrated load $F$ are exerted on the girder, where $q$ is caused by the structure deadweight and $F$ is related to the weight of the hoisted cargo.

    图 8  Mechanical model diagram and sectional dimension.
    Fig. 8  Mechanical model diagram and sectional dimension.

    The parameters $x_i$ $(i = 1, 2, 3, 4, 5)$ are considered to be the design variables, where $6\leq x_1$ , $x_2\leq 30$ , $50\leq x_3$ , $x_4$ $\leq$ $5000$ , $x_5 = x_3 + 2x_2 + 40$ . The parameter $S$ is the span of the bridge crane. Other parameters include the elasticity modulus $E$ , the material density $\rho$ , $q$ $=$ $g(x_1$ , $x_2$ , $x_3$ , $x_4, x_5)$ . The parameters $S$ , $F$ , $E$ and $\rho$ are independent of each other, and they are normal random variables, $S$ $\sim$ ${\rm N}(12, 0.08^2)$ , $F$ $\sim$ , $E$ $\sim$ ${\rm N}(206 000$ , $6180^2)$ , $\rho\sim {\rm N}(7850, 5.6^2)$ .

    Objective function: According to the characteristics of the structural optimization problem, the objective function can be defined as shown in (17).

    $ \begin{align} {\rm min} f(x_1, x_2, x_3, x_4, x_5)=2x_1x_5+2x_2x_4. \end{align} $

    (17)

    Constraint condition: Strength, stiffness and stability are the three basic failure modes of bridge crane. Therefore, the constraint condition can be defined as following:

    1) Strength Constraint: The maximum stress of dangerous point in mid-span section must be smaller than the ultimate stress $f_{rd}$ ;

    $ \begin{align} &h_1(x_1, x_2, x_3, x_4, x_5)=f_{rd}-\sigma\notag\\ &\qquad =f_{rd}-\frac{qS^2+2FS}{8I_Z}\left(\frac{x_4}{2}+x_1\right) \end{align} $

    (18)

    where $f_{rd}$ is determined by the limit state method, and $f_{rd}$ $=$ ${f_{yk}}/{\gamma_m} = {235}/{1.1}=213.64$ MPa, $f_{(yk)}$ = 235 is yield stress, $\gamma_m$ = 1.1 is the resistance coefficient, $I_Z$ is moment of inertia of Section 2.1, $q$ and $I_Z$ are the functions related to design variables $x_i$ ( $i$ = 1, 2, 3, 4, 5).

    2) Stiffness Constraint: The maximum deflection of the structure must be smaller than the allowable value $\gamma_0$ $=$ $S/400$ .

    $ \begin{align} &h_2(x_1, x_2, x_3, x_4, x_5)=\gamma_0-\gamma\notag\\ &\qquad =\gamma_0-\left(\frac{5qS^4}{384EI_Z}+\frac{FS^3}{48EI_Z}\right). \end{align} $

    (19)

    3) Stability Constraint: The depth-width ratio of Section 2.1 must be smaller than 3.

    $ \begin{align} h_3(x_1, x_2, x_3, x_4, x_5)=3-\frac{x_4+2x_1}{x_3+2x_2}. \end{align} $

    (20)

    In conclusion, the optimization model of the bridge crane can be built as (21).

    $ \begin{align} & \min f(x_1, x_2, x_3, x_4, x_5) \notag\\ & {\rm s.t.} \ \ h_k(x_1, x_2, x_3, x_4, x_5)\geq 0, \quad k=1, 2, 3\notag \\ &\qquad 6\leq x_1, \ x_2\leq 30\notag \\ &\qquad 50\leq x_3, \ x_4\leq 5000. \end{align} $

    (21)

    The reliability constraint of structure is added to (21) to achieve the reliability optimization design. The failure of any mode will result in the failure of the structure, so the reliability $R_v$ is defined by (22). The reliability optimization model of bridge crane can be established by (23).

    $ \begin{align} R_v=\prod\limits_{k=1}^3 R_k %(h_k\geq 0) \end{align} $

    (22)

    where $R_k$ , $k=1, 2, 3$ is the probability of the $k$ th failure mode.

    $ \begin{align} & \min f(x_1, x_2, x_3, x_4, x_5)\notag \\ & {\rm s.t.}\ \ h_k(x_1, x_2, x_3, x_4, x_5)\geq 0, \quad k=1, 2, 3\notag\\ &\qquad 6\leq x_1, \ x_2\leq 30\notag\\ &\qquad 50\leq x_3, \ x_4\leq 5000\notag\\ &\qquad R_v-R_0\geq 0. \end{align} $

    (23)

    According to the robust reliability optimization design model which is shown in (14), the index of reliability and robustness are taken into account, the multi-objective optimization model is built by (24).

    $ \begin{align} & \min \omega_1\times f(x_1, x_2, x_3, x_4, x_5)+w_2\times f'(x)\notag \\ & {\rm s.t.} \ \ h_k(x_1, x_2, x_3, x_4, x_5)\geq 0, \quad k=1, 2, 3\notag\\ &\qquad 6\leq x_1, \ x_2\leq 30\notag\\ &\qquad 50\leq x_3, \ x_4\leq 5000\notag\\ &\qquad R_v-R_0\geq 0 \end{align} $

    (24)

    where .

    The three optimization models shown in (21), (23) and (24) are calculated by the SRA, PSO and FOA, respectively. And the results are presented in Table Ⅲ, from which the conclusions can be drawn as follows:

    1) For structural optimization, the results obtained by the three algorithms are 10 704, 11 544 and 11 328, the optimum among the three is 10 704 which is calculated by the SRA, which proves the ability of SRA is higher than PSO and FOA. The reliability results of the three groups of parameters are 0.5071, 0.5314 and 0.5132 respectively, which are unable to meet the requirement of reliability design for the reliability constraint is ignored.

    2) The reliability of the structure can be ensured and the robustness can be improved after reliability optimization design. However, the areas of Section 2.1 are increased to 11 652, 12 334 and 16 576 at the same time, and the best result is also calculated by SRA.

    3) With the requirements of the robustness, the reliability sensitivity index of design variables are significantly reduced, and the robustness of structure is improved notably.

    In this paper, a new optimization algorithm — specular reflection algorithm (SRA) is proposed, which is inspired by the optical property of the mirror. The SRA has a particular searching strategy which is different from the swarm intelligence optimization algorithms. The convergence ability of the SRA is verified by the traditional mathematical method, it converges to the global optimum value with the probability of 100 %. The reasonable values of the control parameters are analysed, and their computational formula is deduced by the method of data fitting, so that the control parameters will vary with the different problems and thus the adaptation and the operability of the SRA will be improved. Four classical numerical test functions are analysed by the SRA, and the results indicate that the ability of the SRA is better than the traditional intelligent optimization algorithms. Then, the theories of the reliability optimization and robust design are combined to establish the mathematical models of the optimization design, reliability optimization design and robust reliability optimization design for the bridge crane as an example system, which are calculated by the SRA and other two optimization methods (PSO and FOA). The conclusions are drawn after the simulation, that the structure designed by the SRA is reliable and robust. The results calculated by the SRA are superior to the PSO and the FOA. All in all, the SRA is the latest research in the area of intelligent optimization, which has the better calculation capability than other optimization algorithms, and the ability for the structure design is verified in this paper. SRA can be widely applied in other fields and create more value.


  • 本文责任编委 王亮
  • 图  1  本文算法流程图

    Fig.  1  Flowchart of our proposed method

    图  2  超像素优化

    Fig.  2  Superpixel optimization

    图  3  初始可靠平面与候选平面

    Fig.  3  Initial reliable planes and candidate planes

    图  4  融合平面夹角先验的平面推断

    Fig.  4  Plane inference based on angle priors

    图  5  图像区域与相应平面的协同优化(不同颜色表示不同的可靠平面)

    Fig.  5  Cooperative optimization of image regions and their related planes (different colors denote different reliable planes)

    图  6  示例图像及初始空间点的投影

    Fig.  6  Sample images and initial 3D points

    图  7  标准数据集重建结果

    Fig.  7  Results on standard data sets

    图  8  标准数据集算法对比(不同等级灰度标记的区域表示不同平面)

    Fig.  8  Results produced by different methods (different regions labeled with different gray levels denote different planes)

    图  9  实拍数据集重建结果

    Fig.  9  Results on real-world datasets

    图  10  实拍数据集算法对比(不同颜色表示不同平面)

    Fig.  10  Results produced by different methods (different colors denote different planes)

    图  11  图 10 (b)~10 (d)中矩形区域内平面结构的放大显示

    Fig.  11  Close-ups of the plane structures in the rectangles in Fig. 10 (b)~10 (d)

    表  1  参数设置

    Table  1  Parameters setting

    参数 默认值 功能描述
    $\gamma$ 0.6 相关性度量
    ${\lambda_{\rm occ}}$ 2 遮挡惩罚量
    ${\lambda_{\rm err}}$ 4 空间可见性冲突惩罚量
    ${\lambda_{\rm dis}}$ 4 空间平面间断惩罚量
    $\mu$ 0.6 场景结构先验松驰量
    $\delta$ 0.5 颜色特征差异截断阈值
    $\vartheta$ 0.9 天空区域语义阈值
    下载: 导出CSV

    表  2  初始化

    Table  2  Initialization

    数据集 空间点 超像素 线段 平面
    Valbonne 561 360 362 17
    Wadham 2 120 1 243 838 38
    City#1 2 234 2 793 1 588 11
    City#2 1 503 2 643 1 297 7
    下载: 导出CSV

    表  3  不同算法获取的结果

    Table  3  Results produced by different methods

    数据集 PSP SP CP 本文算法 文献[8]算法 文献[9]算法
    $M_{\rm 1(Fir)}$ $M_{\rm 1(Fin)}$ $M_2$ $M_1$ $M_2$ $M_1$ $M_2$
    Valbonne 21 1 478 147 0.5259 0.7748 9 0.5145 7 0.6631 7
    Wadham 53 5 889 421 0.6643 0.8046 11 0.3879 7 0.6492 11
    City#1 23 7 110 3 109 0.4608 0.6927 7 0.3390 7 0.4465 6
    City#2 28 6 831 2 612 0.5355 0.7081 6 0.3217 5 0.5977 6
    注: PSP表示已分配初始可靠平面超像素数量, SP与CP分别表示协同优化后超像素与相应平面数量
    下载: 导出CSV

    表  4  不同算法的计算时间(s)

    Table  4  Computation time of different methods (s)

    数据集 本文算法 文献[8]算法 文献[9]算法
    超像素 初始平面 线段检测 结构推断 合计
    Valbonne 1.1 4.9 2.5 21.0 29.5 38.1 221.9
    Wadham 2.4 7.6 4.1 34.8 48.9 73.7 360.3
    City#1 3.7 8.7 10.7 58.5 81.6 92.5 554.9
    City#2 4.2 5.5 9.7 66.4 85.8 97.8 469.4
    下载: 导出CSV
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    • 收稿日期:  2017-08-17
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