混沌SSA优化多重熵阈值的骨料图像自动分割

王梦菲; 王卫星; 李理敏

doi:10.37188/OPE.20233113.1973

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混沌SSA优化多重熵阈值的骨料图像自动分割

信息科学 | 更新时间：2023-07-17

- 混沌SSA优化多重熵阈值的骨料图像自动分割
- Automatic segmentation of aggregate images with MET optimized by chaos SSA
- 光学精密工程
- 作者机构：
  
  1.长安大学信息学院，陕西西安 710064
  2.温州大学电气与电子工程学院，浙江温州 325035
- 作者简介：
  
  [ "王梦菲（1996-），女，安徽人，博士研究生，主要从事数字图像处理和计算机视觉领域的研究。E-mail： m.f.wang@foxmail.com" ]
  
  王梦菲（1996-），女，安徽人，博士研究生，主要从事数字图像处理和计算机视觉领域的研究。E-mail： m.f.wang@foxmail.com
  [ "王卫星（1959-），男，黑龙江人，教授，博士生导师，1997年于瑞典皇家工学院获得博士学位，主要从事图像处理、模式识别和机器视觉等领域的研究。E-mail： wxwang@chd.edu.cn" ]
  
  王卫星（1959-），男，黑龙江人，教授，博士生导师，1997年于瑞典皇家工学院获得博士学位，主要从事图像处理、模式识别和机器视觉等领域的研究。E-mail： wxwang@chd.edu.cn
  [ "李理敏（1984-），男，浙江人，博士，副教授，2006年于浙江大学获得学士学位，2012年于中科院上海微系统所获得博士学位，主要从事信号检测、图像处理等方面的研究。E-mail： lilimin@wzu.edu.cn" ]
  
  李理敏（1984-），男，浙江人，博士，副教授，2006年于浙江大学获得学士学位，2012年于中科院上海微系统所获得博士学位，主要从事信号检测、图像处理等方面的研究。E-mail： lilimin@wzu.edu.cn
- 基金信息：
  
  国家自然科学基金资助项目(61170147);国家自然科学基金资助项目(U1401252);浙江省教育部科研项目(Y202146796);浙江省自然科学基金资助项目(LTY22F020003);温州中国重大科技创新项目(ZG2021029)
- DOI：10.37188/OPE.20233113.1973
  中图分类号： TP394.1;TH691.9
- 纸质出版日期：2023-07-10，
  
  收稿日期：2022-08-30，
  
  修回日期：2022-10-11，
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引用本文
王梦菲,王卫星,李理敏.混沌SSA优化多重熵阈值的骨料图像自动分割[J].光学精密工程,2023,31(13):1973-1987.

WANG Mengfei,WANG Weixing,LI Limin.Automatic segmentation of aggregate images with MET optimized by chaos SSA[J].Optics and Precision Engineering,2023,31(13):1973-1987.
王梦菲,王卫星,李理敏.混沌SSA优化多重熵阈值的骨料图像自动分割[J].光学精密工程,2023,31(13):1973-1987. DOI： 10.37188/OPE.20233113.1973.

WANG Mengfei,WANG Weixing,LI Limin.Automatic segmentation of aggregate images with MET optimized by chaos SSA[J].Optics and Precision Engineering,2023,31(13):1973-1987. DOI： 10.37188/OPE.20233113.1973.

混沌SSA优化多重熵阈值的骨料图像自动分割 Automatic segmentation of aggregate images with MET optimized by chaos SSA 1 first-author 王梦菲（1996-），女，安徽人，博士研究生，主要从事数字图像处理和计算机视觉领域的研究。E-mail： m.f.wang@foxmail.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716175&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716178&type= 22.01332855 32.00399780 王梦菲（1996-），女，安徽人，博士研究生，主要从事数字图像处理和计算机视觉领域的研究。E-mail： m.f.wang@foxmail.com m.f.wang@foxmail.com 1 corresp E-mail： wxwang@chd.edu.cn， lilimin@wzu.edu.cn E-mail： wxwang@chd.edu.cn， lilimin@wzu.edu.cn 王卫星（1959-），男，黑龙江人，教授，博士生导师，1997年于瑞典皇家工学院获得博士学位，主要从事图像处理、模式识别和机器视觉等领域的研究。E-mail： wxwang@chd.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716188&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716184&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716182&type= 22.01332855 32.00399780 王卫星（1959-），男，黑龙江人，教授，博士生导师，1997年于瑞典皇家工学院获得博士学位，主要从事图像处理、模式识别和机器视觉等领域的研究。E-mail： wxwang@chd.edu.cn wxwang@chd.edu.cn 2 corresp E-mail： wxwang@chd.edu.cn， lilimin@wzu.edu.cn E-mail： wxwang@chd.edu.cn， lilimin@wzu.edu.cn 李理敏（1984-），男，浙江人，博士，副教授，2006年于浙江大学获得学士学位，2012年于中科院上海微系统所获得博士学位，主要从事信号检测、图像处理等方面的研究。E-mail： lilimin@wzu.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716193&type= 22.01333237 32.00400162 李理敏（1984-），男，浙江人，博士，副教授，2006年于浙江大学获得学士学位，2012年于中科院上海微系统所获得博士学位，主要从事信号检测、图像处理等方面的研究。E-mail： lilimin@wzu.edu.cn lilimin@wzu.edu.cn 长安大学信息学院，陕西西安 710064 长安大学信息学院，陕西西安 710064 College of Information， Chang'an University， Xi'an 710064， China College of Information， Chang'an University， Xi'an 710064， China 温州大学电气与电子工程学院，浙江温州 325035 温州大学电气与电子工程学院，浙江温州 325035 School of Electrical and Electronic Engineering， Wenzhou University， Wenzhou 325035， China School of Electrical and Electronic Engineering， Wenzhou University， Wenzhou 325035， China 多重熵阈值（MET）随阈值个数 <math id="M1"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716218&type= 2.62466669 2.37066650 的增加其运算时间成倍增长，而相关优化策略的精度与稳定性低，使得分割的骨料图像缺失大量表面粗糙度与边缘等特征信息。为了克服这一问题，提出了一种基于混沌麻雀搜索算法（SSA）优化MET的骨料自动分割模型。为增强SSA的全局优化能力和鲁棒性，在种群位置初始化时加入Logistic混沌映射均匀麻雀分布，并提出扩张参数扩大全局搜索，控距精英变异及时跳出局部停滞，将该算法称为LSSA，可以在不降低收敛速度的情况下提升求解质量。LSSA用于MET参数的自动选取，以Renyi熵、对称交叉熵和Kapur熵作为目标函数，快速确定最佳阈值集。对不同特征的骨料图像进行了分割实验，通过对比6类组合算法与FCM算法，证明了LSSA-MET的有效性，随着 <math id="M2"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716218&type= 2.62466669 2.37066650 的增加仍然保持着较快的运行速度，即使 <math id="M3"><mi>K</mi><mo>=</mo><mn mathvariant="normal">6</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716224&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716205&type= 8.21266651 2.37066650 时，平均分割一张图片也只需1.532 s。其中LSSA-Renyi熵表现最佳，峰值信噪比、结构相似性和特征相似度分别提高了29.92%，10.67%和5.16%，有效地保留了骨料表面纹理和边缘特征，同时达到了精度与速度的最佳平衡。 Multiple entropy thresholding （MET） increases exponentially with an increase in the number of thresholds K . Related optimization strategies exhibit low accuracy and stability with the segmented aggregate images lacking considerable feature information such as surface roughness and edges. To overcome these problems， an automatic image segmentation model based on a chaotic sparrow search algorithm （SSA） was developed to optimize MET. SSA is a newer intelligent optimization algorithm. To enhance the global optimization capability and robustness of SSA， a logistic map is added to the uniform sparrow distribution at the time of population position initialization， an expansion parameter is applied to expand the global search， and temporal local stagnation is avoided by range-control elite mutation jumps. This algorithm is called logistic SSA （LSSA） and can improve the solution quality without reducing convergence speed. LSSA is used for the automatic selection of MET parameters， with the Renyi entropy， symmetric-cross entropy， and Kapur entropy as objective functions to quickly determine the correct thresholds. In this study， image segmentation and algorithm comparison experiments are conducted on aggregate images with different characteristics. The effectiveness of LSSA-MET was demonstrated by comparing six types of combined algorithms with the fuzzy C-means （FCM） algorithm. The proposed algorithm maintains a relatively high speed with an increase in K ， taking 1.532 s to split an image on average even when K =6. Among the variousm entropies， LSSA-Renyi entropy performed the best， achieving 29.92%， 10.67%， and 5.16% accuracy improvements in peak signal-to-noise ratio （PSNR）， structural similarity （SSIM）， and feature similarity （FSIM）， respectively， thereby effectively retaining the aggregate surface texture and edge characteristics while achieving the optimum balance between precision and speed. 图像分割骨料多重熵阈值优化麻雀搜索算法 Logistic映射 image segmentation aggregate multiple entropy thresholding optimization sparrow search algorithm logistic map 1 引言 1 骨料在道路交通、铁路修建和房屋建筑等土木工程中有着广泛的应用，高精度的形态测量与分析是保障生产质量的关键［ 1 1 ］。由于骨料粒径大小在毫米级，形状不规则，表面粗糙度不均匀，而且颗粒互相接触，骨料的图像处理比其他颗粒的图像处理更有挑战。多类分割算法是现代辅助骨料检测的常用工具，根据不连续性、颜色或灰度相似性、纹理等特征将图像划分为多类同质的区域［ 2 2 ］。如阈值、区域、聚类和语义分割算法，适用的骨料图像不尽相同。分水岭［ 3 3 ］、区域生长［ 4 4 ］与聚类算法［ 5 5 ］都适合边缘清晰的骨料图像，但是骨料表面到边缘处灰度过渡平缓，骨料覆盖或接触容易欠分割使粒径偏大，还会出现灰度融合现象丢失粗糙特征。语义分割近年在粒径检测上成果显著，不足的是它在表面纹理的并行分割上还有困难［ 6 6 ］。阈值分割操作简单、鲁棒性高，根据像素的强度和一组阈值将像素分为有限数量的类，适用于各类嘈杂图像。双阈值可以很好地分离背景、边缘与表面［ 7 7 ］。多阈值与单阈值相比能在分离相连的骨料的同时保持边缘、粗糙度和其他细节［ 8 8 ］。自动化阈值确定方法包括OTSU ［ 9 9 ］、Renyi熵［ 10 10 ］、Kapur熵［ 11 11 ］、对称交叉熵［ 12 12 ］、Tsallis熵、指数熵与模糊熵［ 13 13 - 14 14 13-14 ］等。基于熵的骨料检测可以快速得到骨料表面信息［ 15 15 ］，Kapur熵也叫最大熵，可以良好地分割前景与背景具有相同的概率分布时的图像。Renyi和Tsallis熵都是香农熵的推广，广泛应用于目标识别与图像分割。但随着阈值数量的增加，穷举法会使计算时间呈指数倍增长。优化算法可作为参数选择器辅助上述多重熵阈值（Multiple Entropy Thresholding， MET）快速确定阈值集，加快分割效率［ 16 16 - 17 17 16-17 ］。流行的有粒子群优化（Particle Swarm Optimization, PSO）［ 18 18 - 19 19 18-19 ］、布谷鸟算法（CS）、蝙蝠算法（BAT）、灰狼优化（Grey Wolf Optimizer， GWO）［ 20 20 - 21 21 20-21 ］、蚁狮算法（ALO）、鲸鱼优化算法（Whale Optimization Algorithm， WOA）［ 22 22 ］、旗鱼优化算法（SFO）、蜉蝣算法（Mayfly Algorithm， MA）［ 23 23 ］和麻雀搜索算法（Sparrow Search Algorithm, SSA）［ 24 24 - 25 25 24-25 ］等。受种群分布和搜索方式的影响，它们的准确性、速度和稳定性有所不同。例如，WOA的搜索路径为螺旋式，速度较快。GWO按等级划分种群，可以搜索到更优的解。SSA用矩阵存储每一维的位置确保不会重复搜索，划分种群为3种麻雀，共6种搜索方式，效率和鲁稳定性都很高。这些优化算法均存在两大缺点：全局搜索不全与陷入局部最优解，这导致分割精度低或不稳定。针对全局搜索不全的问题，目前的解决办法有多种优化算法的组合［ 26 26 ］、引入反向学习、交叉选择、随机序列和混沌映射等；而针对陷入局部最优解，可以通过Levy飞行、Cauchy-Gaussian变异个体、随机游走和概率扰动等策略跳出局部。Chen等［ 27 27 ］结合Tent混沌、动态自适应权重和Levy飞行（CDLSSA），获得较高的优化精度和稳定性，但分割时间被延长。为了确保速度，Liu等［ 28 28 ］提出用混沌映射、自适应惯性权重全局优化SSA（CASSA），但精度提升不大。这些策略始终无法实现精度与速度的最佳平衡。鉴于以上分析，本文提出一种LSSA-MET自动分割模型。首先，使用Logistic映射、扩张参数和控距精英变异3种策略从全局和局部综合优化SSA，在大幅提高搜索精度的前提下确保其收敛速度。然后，使用LSSA快速准确地确定MET的阈值集，提高分割效率。LSSA-MET具有较高的稳定性，对于各类骨料图像能保留更多的边缘与纹理粗糙度等特征信息。 2 原理 1 2.1　麻雀搜索算法 2 SSA是Xue等［ 24 24 ］在2020年提出的一种模拟麻雀觅食和反捕食行为的优化算法。在觅食期间，将麻雀分为发现者和加入者。发现者具有更好的适应度，它们提供觅食区域和方向。发现者的位置更新规则为： <math display="block" id="M4"><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>·</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced 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时的位置， <math id="M9"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716255&type= 6.18066692 3.72533321 是最大迭代数， <math id="M10"><mi>α</mi><mo>∈</mo><mfenced separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716278&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716260&type= 14.13933277 3.97933316 是一个随机数， <math id="M11"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>∈</mo><mfenced separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716280&type= 16.17133331 3.97933316 是预警值， <math id="M12"><mi>S</mi><mi>T</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0.5,1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716295&type= 20.65866661 3.97933316 是安全值， <math id="M13"><mi>Q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716287&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716284&type= 2.87866688 3.21733332 是服从正态分布的随机数， <math id="M14"><mi mathvariant="bold-italic">L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716291&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716289&type= 2.53999996 2.79399991 是 <math id="M15"><mn mathvariant="normal">1</mn><mo>×</mo><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716292&type= 8.97466660 2.79399991 阶的全1矩阵。加入者跟随发现者来获取食物，但一些非常饿的加入者会改变觅食路径。加入者的位置更新规则为： <math display="block" id="M16"><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>Q</mi><mo>·</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mtext> </mtext><mfenced separators="|"><mrow><mfrac><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup></mrow><mrow><msup><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mo>,</mo></mtd><mtd><mi>i</mi><mo>></mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd></mtr><mtr><mtd><msubsup><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>+</mo><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup></mrow></mfenced><mo>·</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>·</mo><mi>L</mi><mo>,</mo></mtd><mtd><mi>i</mi><mo>≤</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716307&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716304&type= 65.53199768 20.48933220 ，其中： <math id="M17"><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716326&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716310&type= 5.67266655 3.72533321 是最优位置， <math id="M18"><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716313&type= 6.85799980 3.72533321 是最差位置， <math id="M19"><mi mathvariant="bold-italic">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716315&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716314&type= 2.96333337 2.79399991 是 <math id="M20"><mn mathvariant="normal">1</mn><mo>×</mo><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716318&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716333&type= 8.80533314 2.79399991 阶矩阵，它的元素值为 <math id="M21"><mo>±</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716321&type= 5.24933338 2.87866688 ，且 <math id="M22"><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">A</mi><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716358&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716342&type= 27.26266861 5.16466665 。在反捕食期间，随机产生预警麻雀。边缘的麻雀迅速移动到安全的位置，剩下的麻雀互相靠近。预警者的位置更新规则为： <math display="block" id="M23"><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>+</mo><mi>β</mi><mo>·</mo><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow><mrow><mi>t</mi></mrow></msubsup></mrow></mfenced><mo>,</mo></mtd><mtd><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><msub><mrow><mi>f</mi></mrow><mrow><mi>g</mi></mrow></msub></mtd></mtr><mtr><mtd><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>+</mo><mi>K</mi><mo>·</mo><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow><mrow><mi>t</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mfenced separators="|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>ω</mi></mrow></msub></mrow></mfenced><mo>+</mo><mi>ε</mi></mrow></mfrac><mo>,</mo></mtd><mtd><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>g</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716350&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716348&type= 61.72200012 18.88066673 ，其中： <math id="M24"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716369&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716362&type= 2.28600001 3.64066648 是步长控制参数，服从标准正态分布， <math id="M25"><mi>K</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mo>-</mo><mn mathvariant="normal">1,1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716367&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716364&type= 18.45733452 3.97933316 控制麻雀的移动方向， <math id="M26"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716375&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716374&type= 1.43933344 2.79399991 是最小的常数， <math id="M27"><msub><mrow><mi>f</mi></mrow><mrow><mi>g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716377&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716376&type= 2.96333337 4.57200003 是最佳适应度值， <math id="M28"><msub><mrow><mi>f</mi></mrow><mrow><mi>ω</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716378&type= 3.30200005 4.57200003 是最差适应度值。当SSA作为参数选择器时，让所有麻雀随机分布在搜索空间内，根据上述3个位置更新公式快速收敛至最优值，即为最终的寻优结果。 2.2　LSSA 2 2 . 2 . 1　Logistic映射 3 在SSA中，麻雀的初始位置是随机的，麻雀聚集对全局搜索非常不利。而混沌映射可以将一般序列转换为遍历的、均匀的、随机的非线性混沌序列。Logistic ［ 29 29 ］是一个典型的混沌系统，可描述为： <math display="block" id="M29"><mi>x</mi><mfenced separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mi>a</mi><mo>·</mo><mi>x</mi><mfenced separators="|"><mrow><mi>k</mi></mrow></mfenced><mo>·</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>x</mi><mfenced separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716385&type= 48.51399994 4.74133301 ，其中： <math id="M30"><mi>x</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>∉</mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">0,0.25,0.5,0.75,1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716400&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716399&type= 46.31266785 3.97933316 ， <math id="M31"><mi>x</mi><mfenced separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>∈</mo><mfenced separators="|"><mrow><mn mathvariant="normal">0,1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716393&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716391&type= 26.41600037 4.06400013 ， <math id="M32"><mi>a</mi><mo>∈</mo><mfenced separators="|"><mrow><mn mathvariant="normal">0,4</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716404&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716403&type= 14.22399998 3.97933316 为控制参数。 <math id="M33"><mi>a</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716396&type= 1.86266661 2.79399991 与 <math id="M34"><mi>x</mi><mfenced separators="|"><mrow><mi>k</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716406&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716414&type= 7.78933382 3.97933316 的关系如图1 图1 （a）所示， <math id="M35"><mi>a</mi><mo>></mo><mn mathvariant="normal">3.57</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716410&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716409&type= 14.30866718 2.79399991 时产生混沌； <math id="M36"><mi>a</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">3.9,4</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716412&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716417&type= 17.86466599 3.97933316 时存在稳定的混沌，均匀分布在0~1之间，不收敛且非周期性。当 <math id="M37"><mi>a</mi><mo>=</mo><mn mathvariant="normal">4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716430&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716428&type= 8.63599968 2.79399991 ， <math id="M38"><mi>x</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0.1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716422&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716420&type= 18.37266731 3.97933316 时的混沌序列分布如图1 图1 （b）所示。使用Logistic混沌映射种群中每只麻雀在每个维度下的初始位置 <math id="M39"><mi>x</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716435&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716445&type= 7.78933382 3.97933316 ，它可以在不增加优化时间的情况下快速干扰种群，让麻雀更均匀地分布，随机性高且系统稳定。 2 . 2 . 2　扩张参数 3 发现者受函数 <math id="M40"><mi>y</mi><mo>=</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mi>x</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716448&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716447&type= 11.43000031 3.89466691 的影响。随着迭代次数的增加，搜索范围迅速缩小，降低了全局搜索能力，容易陷入局部，从而降低了寻优精度，所以提出扩张参数用于增大搜索范围。在迭代的前期 <math id="M41"><mi>t</mi><mo><</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716438&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716449&type= 12.86933327 7.87400007 时，使用较大的参数，尽可能地分散麻雀。在迭代的后期 <math id="M42"><mi>t</mi><mo>≥</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716441&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716451&type= 12.86933327 7.87400007 ，麻雀集中，使用较小的参数，可表示为： <math display="block" id="M43"><mi>σ</mi><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><mfenced separators="|"><mrow><mfrac><mrow><mi>t</mi><mi mathvariant="normal">π</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo></mtd><mtd><mi>t</mi><mo><</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><mfenced separators="|"><mrow><mfrac><mrow><mi>t</mi><mi mathvariant="normal">π</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo></mtd><mtd><mi>t</mi><mo>≥</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716452&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716459&type= 60.62133789 29.71800041 ，其中： <math id="M44"><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716453&type= 5.33400011 3.72533321 是迭代开始的参数， <math id="M45"><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716481&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716479&type= 4.91066647 3.72533321 是迭代结束的参数。经多次实验，当 <math id="M46"><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716476&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716475&type= 5.33400011 3.72533321 接近 <math id="M47"><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716463&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716462&type= 1.86266661 2.79399991 ， <math id="M48"><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716481&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716479&type= 4.91066647 3.72533321 接近 <math id="M49"><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716465&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716483&type= 5.33400011 2.79399991 时扩张效果最好。取 <math id="M50"><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0.99</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716486&type= 17.69533348 3.72533321 ， <math id="M51"><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo>=</mo><mo>-</mo><mn mathvariant="normal">0.99</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716468&type= 20.91266632 3.72533321 。将扩张参数引入发现者的位置更新，那么它们的位置更新规则变为： <math display="block" id="M52"><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>·</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced separators="|"><mrow><mo>-</mo><mfrac><mrow><mi>i</mi></mrow><mrow><mi>σ</mi><mo>·</mo><mi>α</mi><mo>·</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow></mfrac></mrow></mfenced><mo>,</mo></mtd><mtd><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo><</mo><mi>S</mi><mi>T</mi></mtd></mtr><mtr><mtd><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>+</mo><mi>Q</mi><mo>·</mo><mi mathvariant="bold-italic">L</mi><mo>,</mo></mtd><mtd><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≥</mo><mi>S</mi><mi>T</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716490&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716489&type= 62.73799896 15.57866764 . 2 . 2 . 3　控距精英变异 3 麻雀在搜索中会陷入许多的局部最优解，如果个体无法及时逃出，会过分消耗种群，导致更优的解被漏检。现有的策略都不确保麻雀跳出局部，如Cauchy-Gaussian变异、Levy飞行与随机游走，而其中一些需要更多的时间。因此，提出一种控距精英变异策略，它可以被快速实施，同时提升解的精度。每次迭代选择一只精英麻雀使它跳出局部区域，控制它移动的距离。为确保麻雀选择准确，在每次迭代后期对麻雀适应度值排序，筛选出适应度值最好的麻雀作为精英麻雀，其位置为 <math id="M53"><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716492&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716491&type= 5.67266655 3.72533321 。当 <math id="M54"><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716525&type= 5.67266655 3.72533321 与适应度值最低的 <math id="M55"><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716517&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716516&type= 6.85799980 3.72533321 距离较远时，麻雀依然分散，让 <math id="M56"><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716525&type= 5.67266655 3.72533321 在 <math id="M57"><mo stretchy="false">(</mo><mi>l</mi><mi>b</mi><mo>,</mo><mtext> </mtext><mi>u</mi><mi>b</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716520&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716500&type= 12.27666664 3.55599999 内随机变异。相反地，当 <math id="M58"><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716525&type= 5.67266655 3.72533321 和 <math id="M59"><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716517&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716516&type= 6.85799980 3.72533321 距离较近时麻雀聚集，控制 <math id="M60"><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716525&type= 5.67266655 3.72533321 变异到 <math id="M61"><mi>r</mi><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716531&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716529&type= 23.28333473 3.72533321 的矩形区域外，并在 <math id="M62"><mfenced separators="|"><mrow><mi>l</mi><mi>b</mi><mo>,</mo><mtext> </mtext><mi>u</mi><mi>b</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716533&type= 12.27666664 3.97933316 的区域内。这是为了保证精英麻雀在避免该局部的同时继续全局寻优。精英麻雀的位置更新规则为： <math display="block" id="M63"><msubsup><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msubsup><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>d</mi><mo>·</mo><mi>r</mi><mi>a</mi><mi>n</mi><mi>d</mi><mi>n</mi><mo>+</mo><mi>l</mi><mi>b</mi><mo>,</mo></mtd><mtd><mi>r</mi><mo>></mo><mfrac><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub><mo>+</mo><mi>r</mi><mo>+</mo><mfenced separators="|"><mrow><mi>d</mi><mo>·</mo><mi>r</mi><mi>a</mi><mi>n</mi><mi>d</mi><mi>n</mi><mo>+</mo><mi>l</mi><mi>b</mi></mrow></mfenced><mi>r</mi><mi>a</mi><mi>n</mi><mi>d</mi><mi>n</mi><mo>,</mo></mtd><mtd><mi>r</mi><mo>≤</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716536&type= 73.23667145 18.71133423 ，其中： <math id="M64"><mi>d</mi><mo>=</mo><mi>u</mi><mi>b</mi><mo>-</mo><mi>l</mi><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716550&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716549&type= 18.54199982 2.79399991 ， <math id="M65"><mi>r</mi><mo>=</mo><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub><mo stretchy="false">|</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716551&type= 24.72266769 4.40266657 。这种方式可以很大程度地提高跳出局部最优解的几率，最大限度地减少迭代消耗，不会像传统策略跳出的麻雀重回该局部消耗迭代时间。 2.3　多重熵阈值 2 MET借助直方图将具有不同灰度强度的像素分类，并为每个类别分配最近的灰度值，从而实现分割。Renyi熵、对称交叉熵与Kapur熵对信息量的计算方法有所不同。当阈值个数为 <math id="M66"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716930&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716929&type= 3.04800010 2.79399991 时，直方图被分为 <math id="M67"><mi>K</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716800&type= 9.56733322 2.96333337 个区域，记每个区域对应的信息量分别为 <math id="M68"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi>H</mi></mrow><mrow><mi>K</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716564&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716556&type= 28.44799995 4.14866638 ，那么所有区域的信息总和为 <math id="M69"><mi>E</mi><mo>=</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>K</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716559&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716566&type= 42.33333206 3.80999994 。 Renyi熵、对称交叉熵与Kapur熵在第 <math id="M70"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716585&type= 1.69333339 2.79399991 个区域的信息量可分别表达为： <math display="block" id="M71"><msub><mrow><mi>H</mi></mrow><mrow><mi>k</mi><mo>-</mo><mi>R</mi></mrow></msub><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfrac><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><msup><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></munderover></mstyle><mrow><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mfenced></mrow></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716574&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716572&type= 54.44067001 11.76866722 ， <math display="block" id="M72"><msub><mrow><mi>H</mi></mrow><mrow><mi>k</mi><mo>-</mo><mi>S</mi></mrow></msub><mfenced separators="|"><mrow><mi>l</mi></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></munderover></mstyle><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mfenced separators="|"><mrow><mi>i</mi><mo>·</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mfrac><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced separators="|"><mrow><mi>l</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>·</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced separators="|"><mrow><mi>l</mi></mrow></mfenced></mrow><mrow><mi>i</mi></mrow></mfrac></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716577&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716576&type= 72.98266602 11.26066685 ， <math display="block" id="M73"><msub><mrow><mi>H</mi></mrow><mrow><mi>k</mi><mo>-</mo><mi>K</mi></mrow></msub><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></munderover></mstyle><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716579&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716561&type= 53.34000015 9.65200043 ，其中： <math id="M74"><mo stretchy="false">[</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716583&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716581&type= 13.29266548 4.48733330 是第 <math id="M75"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716585&type= 1.69333339 2.79399991 个区域的灰度范围， <math id="M76"><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716595&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716588&type= 2.70933342 3.72533321 是灰度值为 <math id="M77"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716599&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716598&type= 1.10066664 2.79399991 的频数， <math id="M78"><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716592&type= 2.87866688 3.72533321 是该区域的灰度均值， <math id="M79"><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716611&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716610&type= 3.30200005 3.72533321 是像素的灰度值为 <math id="M80"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716599&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716598&type= 1.10066664 2.79399991 的概率， <math id="M81"><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716601&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716614&type= 3.55599999 3.72533321 是该区域灰度值的概率总和， <math id="M82"><mi>α</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716617&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716602&type= 15.40933418 3.55599999 是可调参数，本文 <math id="M83"><mi>α</mi><mo>=</mo><mn mathvariant="normal">0.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716622&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716620&type= 12.53066730 2.79399991 。利用Renyi熵、对称交叉熵或Kapur熵求解阈值集，即计算使 <math id="M84"><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716626&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716624&type= 2.70933342 2.79399991 达到最大或最小值的一组 <math id="M85"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716630&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716628&type= 1.26999998 2.79399991 ，得到： <math display="block" id="M86"><msub><mrow><mi>l</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>K</mi></mrow></mfenced><mo>-</mo><mi>R</mi></mrow></msub><mo>=</mo><mi>a</mi><mi>r</mi><mi>g</mi><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">{</mo><mi>E</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716631&type= 38.26933289 4.14866638 ， <math display="block" id="M87"><msub><mrow><mi>l</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>K</mi></mrow></mfenced><mo>-</mo><mi>S</mi></mrow></msub><mo>=</mo><mi>a</mi><mi>r</mi><mi>g</mi><mi>m</mi><mi>i</mi><mi>n</mi><mo stretchy="false">{</mo><mi>E</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716604&type= 36.99933624 4.14866638 ， <math display="block" id="M88"><msub><mrow><mi>l</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>K</mi></mrow></mfenced><mo>-</mo><mi>K</mi></mrow></msub><mo>=</mo><mi>a</mi><mi>r</mi><mi>g</mi><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">{</mo><mi>E</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716607&type= 38.26933289 4.14866638 . 以上3种熵的分割效果略有不同，如图2 图2 所示，主要原因是骨料图像的颜色、纹理与目标占比带来的直方图的变化。 1;2; 骨料图像直方图在波谷处通常很平缓，但有非常多的极值点，而且它们距离接近。所以，即使阈值仅相差一个灰度值，分割结果也可能有着较大的差异。因此，优化算法的性能直接影响分割精度和稳定性。 2.4　LSSA-MET 2 在分割图像时，LSSA把熵求解阈值的公式作为目标函数，直方图作为搜索空间。LSSA使均匀分布的麻雀快速收敛至最优解。对应的麻雀位置就是分割阈值，该模型被称为LSSA-MET。用LSSA-MET分割骨料图像的相关流程见图3 图3 。 3 实验与结果分析 1 3.1　优化算法性能测试 2 基准函数实验用于测试优化算法的性能，是一种公平且全面的评估方法。单峰函数只有一个最优值，检验全局搜索能力；而多峰函数有多个局部最优值和一个全局最优值，检测局部收敛能力，同时可以评估其收敛过程及收敛效率等。基准测试函数的表达式如下： <math display="block" id="M91"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>D</mi></mrow></msubsup><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716646&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716645&type= 26.50066566 4.74133301 ， <math display="block" id="M92"><msup><mrow><mo stretchy="false">[</mo><mo>-</mo><mn mathvariant="normal">100,100</mn><mo stretchy="false">]</mo></mrow><mrow><mi>D</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716648&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716647&type= 20.82799911 3.80999994 ， <math display="block" id="M93"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo stretchy="false">{</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>,</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>D</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716672&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716649&type= 44.02666473 3.97933316 ， <math display="block" id="M94"><msup><mrow><mo stretchy="false">[</mo><mo>-</mo><mn mathvariant="normal">100,100</mn><mo stretchy="false">]</mo></mrow><mrow><mi>D</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716650&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716673&type= 20.82799911 3.80999994 ， <math display="block" id="M95"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>D</mi></mrow></msubsup><mrow><mfenced open="[" close="" separators="|"><mrow><mn mathvariant="normal">100</mn></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">]</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716651&type= 65.95532990 5.50333309 ， <math display="block" id="M96"><msup><mrow><mo stretchy="false">[</mo><mo>-</mo><mn mathvariant="normal">30,30</mn><mo stretchy="false">]</mo></mrow><mrow><mi>D</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716677&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716675&type= 17.10266685 3.80999994 ， <math display="block" id="M97"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>D</mi></mrow></msubsup><mrow><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>·</mo><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mroot><mrow><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo></mrow><mrow/></mroot><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716680&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716678&type= 44.87333298 5.33400011 ， <math display="block" id="M98"><msup><mrow><mo stretchy="false">[</mo><mo>-</mo><mn mathvariant="normal">500,500</mn><mo stretchy="false">]</mo></mrow><mrow><mi>D</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716685&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716682&type= 20.82799911 3.80999994 ， <math display="block" id="M99"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>D</mi></mrow></msubsup><mrow><mo stretchy="false">[</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mn mathvariant="normal">10</mn><mi>c</mi><mi>o</mi><mi>s</mi><mtext> </mtext><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi>π</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>+</mo><mn mathvariant="normal">10</mn><mo stretchy="false">]</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716688&type= 63.33066940 4.74133301 ， <math display="block" id="M100"><msup><mrow><mo stretchy="false">[</mo><mo>-</mo><mn mathvariant="normal">5.12,5.12</mn><mo stretchy="false">]</mo></mrow><mrow><mi>D</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716692&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716691&type= 24.80733490 3.80999994 ， <math id="M101"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mi>D</mi></mrow></mfrac><mo stretchy="false">{</mo><mn mathvariant="normal">10</mn><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi mathvariant="normal">π</mi><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>D</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716693&type= 67.64866638 7.87400007 · <math id="M102"><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mn mathvariant="normal">10</mn><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi mathvariant="normal">π</mi><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>D</mi></mrow></msubsup><mrow><mi>u</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">10,100,4</mn><mo stretchy="false">)</mo></mrow><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716696&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716695&type= 71.71266174 5.58799982 ， <math id="M103"><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>/</mo><mn mathvariant="normal">4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716713&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716697&type= 26.41600037 4.91066647 ， <math display="block" id="M104"><mi>u</mi><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>a</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi></mrow></mfenced><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>m</mi></mrow></msup></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><mi>a</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mi>a</mi><mo>≤</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mi>a</mi></mtd></mtr><mtr><mtd><mi>k</mi><mo stretchy="false">(</mo><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mo>-</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>m</mi></mrow></msup></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo><</mo><mo>-</mo><mi>a</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716715&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716714&type= 62.90732956 15.49400043 ， <math display="block" id="M105"><msup><mrow><mo stretchy="false">[</mo><mo>-</mo><mn mathvariant="normal">50,50</mn><mo stretchy="false">]</mo></mrow><mrow><mi>D</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716701&type= 17.10266685 3.80999994 。其中， F 1 ~ F 6 的最优值分别为0，0，0， - 418.982 9D，0，0。维度 <math id="M106"><mi>D</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716705&type= 9.65200043 2.79399991 时 F 1 和 F 4 的示意图见图4 图4 。将LSSA和PSO，GWO，WOA，MA，SSA，CDLSSA及CASSA进行对比。为保证公平，参阅大量文献后设置的各个优化算法的参数如下［ 18 18 - 24 24 18-24 ， 27 27 - 28 28 27-28 ］：PSO中 <math id="M113"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716724&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716722&type= 22.86000061 3.72533321 ， <math id="M114"><mi>ω</mi><mo>=</mo><mn mathvariant="normal">0.74</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716735&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716726&type= 14.90133286 2.79399991 ，GWO中 <math id="M115"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716737&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716736&type= 1.94733346 2.79399991 从2线性减小至0， <math id="M116"><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716743&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716738&type= 20.65866661 4.48733330 ，WOA中 <math id="M117"><mi>α</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0,2</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716745&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716744&type= 14.73200035 3.55599999 ， <math id="M118"><mi>b</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716746&type= 8.63599968 2.79399991 ， <math id="M119"><mi>l</mi><mo>∈</mo><mo stretchy="false">[</mo><mo>-</mo><mn mathvariant="normal">1,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716740&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716748&type= 16.51000023 3.55599999 ，MA中 <math id="M120"><mi>g</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716742&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716741&type= 8.72066593 3.64066648 ， <math id="M121"><msub><mrow><mi>g</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">p</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716750&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716749&type= 13.29266548 4.57200003 ， <math id="M122"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716759&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716758&type= 9.82133293 3.72533321 ， <math id="M123"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716760&type= 21.67466736 3.72533321 。SSA，CDLSSA，CASSA和LSSA的发现者比例 <math id="M124"><mi mathvariant="normal">P</mi><mi mathvariant="normal">D</mi><mo>=</mo><mn mathvariant="normal">20</mn><mi mathvariant="normal">%</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716763&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716762&type= 17.35666656 3.47133350 ，预警者比例 <math id="M125"><mi>S</mi><mi>D</mi><mo>=</mo><mn mathvariant="normal">10</mn><mi mathvariant="normal">%</mi><mo>,</mo><mi>S</mi><mi>T</mi><mo>=</mo><mn mathvariant="normal">0.8</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716751&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716764&type= 34.29000092 3.55599999 ，在CDLSSA中， <math id="M126"><mi mathvariant="normal">l</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">v</mi><mi mathvariant="normal">y</mi><mtext> </mtext><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mo>=</mo><mn mathvariant="normal">1.5</mn><mo>,</mo><mi>K</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716753&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716752&type= 35.30599976 3.64066648 ，而在CASSA中 <math id="M127"><mi>S</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716766&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716765&type= 8.89000034 2.79399991 。此外，种群大小为60，最大迭代次数为600，基准函数维度 <math id="M128"><mi>D</mi><mo>=</mo><mn mathvariant="normal">30</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716755&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716754&type= 11.51466751 2.79399991 ，实验次数为60。从算法的精度、稳定性和效率考虑，本文采用寻优的最佳值（Best）、平均值（AVG）、标准差（SD）和总时间（ T ）作为评估参数，统计结果见表1 表1 。 2 方法 4 F 1 4 F 2 Best AVG SD T /s Best AVG SD T /s PSO 8.75×10 -11 3.47×10 -9 6.08×10 -9 69.945 2.64×10 -1 1.56 8.07×10 -1 68.919 GWO 4.93×10 -45 4.01×10 -43 9.02×10 -43 64.120 4.12×10 -12 6.52×10 -11 8.46×10 -11 60.733 WOA 1.07×10 -126 9.61×10 -111 6.10×10 -110 23.023 2.07×10 -1 4.79×10 2.44×10 23.022 MA 9.48×10 -12 1.43×10 -10 3.42×10 -10 174.036 2.81×10 4.24×10 5.54 170.238 SSA 3.10×10 -258 2.22×10 -50 1.71×10 -49 20.316 1.03×10 -53 1.96×10 -10 1.25×10 -9 20.210 CASSA 0 9.93×10 -53 6.72×10 -39 19.162 3.76×10 -27 6.38×10 -13 3.79×10 -11 20.661 CDLSSA 0 0 0 41.614 0 1.43×10 -204 0 40.259 LSSA 0 0 0 21.367 0 0 0 20.497 2 方法 4 F 3 4 F 4 Best AVG SD T /s Best AVG SD T /s PSO 9.17 5.51×10 8.35×10 +1 72.211 -9.07×10 3 -7.86×10 3 6.17×10 2 72.674 GWO 2.52×10 2.65×10 7.21×10 -1 64.341 -7.23×10 3 -6.02×10 3 5.78×10 2 64.946 WOA 2.60×10 2.70×10 7.08×10 -1 26.039 -1.26 ×10 4 -9.25×10 3 1.46×10 3 25.942 MA 8.06×10 -1 4.68×10 5.19×10 +1 183.103 -1.11×10 4 -1.02×10 4 3.19×10 2 185.479 SSA 1.21×10 -7 4.91×10 -4 6.97×10 -3 22.811 -9.91×10 3 -8.86×10 3 1.73×10 3 22.903 CASSA 1.19×10 -7 7.40×10 -4 2.01×10 -3 22.548 -9.29×10 3 -9.76×10 3 1.86×10 3 22.432 CDLSSA 1.03×10 -7 1.19×10 -3 5.70×10 -3 44.432 -1.26 × 10 4 -1.04×10 4 1.03×10 2 43.501 LSSA 1.16 × 10 -8 4.45 × 10 -4 1.91 × 10 -3 23.025 -1.26 × 10 4 -1.08 × 10 4 1.00 × 10 2 23.011 2 方法 4 F 5 4 F 6 Best AVG SD T /s Best AVG SD T /s PSO 2.49×10 5.63×10 1.36×10 70.819 1.04×10 -11 1.07×10 -1 1.98×10 -1 100.284 GWO 0 7.56×10 -1 2.06 62.135 1.68×10 -6 2.53×10 -2 1.52×10 -2 94.983 WOA 0 0 0 23.341 6.67×10 -4 1.66×10 -2 1.13×10 -2 50.255 MA 4.98 2.09×10 1.06×10 176.503 2.07×10 -9 9.10×10 -2 2.12×10 -1 256.946 SSA 0 0 0 20.350 2.39×10 -10 1.21×10 -6 2.46×10 -6 48.751 CASSA 0 0 0 20.543 9.53×10 -11 5.80×10 -7 2.56×10 -6 48.613 CDLSSA 0 0 0 40.018 2.58×10 -11 5.10×10 -7 1.65×10 -6 88.256 LSSA 0 0 0 20.155 1.09 × 10 -12 3.70 × 10 -8 9.65 ×10 -7 48.639 表1 表1 中，横向对比的最优值用粗体字突出表示，可以看出，LSSA的总体性能更优。在单峰函数 <math id="M129"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716768&type= 3.72533321 3.72533321 ~ <math id="M130"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716776&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716775&type= 3.72533321 3.72533321 上，LSSA的BEST，AVG和SD远远高于其他算法，而它的 T 与最优的 T 相比差距很小。在多峰函数 <math id="M131"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716778&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716777&type= 3.72533321 3.72533321 ~ <math id="M132"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716783&type= 3.72533321 3.72533321 上，LSSA依然优越，且 T 在 <math id="M133"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716782&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716781&type= 3.72533321 3.72533321 上达到最优。总的来说，WOA，SSA，CASSA与LSSA速度较快，而CDLSSA与LSSA的精度和稳定性最高，但CDLSSA的 T 约是LSSA的2倍。综合来看，LSSA达到了速度与精度的最佳平衡。图5 图5 展示了LSSA和其他优化算法的收敛曲线。对于30维的函数，LSSA精度最高且迭代次数较少。在 F 1 ， F 2 和 F 5 上CDLSSA与LSSA的精度相同，但CDLSSA一次迭代时间过长，效率低。在 <math id="M134"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716783&type= 3.72533321 3.72533321 上，LSSA的初始值总是更靠近最优值，这归功于在种群初始化时加入了Logistic映射，麻雀的位置变得更均匀。在 <math id="M135"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716768&type= 3.72533321 3.72533321 和 <math id="M136"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716770&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716769&type= 3.72533321 3.72533321 迭代200次的地方可以看出，LSSA的搜索范围更广泛，在加入扩张参数后，局部最优值不断更新，靠近全局最优值。而LSSA的收敛曲线均有折线，这是由于加入了控距精英变异，即使在 <math id="M137"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716771&type= 3.72533321 3.72533321 这种困境下也可轻易跳出局部。虽然CASSA也有类似的折线，但逃脱效果较差。整体来看，当3个策略共同作用，LSSA可以达到精度和速度的最佳平衡。 1;2; 3.2　分割模型实验评估 2 实验所用的骨料图像均从Key Laboratory of Road Construction Technology and Equipment， MOE中获得。本研究测试了100多张图像，对比了LSSA-Renyi熵、LSSA-Kapur熵、LSSA-对称交叉熵、SSA-Renyi熵、SSA-Kapur熵、SSA-对称交叉熵和聚类算法模糊C均值（Fuzzy C Mean， FCM）。图6 图6 显示了4帧像素尺寸为512×512的RGB骨料图像及其直方图。直方图在波谷处很平缓，但有非常多的距离相近的极值点。骨料表面粗糙，分割结果呈离散的颗粒状，即使阈值接近，也会降低原粗糙水平，所以优化算法的性能会直接影响分割精度。骨料颗粒包含丰富的特征，如形状、尺寸、颜色、边缘和表面粗糙度等。这些特征用于判断碎石与卵石、粗细骨料、分析矿物种类、判断打磨程度等。根据应用场景确定分割类别。设置阈值个数 <math id="M138"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716930&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716929&type= 3.04800010 2.79399991 ，图像被分为 <math id="M139"><mi>K</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716800&type= 9.56733322 2.96333337 类，大多数相关文献 <math id="M140"><mi>K</mi><mo>=</mo><mn mathvariant="normal">2</mn><mo>~</mo><mn mathvariant="normal">6</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716802&type= 14.73200035 2.79399991 ，能在满足工程基本需求的同时评估阈值的准确性［ 17 17 ］。限于篇幅，本文仅展示 <math id="M141"><mi>K</mi><mo>=</mo><mn mathvariant="normal">2,4</mn><mo>,</mo><mn mathvariant="normal">6</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716907&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716906&type= 16.93333435 3.21733332 时的部分分割结果，见图7 图7 。 1;2; 主观来看，当 <math id="M142"><mi>K</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716928&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716927&type= 9.56733322 2.79399991 时，FCM的分割效果最好，其次是Renyi熵。随着 <math id="M143"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716930&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716929&type= 3.04800010 2.79399991 增加到4时，基于熵的分割结果精度不断提高，而FCM愈加不稳定，出现相邻像素灰度相融现象。在 <math id="M144"><mi>K</mi><mo>=</mo><mn mathvariant="normal">6</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716931&type= 9.56733322 2.79399991 时，Renyi熵对于尺寸小、接触或重叠颗粒的骨料图像仍然表现良好，其次是Kapu熵，而对称交叉熵整体灰度值偏亮，FCM虽边缘清晰，但灰度偏差大，表面粗糙度特征丢失。整体来看，LSSA-MET的分割精度和稳定性高于SSA-MET，这是由于LSSA可以获取到更准确的阈值。衡量优化算法精度的标准是看它优化的目标函数是否能取得较高的适应度值。在 <math id="M145"><mi>K</mi><mo>=</mo><mn mathvariant="normal">6</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716931&type= 9.56733322 2.79399991 时，LSSA与SSA获取的适应度值和其对应的阈值如表2 表2 所示，将两个适应度值中更好的一个用粗体字突出显示。 2 图像 2 参数 2 Renyi熵 2 对称交叉熵 2 Kapur熵 LSSA SSA LSSA SSA LSSA SSA 2 No.1 阈值 41 74 115 150 186 227 35 62 95 131 171 228 30 56 92 121 154 185 32 55 92 113 149 172 41 73 102 133 161 192 39 72 94 124 169 191 适应度值 24.356 6 24.237 9 241 731.634 266 481.176 9 24.124 9 23.974 2 No.2 阈值 44 76 108 167 142 199 45 71 92 114 137 189 32 65 101 131 157 183 36 73 109 133 150 178 37 75 106 134 162 194 40 61 80 118 151 190 适应度值 24.305 9 23.971 1 239 635.992 8 262 853.106 6 24.091 3 23.926 6 2 No.3 阈值 37 71 140 14 173 208 39 71 104 137 169 201 25 43 81 109 143 173 29 53 91 126 139 166 38 70 105 136 168 199 43 77 100 124 150 185 适应度值 24.416 4 24.409 5 263 073.381 9 284 278.489 6 24.142 1 23.989 2 No.4 阈值 36 69 100 131 163 193 53 80 101 122 149 176 35 65 95 126 157 183 37 63 95 126 154 183 41 72 101 129 159 190 43 85 109 134 160 183 适应度值 24.088 9 23.795 7 183 027.266 6 183 338.434 2 23.824 8 23.693 1 表2 表2 中，LSSA与SSA取得更优的适应度值的比例为12∶0，这证明LSSA具有更好的寻优能力。实验发现，当 <math id="M148"><mi>K</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716928&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716927&type= 9.56733322 2.79399991 时，LSSA与SSA在阈值不同时也有可能取得同样的适应度值，但分割结果不同。然而，随着 <math id="M149"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716810&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716809&type= 3.04800010 2.79399991 的增加，LSSA性能愈发优越，尤其是 <math id="M150"><mi>K</mi><mo>=</mo><mn mathvariant="normal">6</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716931&type= 9.56733322 2.79399991 时，无论是适应度值还是分割质量，LSSA-MET都普遍优于SSA-MET。图7 图7 中的图像尺寸太小，无法看清LSSA-MET与SSA-MET的分割结果的差异，所以截取No.2在 <math id="M151"><mi>K</mi><mo>=</mo><mn mathvariant="normal">6</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716813&type= 9.56733322 2.79399991 时的局部分割结果，见图8 图8 。LSSA-MET的分割图像特征更加精细，灰度更接近原图，而SSA-MET的分割图像缺失了许多细节，尤其是骨料表面的粗糙点。分割图像的质量评估需要考虑每个像素的所属类别。肉眼无法标注每一种 <math id="M154"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716930&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716929&type= 3.04800010 2.79399991 下的真值图像，所以将分割图像与原始图像进行对比。以峰值信噪比（Peak Signal-to-Noise Ratio,PSNR）、结构相似性（Structural Similarity, SSIM）、特征相似度（Feature Similarity, FSIM）和耗时（ T ）作为评价指标［ 30 30 - 31 31 30-31 ］。由于LSSA与SSA在位置初始化时具有随机性，采用60次实验的平均值作为最终实验结果。（1）PSNR是测量图像噪声的度量标准。PSNR越大，包含的信息越多，噪声越小，分割效果越好。 <math display="block" id="M155"><mi>P</mi><mi>S</mi><mi>N</mi><mi>R</mi><mo>=</mo><mn mathvariant="normal">20</mn><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">255</mn></mrow><mrow><mi>R</mi><mi>M</mi><mi>S</mi><mi>E</mi></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716853&type= 41.14799881 9.05933285 ，其中RMSE均方根误差代表像素间的距离。 <math display="block" id="M156"><mi>R</mi><mi>M</mi><mi>S</mi><mi>E</mi><mo>=</mo><mroot><mrow><mfrac><mrow><mo>∑</mo><mo>∑</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>-</mo><mover accent="true"><mi>f</mi><mo>^</mo></mover><msup><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>M</mi><mo>×</mo><mi>M</mi></mrow></mfrac></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716855&type= 56.13399887 10.66800022 ，其中： <math id="M157"><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716858&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716857&type= 10.15999985 3.97933316 是原图， <math id="M158"><mover accent="true"><mi>f</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716837&type= 10.15999985 4.57200003 是分割后的图像， <math id="M159"><mi>M</mi><mo>×</mo><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716840&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716839&type= 12.10733318 2.87866688 是图像像素尺寸。（2）SSIM用于比较分割图像与原始图像的整体相似性。SSIM∈ <math id="M160"><mo stretchy="false">[</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716842&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716841&type= 8.97466660 3.55599999 ，SSIM越高，说明分割图像失真越小。 <math display="block" id="M161"><mi>S</mi><mi>S</mi><mi>I</mi><mi>M</mi><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>μ</mi></mrow><mrow><mi>f</mi></mrow></msub><msub><mrow><mi>μ</mi></mrow><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover></mrow></msub><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>f</mi><mo>,</mo><mover accent="true"><mi>f</mi><mo>^</mo></mover></mrow></msub><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><msup><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716843&type= 67.81800079 10.49866676 ，其中： <math id="M162"><msub><mrow><mi>μ</mi></mrow><mrow><mi>f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716846&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716845&type= 2.87866688 4.57200003 是原图的灰度平均值， <math id="M163"><msub><mrow><mi>μ</mi></mrow><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716878&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716847&type= 2.87866688 4.57200003 是分割图像的灰度平均值， <math id="M164"><msub><mrow><mi>σ</mi></mrow><mrow><mi>f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716879&type= 2.45533323 3.72533321 是原图的标准差， <math id="M165"><msub><mrow><mi>σ</mi></mrow><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716882&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716881&type= 2.45533323 3.72533321 是分割图像的标准差， <math id="M166"><msub><mrow><mi>σ</mi></mrow><mrow><mi>f</mi><mo>,</mo><mover accent="true"><mi>f</mi><mo>^</mo></mover></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716884&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716883&type= 3.97933316 3.72533321 为原图像和分割后图像的协方差， <math id="M167"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716886&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716885&type= 3.72533321 3.72533321 和 <math id="M168"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716887&type= 3.72533321 3.72533321 是常量，避免系统 <math id="M169"><msup><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≈</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716889&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716859&type= 20.23533440 4.82600021 时不稳定，本文取值 <math id="M170"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716890&type= 16.34066582 3.72533321 6.45。（3）FSIM比较分割前后图像的特征差别，通过相位一致性（PC）提取局部结构中的稳定特征，用梯度特征（GM）来刻画方向。 <math id="M171"><mi mathvariant="normal">F</mi><mi mathvariant="normal">S</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">M</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716892&type= 22.09799957 3.55599999 ，FSIM越高，分割图像的质量越好。 <math display="block" id="M172"><mi>F</mi><mi>S</mi><mi>I</mi><mi>M</mi><mo>=</mo><mfrac><mrow><mo>∑</mo><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><mo>×</mo><mi>P</mi><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></mrow><mrow><mo>∑</mo><mi>P</mi><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716895&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716894&type= 33.10466385 8.38199997 ，其中： <math id="M173"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716897&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716896&type= 3.64066648 3.72533321 是相似性分数， <math id="M174"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">C</mi></mrow></msub><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716899&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716898&type= 24.12999916 4.48733330 ， <math id="M175"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">C</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi>P</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>P</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>P</mi><msubsup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mi>P</mi><msubsup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716860&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716900&type= 36.66066742 8.63599968 ， <math id="M176"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>G</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msubsup><mrow><mi>G</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716862&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716861&type= 31.32666588 8.63599968 ， <math id="M177"><mi>G</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716909&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716908&type= 2.87866688 2.79399991 是图像梯度， <math id="M178"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716911&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716910&type= 3.72533321 3.72533321 和 <math id="M179"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716913&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716912&type= 3.72533321 3.72533321 为常数， <math id="M180"><mi>P</mi><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo stretchy="false">(</mo><mi>P</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><mi>P</mi><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716915&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716914&type= 36.83000183 4.48733330 ，本文取值 <math id="M181"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0.85</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716917&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716916&type= 16.17133331 3.72533321 ， <math id="M182"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">160</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716919&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716918&type= 14.13933277 3.72533321 。（4） T 是平均分割一张图像的时间，用于衡量算法效率，判断能否满足工程需求。相关统计数据见表3 表3 ~ 表6 表6 ，其中粗体是每组LSSA与SSA之间更优的参数值。 2 图像 2 K 2 Renyi Entropy 2 Symmetric Cross Entropy 2 Kapur Entropy 2 FCM LSSA SSA LSSA SSA LSSA SSA 3 No.1 2 13.0 13.1 12.0 12.0 12.4 12.4 12.8 4 17.9 17.2 15.8 15.6 17.3 16.4 12.3 6 24.4 22.7 17.3 15.4 19.2 19.4 8.46 3 No 2 2 13.3 13.2 12.4 12.4 13.1 13.1 11.3 4 17.6 17.0 15.8 14.2 16.9 16.5 10.3 6 20.4 15.7 17.4 16.4 19.2 16.8 7.51 3 No.3 2 13.5 13.5 11.5 11.5 12.6 12.6 12.4 4 18.3 17.6 14.1 13.7 17.6 16.5 11.0 6 22.9 21.6 14.7 14.7 21.3 17.0 6.41 3 No.4 2 13.0 13.0 13.6 13.5 12.6 12.6 12.0 4 18.2 17.9 17.4 17.4 17.3 17.1 10.7 6 21.5 16.8 19.0 18.7 20.2 19.0 7.69 2 优值个数 9 1 7 0 7 1 — 2 图像 2 K 2 Renyi Entropy 2 Symmetric Cross Entropy 2 Kapur Entropy 2 FCM LSSA SSA LSSA SSA LSSA SSA 3 No.1 2 3.29×10 -1 3.31 ×10 -1 3.15×10 -1 3.15×10 -1 3.20×10 -1 3.20×10 -1 3.66×10 -1 4 5.19 × 10 -1 5.11×10 -1 5.14 × 10 -1 5.09×10 -1 5.20 × 10 -1 5.12×10 -1 2.39×10 -1 6 6.21 × 10 -1 5.95×10 -1 6.35 × 10 -1 5.97×10 -1 6.32 × 10 -1 6.11×10 -1 2.68×10 -2 3 No 2 2 3.72 × 10 -1 3.70×10 -1 3.61×10 -1 3.62 × 10 -1 3.68×10 -1 3.68×10 -1 1.24×10 -1 4 5.61 × 10 -1 5.56×10 -1 5.56 × 10 -1 5.30×10 -1 5.59 × 10 -1 5.56×10 -1 3.81×10 -1 6 6.67 × 10 -1 6.03×10 -1 6.63 × 10 -1 6.27×10 -1 6.69 × 10 -1 6.36×10 -1 2.28×10 -2 3 No.3 2 3.52×10 -1 3.52×10 -1 3.31×10 -1 3.31×10 -1 3.48×10 -1 3.48×10 -1 6.17×10 -1 4 5.48×10 -1 5.49 × 10 -1 5.40 × 10 -1 5.32×10 -1 5.50 × 10 -1 5.38×10 -1 2.29×10 -2 6 6.78 × 10 -1 6.74×10 -1 6.47 × 10 -1 6.21×10 -1 6.74 × 10 -1 6.46×10 -1 3.47×10 -3 3 No.4 2 3.06×10 -1 3.06×10 -1 3.21 × 10 -1 3.18×10 -1 2.96×10 -1 2.96×10 -1 2.84×10 -1 4 5.18 × 10 -1 5.09×10 -1 5.10 × 10 -1 5.02×10 -1 5.09 × 10 -1 5.04×10 -1 1.46×10 -1 6 6.29 × 10 -1 6.02×10 -1 6.26 × 10 -1 6.18×10 -1 6.29 × 10 -1 6.14×10 -1 1.11×10 -2 2 优值个数 8 2 9 1 8 0 — 2 图像 2 K 2 Renyi Entropy 2 Symmetric Cross Entropy 2 Kapur Entropy 2 FCM LSSA SSA LSSA SSA LSSA SSA 3 No.1 2 7.40×10 -1 7.41 × 10 -1 7.32×10 -1 7.32×10 -1 7.32×10 -1 7.32×10 -1 7.67×10 -1 4 8.81 × 10 -1 8.77×10 -1 8.76 × 10 -1 8.66×10 -1 8.81 × 10 -1 8.69×10 -1 7.08×10 -1 6 9.33 × 10 -1 9.09×10 -1 9.27 × 10 -1 8.96×10 -1 9.24 × 10 -1 9.12×10 -1 6.40×10 -1 3 No 2 2 7.59 × 10 -1 7.57×10 -1 7.49×10 -1 7.50 × 10 -1 7.52×10 -1 7.53 × 10 -1 7.66×10 -1 4 8.92 × 10 -1 8.84×10 -1 8.83 × 10 -1 8.61×10 -1 8.87 × 10 -1 8.80×10 -1 7.64×10 -1 6 9.32 × 10 -1 8.87×10 -1 9.19 × 10 -1 8.89×10 -1 9.33 × 10 -1 9.17×10 -1 6.90×10 -1 3 No.3 2 7.55×10 -1 7.55×10 -1 7.29×10 -1 7.29×10 -1 7.44×10 -1 7.44×10 -1 7.05×10 -1 4 8.91 × 10 -1 8.84×10 -1 8.77 × 10 -1 8.69×10 -1 8.90 × 10 -1 8.76×10 -1 6.41×10 -1 6 9.46 × 10 -1 9.22×10 -1 9.16 × 10 -1 8.92×10 -1 9.41 × 10 -1 9.11×10 -1 6.05×10 -1 3 No.4 2 7.21×10 -1 7.21×10 -1 7.34 × 10 -1 7.33×10 -1 7.12 × 10 -1 7.11×10 -1 6.93×10 -1 4 8.71 × 10 -1 8.64×10 -1 8.66 × 10 -1 8.61×10 -1 8.65 × 10 -1 8.58×10 -1 6.80×10 -1 6 9.28 × 10 -1 8.87×10 -1 9.21 × 10 -1 9.11×10 -1 9.25 × 10 -1 9.02×10 -1 6.72×10 -1 2 优值个数 9 1 9 1 9 1 — 2 K 2 Renyi Entropy 2 Symmetric Cross Entropy 2 Kapur Entropy 2 FCM LSSA SSA LSSA SSA LSSA SSA 2 1.421 s 1.422 s 1.498 s 1.498 s 1.440 s 1.440 s 1.667 s 4 1.461 s 1.453 s 1.546 s 1.564 s 1.496 s 1.476 s 4.386 s 6 1.472 s 1.491 s 1.618 s 1.603 s 1.507 s 1.489 s 8.602 s 优值个数 2 1 1 1 0 2 — 由表3 表3 ~ 表6 表6 可知，LSSA-MET对比SSA-MET，更高的PSNR值的比例为23∶2，SSIM的比例为25∶3，FSIM的比例为27∶3， T 的比例为3∶4。这意味着LSSA优化MET的能力在骨料图像分割上有精度优势，而且随着阈值个数的增多，优势越大，这表明LSSA-MET可以分割出更多的细节。与FCM相比，LSSA-MET随阈值个数的增加运行时间不再成倍增长，即使 <math id="M183"><mi>K</mi><mo>=</mo><mn mathvariant="normal">6</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716931&type= 9.56733322 2.79399991 时，平均分割一张图片只需1.532 s，满足工程需求。从阈值角度来看，当 <math id="M184"><mi>K</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716928&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716927&type= 9.56733322 2.79399991 时，FCM的评估参数值最高，但受到聚类中心与邻域像素灰度值的影响，FCM在 <math id="M185"><mi>K</mi><mo>=</mo><mn mathvariant="normal">4,6</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716905&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716904&type= 13.29266548 3.21733332 时，表现愈发不佳。相反的，LSSA-Renyi熵在4个参数上的表现则一直稳定且突出，PSNR，SSIM，FSIM分别提升了29.92%，10.67%和5.16%。其次是Kapur熵，对称交叉熵的结果稍差。对比Renyi熵、对称交叉熵和Kapur熵，表7 表7 是LSSA与3种熵结合的组合算法在4个参数上的统计平均值，包含 <math id="M186"><mi>K</mi><mo>=</mo><mn mathvariant="normal">2,4</mn><mo>,</mo><mn mathvariant="normal">6</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716907&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716906&type= 16.93333435 3.21733332 的所有样本数据。其中最优值用粗体表示，结果表明LSSA-Renyi熵更适合骨料图像的分割。算法 PSNR SSIM FSIM T /s LSSA-Renyi熵 1.78 ×10 5.08 × 10 -1 8.54 × 10 -1 1.451 LSSA-对称交叉熵 1.51×10 5.02×10 -1 8.44×10 -1 1.554 LSSA-Kapur熵 1.66×10 5.06×10 -1 8.49×10 -1 1.481 4 结论 1 本文提出了一种基于混沌SSA的多重熵阈值分割模型LSSA-MET，旨在克服MET在多类分割骨料图像时运行时间长、精度低的缺点。首先，引入Logistic混沌映射扰动种群的初始位置，使其变得更均匀随机，克服了原SSA的麻雀聚集不利于搜索全局最优解的情况；然后，利用一种扩张参数来改善麻雀急剧收敛的问题，有利于更广泛的全局搜索；最后，提出了一种跳出局部最优解的方法，即控距精英变异，可以极大概率地跳出局部避免种群在迭代中的过度消耗。这3种策略全面地提高了SSA的优化性能，称之为LSSA。基准函数实验验证了LSSA的优化能力。 LSSA用于快速确定MET的阈值。由于骨料图像在波谷处平缓且有非常多的极值点，LSSA善于搜索到更精确的阈值，而且具有较高的稳定性和鲁棒性。同时，LSSA-MET具有较高的效率，随阈值个数 <math id="M187"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716930&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716929&type= 3.04800010 2.79399991 的增加，运行时间不再成倍增长， <math id="M188"><mi>K</mi><mo>=</mo><mn mathvariant="normal">6</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45716931&type= 9.56733322 2.79399991 时的平均耗时只有1.523 s。在3种MET中，Renyi熵的PSNR，SSIM，FSIM与 T 值普遍最优，前3个评估参数分别提高了29.92%，10.67%和5.16%，完整地保留了骨料的表面纹理、形状和尺寸等几何特征，实现了精度与速度的最佳平衡。参考文献 ZHAO Y 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DOI：10.37188/OPE.20233113.1973