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平稳小波域多尺度乘积下的指数Renyi熵自动阈值选择方法
信息科学 | 更新时间：2023-07-17
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平稳小波域多尺度乘积下的指数Renyi熵自动阈值选择方法
Automatic threshold selection method using exponential Renyi entropy under multi-scale product in stationary wavelet domain
- 光学精密工程
- 作者机构：
1.湖北省水电工程智能视觉监测重点实验室（三峡大学），湖北 宜昌 443002
2.三峡大学 大数据研究中心，湖北 宜昌 443002
3.三峡大学 计算机与信息学院，湖北 宜昌 443002
- 作者简介：
[ "邹耀斌（1978-），男，江西鹰潭人，博士，2001年于中南大学获得学士学位，2007年于武汉理工大学获得硕士学位，2011年于华中科技大学获得博士学位，主要从事数字图像处理、机器学习等方面的研究。E-mail： zyb@ctgu.edu.cn" ]
邹耀斌（1978-），男，江西鹰潭人，博士，2001年于中南大学获得学士学位，2007年于武汉理工大学获得硕士学位，2011年于华中科技大学获得博士学位，主要从事数字图像处理、机器学习等方面的研究。E-mail： zyb@ctgu.edu.cn
孟祥丹（1996-），男，湖北十堰人，硕士研究生，2019年于三峡大学科技学院获得学士学位，主要从事数字图像处理方面的研究。E-mail： mxd19960111@163.com
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- 基金信息：国家自然科学基金资助项目(61871258)
DOI：10.37188/OPE.20233112.1841
中图分类号： TP391纸质出版日期：2023-06-25，
收稿日期：2022-07-18，
修回日期：2022-09-22，
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引用本文
邹耀斌,孟祥丹,孙水发等.平稳小波域多尺度乘积下的指数Renyi熵自动阈值选择方法[J].光学精密工程,2023,31(12):1841-1858.
ZOU Yaobin,MENG Xiangdan,SUN Shuifa,et al.Automatic threshold selection method using exponential Renyi entropy under multi-scale product in stationary wavelet domain[J].Optics and Precision Engineering,2023,31(12):1841-1858.
邹耀斌,孟祥丹,孙水发等.平稳小波域多尺度乘积下的指数Renyi熵自动阈值选择方法[J].光学精密工程,2023,31(12):1841-1858. DOI： 10.37188/OPE.20233112.1841.
ZOU Yaobin,MENG Xiangdan,SUN Shuifa,et al.Automatic threshold selection method using exponential Renyi entropy under multi-scale product in stationary wavelet domain[J].Optics and Precision Engineering,2023,31(12):1841-1858. DOI： 10.37188/OPE.20233112.1841.
平稳小波域多尺度乘积下的指数Renyi熵自动阈值选择方法
平稳小波域多尺度乘积下的指数Renyi熵自动阈值选择方法 Automatic threshold selection method using exponential Renyi entropy under multi-scale product in stationary wavelet domain 1 2 3 corresp first-author E-mail： zyb@ctgu.edu.cn E-mail： zyb@ctgu.edu.cn 邹耀斌（1978-），男，江西鹰潭人，博士，2001年于中南大学获得学士学位，2007年于武汉理工大学获得硕士学位，2011年于华中科技大学获得博士学位，主要从事数字图像处理、机器学习等方面的研究。E-mail： zyb@ctgu.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45717916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45717910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45717918&type= 22.01332855 32.00399780 邹耀斌 （1978-），男，江西鹰潭人，博士，2001年于中南大学获得学士学位，2007年于武汉理工大学获得硕士学位，2011年于华中科技大学获得博士学位，主要从事数字图像处理、机器学习等方面的研究。E-mail： zyb@ctgu.edu.cn zyb@ctgu.edu.cn 2 孟祥丹（1996-），男，湖北十堰人，硕士研究生，2019年于三峡大学科技学院获得学士学位，主要从事数字图像处理方面的研究。E-mail： mxd19960111@163.com 孟祥丹（1996-），男，湖北十堰人，硕士研究生，2019年于三峡大学科技学院获得学士学位，主要从事数字图像处理方面的研究。E-mail： mxd19960111@163.com mxd19960111@163.com 1 3 1 2 3 湖北省水电工程智能视觉监测重点实验室（三峡大学），湖北 宜昌 443002 湖北省水电工程智能视觉监测重点实验室（三峡大学），湖北 宜昌 443002 Hubei Key Laboratory of Intelligent Vision Based Monitoring for Hydroelectric Engineering （China Three Gorges University）， Yichang 443002， China Hubei Key Laboratory of Intelligent Vision Based Monitoring for Hydroelectric Engineering （China Three Gorges University）， Yichang 443002， China 三峡大学 大数据研究中心，湖北 宜昌 443002 三峡大学 大数据研究中心，湖北 宜昌 443002 Center for Big Data， China Three Gorges University， Yichang 443002， China Center for Big Data， China Three Gorges University， Yichang 443002， China 三峡大学 计算机与信息学院，湖北 宜昌 443002 三峡大学 计算机与信息学院，湖北 宜昌 443002 College of Computer and Information Technology， China Three Gorges University， Yichang 443002， China College of Computer and Information Technology， China Three Gorges University， Yichang 443002， China 受目标和背景的大小比例、噪声或者随机细节等因素影响，灰度图像的灰度直方图可能呈现出无峰、单峰、双峰或者多峰模式。为了在统一框架内处理这4种不同直方图模式下的自动阈值选择问题，提出了一种平稳小波域多尺度乘积下的指数Renyi熵自动阈值选择方法。该方法首先利用平稳小波对原始图像进行水平、垂直和对角方向的多尺度变换，再对各个方向的高频子带进行多尺度乘积运算构造出融合图像；然后通过内外轮廓图像对融合图像进行采样重构一维直方图；最后计算重构直方图所对应的指数Renyi熵，并取最大值对应的阈值作为最终分割阈值。提出的方法与4种自动阈值分割方法、2种聚类分割方法以及2种活动轮廓分割方法进行了比较。在16幅合成图像和50幅真实世界图像上的实验结果表明：在合成图像集中，相比于分割精度第2的方法，平均马修斯相关系数提高了41.2%；在真实世界图像集中，相比于分割精度第二的方法，平均马修斯相关系数提高了20.8%。提出的方法具有更稳健的分割适应性，且在分割精度方面明显优于其他8个分割方法。 When a gray-level image is affected by different factors， such as the size ratio of the target to the background， noise， or random details， its gray-level histogram exhibits peakless， unimodal， bimodal， or multimodal patterns. To deal with the issue of automatic threshold selection in these four situations within a unified framework， an automatic threshold selection method using the exponential Rényi entropy under the multi-scale product in the stationary wavelet domain is proposed. First， stationary wavelet multi-scale transformation is applied to the original gray-level image in the horizontal， vertical， and diagonal directions， and a fused image is constructed via the multi-scale multiplication of high-frequency sub-bands in each direction. Then， the fused image is sampled by the inner and outer contour image to construct a one-dimensional gray-level histogram. Finally， the exponential Rényi entropy corresponding to the constructed histogram is calculated， and the threshold corresponding to the maximum exponential Rényi entropy is taken as the final threshold. The proposed method was compared with four automatic threshold segmentation methods， two clustering segmentation methods， and two active contour segmentation methods. The experimental results for 16 synthetic images and 50 real-world images indicated that with regard to the segmentation accuracy， the proposed method outperformed the second-best method by 41.2% and 20.8% in terms of the average Matthews correlation coefficient for the synthetic and real-world images， respectively. Although the proposed method has no advantage with regard to computational efficiency， it has more robust segmentation adaptability and a higher segmentation accuracy than the other eight segmentation methods. 阈值分割 指数Renyi熵 平稳小波域 多尺度乘积 image thresholding exponential Renyi entropy stationary wavelet domain multi-scale product 1 引 言 1 阈值分割是图像分割中最早被研究和使用的一类方法，其具有物理意义明确、易于实现、分割效率高等特点 ［ 1 1 ］ 。阈值分割在诸如工业无损检测 ［ 2 2 ］ 、舰船目标检测 ［ 3 3 ］ 、行人视频监控 ［ 4 4 ］ 、生物细胞分析 ［ 5 5 ］ 等实际应用领域里仍然是最佳的候选方法之一。阈值分割将待分割图像中的每个像素划分为目标或者背景，其划分的基本原则是比较像素灰度值和分割阈值的大小。分割阈值的选择由此成为精确阈值分割的关键步骤，而让所选阈值尽可能地接近最优阈值则是阈值分割的核心目标。 在众多阈值分割方法中，基于熵的阈值分割方法颇具代表性 ［ 6 6 ］ 。罗希平等人从信息论角度分析了熵阈值分割法，认为这类方法所选分割阈值能获得更大的信息量 ［ 7 7 ］ 。Chang等人不但指出熵阈值分割法与信息论的紧密关系，并对使用不同类型熵所设计的目标函数进行了比较分析 ［ 8 8 ］ 。熵阈值分割法由于得到信息论的相关理论支撑，且相对易于实现而长期受到国内外研究人员的关注 ［ 6 6 ］ 。早期已经提出了许多基于熵概念的分割方法，比如Pun提出的最大后验熵上界法 ［ 9 9 ］ ，Kapur等人提出的基于直方图形状的一维熵阈值法 ［ 10 10 ］ ，Abutaleb等人提出的基于局部信息的二维熵阈值法 ［ 11 11 ］ 。在早期的不少熵阈值方法中，熵计算模型主要采用了Shannon熵 ［ 12 12 ］ 。Shannon熵虽然具有可加性，但它忽略了子系统间的相互作用，这限制了其对概率分布的表达能力。Tsallis熵 ［ 13 13 ］ 和Renyi熵 ［ 14 14 ］ 则考虑了子系统间的相互作用，并且具有伪可加性。Renyi熵和Tsallis熵均是Shannon熵的广义形式，并且Renyi熵和Tsallis熵不仅在形式上存在转换关系，在图像分割中也具有特殊的等价关系 ［ 15 15 ］ 。 在基于Renyi熵的图像分割方法中，Sahoo等人先后提出一维Renyi熵方法 ［ 14 14 ］ 和二维Renyi熵方法 ［ 16 16 ］ 。与一维Renyi熵方法相比，二维Renyi熵方法综合了灰度分布的空间特征，但增加了计算复杂度。文献［ 17 17 ］也提出了一种基于二维窗口的组合熵阈值分割方法，该方法将Renyi熵和Tsallis熵线性组合，并利用遗传算法进行优化求解。该方法对于低照度图像具有较好的分割效果，但因其窗口数量和大小为固定值，方法的分割适应性不够稳健。 新近的一些研究进一步考虑将Renyi熵与其它理论或方法相结合。Abdel-Khalek等人提出了一种基于Renyi熵的二维图像分割方法 ［ 18 18 ］ ，该方法使用Renyi熵来测量图像二维直方图所包含的信息量，并利用遗传算法来最大化Renyi熵，以便从图像中分隔开目标和背景。在最大化Renyi熵分割的基础上，刘松涛等人结合模糊隶属度函数与模糊Renyi熵理论，提出了基于模糊Renyi熵和区域增长的分割方法 ［ 19 19 ］ ，但因其目标数量无法自适应确定，难以有效分割出全部目标，且存在像素距离误差较高的不足。Wang等人提出了基于局部Renyi熵的显著性水平集图像分割方法 ［ 20 20 ］ ，该方法通过显著性滤波算法提取原始图像的显著图。He等人构建了一种由灰度和局部熵组成的二维直方图，通过引入局部熵来描述像素之间的空间信息，并采用Renyi熵确定阈值 ［ 21 21 ］ 。Lei等人提出了一种嵌套优化的自适应粒度Renyi粗糙熵阈值分割方法 ［ 22 22 ］ ，该方法通过最小化Renyi粗糙熵来确定图像分割的最优阈值。 受噪声或随机细节，目标和背景的大小比例等多种因素影响，图像的灰度直方图可能呈现出无峰、单峰、双峰或者多峰模式。其次，构成每种直方图模式的基本分布除了高斯分布外，还有极值分布、伽玛分布、瑞利分布、贝塔分布或均匀分布等。当图像的灰度分布由一种或者多种分布组合而成时，从这些图像中自动选择出合理分割阈值将成为一项颇具挑战性的任务。 为了能在统一的方法框架内从上述不同直方图模式的图像中自动选择出合理的分割阈值，本文提出的解决思路是将不同直方图模式转换为一种统一的模式，然后在统一直方图模式的基础上设计稳健的自动阈值选择目标函数。按照该思路提出了一种平稳小波域多尺度乘积下的指数Renyi熵自动阈值选择方法（Exponential Renyi Entropy Maximization，EREM）。EREM方法应用平稳小波对灰度图像进行水平、垂直和对角三个方向的多尺度变换；再在三个方向的多尺度变换图像上分别进行多尺度乘积运算，然后将三个方向的多尺度乘积运算结果图像融合成一幅图像；最后在融合图像上定义指数Renyi熵目标函数，并通过最大化该目标函数选取最终分割阈值。 2 平稳小波变换的多尺度乘积效应 1 平稳小波变换在每次分解过程中不再进行下采样操作，具备良好的平移不变性 ［ 23 23 ］ 。平稳小波变换也是小波变换的一种快速实现，能够提供一组时域和频域带宽均较窄的线性滤波器 ［ 24 24 ］ 。使用平稳小波变换对灰度图像进行分解时，其具备的非抽样特性使得各分解层上高频子带的尺寸都与原始图像的尺寸保持一致。给定一幅大小为 <math id="M1"><mi>M</mi><mo>×</mo><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45717947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45717932&type= 11.59933376 2.87866688 的灰度图像 <math id="M2"><mi>f</mi></math> 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28.02466965 3.21733332 且 <math id="M6"><mi>n</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45717994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45717979&type= 26.50066566 3.21733332 。 <math id="M7"><msubsup><mrow><mi>ϕ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45717999&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45717985&type= 3.47133350 4.82600021 和 <math id="M8"><msubsup><mrow><mi>ϕ</mi></mrow><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718007&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718018&type= 3.30200005 4.82600021 是低通滤波器， <math id="M9"><msubsup><mrow><mi>ζ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718025&type= 2.79399991 4.82600021 和 <math id="M10"><msubsup><mrow><mi>ζ</mi></mrow><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718016&type= 2.70933342 4.82600021 是高通滤波器， <math id="M11"><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718041&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718039&type= 9.05933285 4.82600021 是滤波器的大小。 <math id="M12"><msup><mrow><mi>L</mi></mrow><mrow><mi>j</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718053&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718043&type= 3.13266683 3.04800010 和 <math id="M13"><msup><mrow><mi>L</mi></mrow><mrow><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718056&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718054&type= 6.51933336 3.04800010 分别表示第 <math id="M14"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718069&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718059&type= 1.60866666 3.64066648 层和第 <math id="M15"><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718102&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718100&type= 8.12800026 3.64066648 层的低频子带。 <math id="M16"><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718083&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718074&type= 7.53533268 3.97933316 ， <math id="M17"><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718091&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718076&type= 7.53533268 3.97933316 和 <math id="M18"><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>d</mi></mrow><mrow><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718108&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718095&type= 7.53533268 3.97933316 分别表示第 <math id="M19"><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718102&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718100&type= 8.12800026 3.64066648 层的水平、垂直和对角方向的高频子带。 在平稳小波变换中小波函数的选择与其自身的数学特性密切相关，选取不同的小波函数将产生不同的分解效果。平稳小波变换中常用的小波函数有Haar小波、Symlets小波、Daubechies小波、Fejer-Korovkin小波和Coiflets小波等，考虑到线性相位利于后续多尺度乘积运算，选用具有对称性且有限支撑的Haar小波，其相对优势将通过第3节的比较分析去证实。 对于待分割处理的灰度图像 <math id="M20"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719514&type= 2.45533323 3.64066648 ，其灰度直方图可能呈现出无峰、单峰、双峰或者多峰模式。从二维灰度图像 <math id="M21"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719514&type= 2.45533323 3.64066648 到一维灰度直方图的转化将导致像素位置信息的丢失，从而难以直接从上述4种不同直方图中获得统一的阈值选取线索。如果希望在统一的方法框架内处理不同直方图模式，除了从待分割图像 <math id="M22"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719514&type= 2.45533323 3.64066648 中获取关联像素间的相对位置信息（参考第5节的步骤4），还需要将不同直方图模式转换为某种统一模式。考虑到单峰模式相对于双峰、多峰和无峰模式更为简单，并且是构成双峰、多峰或无峰模式的基础，将不同直方图模式转换为统一的单峰模式就成为优先候选方案。在技术实现上，本文提出利用平稳小波变换相同方向不同尺度下的高频子带进行多尺度乘积变换（Multi-scale Product Transformation，MPT）进行这种统一转换。 为了便于后续的多尺度乘积效应分析，将平稳小波变换作用在灰度图像 <math id="M23"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718140&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718135&type= 2.45533323 3.64066648 上产生的高频子带记为 <math id="M24"><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>j</mi><mo>∈</mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>ω</mi></mrow></mfenced><mo>,</mo><mi>θ</mi><mo>∈</mo><mfenced open="{" close="}" separators="|"><mrow><mi>h</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>d</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718144&type= 55.37200165 4.74133301 ，其中 <math id="M25"><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718198&type= 2.53999996 2.79399991 代表小波分解的层数， <math id="M26"><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718159&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718171&type= 1.94733346 2.79399991 、 <math id="M27"><mi>v</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718163&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718180&type= 1.86266661 2.79399991 和 <math id="M28"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718188&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718186&type= 2.28600001 2.79399991 分别代表水平方向、垂直方向和对角方向。给定 <math id="M29"><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718198&type= 2.53999996 2.79399991 个方向均为 <math id="M30"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718228&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718221&type= 1.86266661 2.79399991 但分解层数不同的高频子带 <math id="M31"><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>j</mi><mo>∈</mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>ω</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718215&type= 34.96733475 4.74133301 ，将 式（2） 式（2） 表示的运算称为在方向 <math id="M32"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718228&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718221&type= 1.86266661 2.79399991 上的多尺度乘积变换，所得图像 <math id="M33"><msub><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718233&type= 3.47133350 3.72533321 称为MPT图像： <math display="block" id="M34"><msub><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>ω</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msubsup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718242&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718239&type= 17.44133186 8.97466660 . 在小波函数选择具有对称性且有限支撑的Haar小波前提下，相同方向不同分解层次的高频子带图像在边缘信号和噪声（或随机细节）方面有不同的特点：随着分解层次的逐渐增大，噪声（或随机细节）在高频子带图像中的响应灰度值衰减较快，而边缘信号在高频子带图像中的响应灰度值则呈现如下特征：边缘中心位置的响应灰度值相对保持良好，并且随着距离中心位置越来越远，响应值逐渐衰减。 基于相同方向不同分解层次的高频子带图像在边缘信号和噪声（或随机细节）方面的上述不同特点，在MPT图像 <math id="M35"><msub><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718248&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718243&type= 3.47133350 3.72533321 的灰度值被规范化到［0，255］的前提下，如果逐渐增大 <math id="M36"><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718406&type= 2.53999996 2.79399991 值，那么灰度图像 <math id="M37"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719514&type= 2.45533323 3.64066648 中的噪声或随机细节的MPT响应结果将逐渐向灰度最小值0趋近，而目标和背景之间的边缘信号的MPT响应结果将散布在0~255之间（参考 图1 图1 中的第2行图像）。结果是MPT图像 <math id="M38"><msub><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718765&type= 3.47133350 3.72533321 的灰度直方图呈现出一种特殊的分布模式，即灰度直方图的众数趋向于灰度值0（参考 图1 图1 中的第3行直方图）。图像灰度直方图的众数趋向于最小灰度值0，而其他灰度值散布在0到255之间，从概率分布的角度而言，这种灰度直方图是一种“单峰右偏灰度直方图”。这里将MPT图像的上述特征表现称为平稳小波变换的多尺度乘积效应。从前述分析可知，MPT技术和多尺度乘积效应主要具有三个作用：（1）适当凸显目标和背景间的差异性；（2）相对抑制目标和背景中的噪声或随机细节；（3）将不同图像 <math id="M39"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719514&type= 2.45533323 3.64066648 可能具有的多样化直方图转化为统一的单峰右偏直方图。 在多尺度乘积变换过程中，如果 <math id="M44"><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718406&type= 2.53999996 2.79399991 值较小将导致MPT图像 <math id="M45"><msub><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718765&type= 3.47133350 3.72533321 在保留边缘信息的同时仍含有大量噪声或者随机细节（参考 图1 图1 中的第2行图像（b）），而 <math id="M46"><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718406&type= 2.53999996 2.79399991 值逐渐变大将使MPT图像 <math id="M47"><msub><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718765&type= 3.47133350 3.72533321 在抑制噪声或者随机细节的同时逐渐弱化边缘（参考 图1 图1 中的第2行图像（c）~（f）），最极端的情况是MPT图像 <math id="M48"><msub><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718399&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718384&type= 3.47133350 3.72533321 的灰度直方图退化为0-1分布。MPT图像 <math id="M49"><msub><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718765&type= 3.47133350 3.72533321 的灰度直方图模式与 <math id="M50"><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718406&type= 2.53999996 2.79399991 值之间的动态关系表明：需要寻找出一个相对恰当的 <math id="M51"><msub><mrow><mi>ω</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718750&type= 3.64066648 3.72533321 值，使得MPT图像 <math id="M52"><msub><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718765&type= 3.47133350 3.72533321 既保留必要的边缘信息，同时抑制噪声或随机细节。接下来的第3节将引出指数Renyi熵差的概念，并分析其重要性质，以便将其用于合理 <math id="M53"><msub><mrow><mi>ω</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718750&type= 3.64066648 3.72533321 值的自动计算。 3 指数Renyi熵差及其最大化导向的 <math id="M54"><msub><mrow><mi>ω</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718436&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718435&type= 4.82600021 4.91066647 值计算 1 对任意给定的一个灰度直方图，可用一个灰度 <math id="M55"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718438&type= 1.26999998 2.79399991 将该直方图分为一左一右两个部分（参考 图2 图2 ）。假设左部直方图共有 <math id="M56"><msub><mrow><mi>ψ</mi></mrow><mrow><mo>-</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718445&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718454&type= 3.80999994 4.48733330 个灰度级，总的频数为 <math id="M57"><msub><mrow><mi>μ</mi></mrow><mrow><mo>-</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718698&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718697&type= 3.80999994 4.57200003 ，灰度级的离散概率分布为 <math id="M58"><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718452&type= 22.35199928 4.57200003 且 <math id="M59"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></munderover></mstyle><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718461&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718469&type= 13.97000027 8.89000034 ；假设右部直方图共有 <math id="M60"><msub><mrow><mi>ψ</mi></mrow><mrow><mo>+</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718465&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718463&type= 3.89466691 4.48733330 个灰度级，总的频数为 <math id="M61"><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718700&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718699&type= 3.89466691 4.57200003 ，灰度级的离散概率分布为 <math id="M62"><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718502&type= 22.86000061 4.57200003 且 <math id="M63"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718495&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718492&type= 13.97000027 8.89000034 。 level histogram 令 <math id="M64"><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718526&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718524&type= 3.89466691 3.72533321 和 <math id="M65"><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718544&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718540&type= 3.97933316 3.72533321 分别表示左部直方图和右部直方图对应的 <math id="M66"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719773&type= 1.94733346 2.79399991 阶Renyi熵。根据Renyi熵的定义 ［ 14 14 ］ ， <math id="M67"><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718526&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718524&type= 3.89466691 3.72533321 和 <math id="M68"><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718544&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718540&type= 3.97933316 3.72533321 的计算式分别为： <math display="block" id="M69"><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub><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><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></munderover></mstyle><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718547&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718545&type= 34.29000092 8.89000034 ， <math display="block" id="M70"><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub><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><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></munderover></mstyle><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718561&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718560&type= 34.79800034 8.89000034 . 用 <math id="M71"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718975&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718972&type= 2.20133328 2.79399991 表示左部和右部总的Renyi熵，其计算式如 式（5） 式（5） 所示： <math display="block" id="M72"><mi>S</mi><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><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mo>-</mo></mrow></msub><mo>+</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></munderover></mstyle><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>o</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718552&type= 35.22133255 8.97466660 ， 其中： <math display="block" id="M73"><msub><mrow><mi>o</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mo>-</mo></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></mfrac><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo></mtd><mtd><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><mfrac><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mo>-</mo></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></mfrac><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo></mtd><mtd><mfenced separators="|"><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mo>-</mo></mrow></msub><mo>+</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mo>-</mo></mrow></msub><mo>+</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718555&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718554&type= 68.74933624 20.57399940 . 对式（3）~ 式（5） 式（5） 左右两边进行以自然常数 <math id="M74"><mi>e</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718556&type= 1.52400005 2.79399991 为底的指数函数变换，变换后的等式依次为： <math display="block" id="M75"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></munderover></mstyle><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718563&type= 27.17799950 8.89000034 ， <math display="block" id="M76"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></munderover></mstyle><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718565&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718577&type= 27.68600082 8.89000034 ， <math display="block" id="M77"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mo>-</mo></mrow></msub><mo>+</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></munderover></mstyle><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>o</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718566&type= 28.70199966 8.97466660 . 定义 <math id="M78"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 为指数Renyi熵差，其中 <math id="M79"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718701&type= 10.32933331 3.21733332 表示右部指数Renyi熵， <math id="M80"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718572&type= 9.31333351 3.21733332 表示左部和右部总的指数Renyi熵。下面4个命题用于分析和论证：为了尽可能提高指数Renyi熵差 <math id="M81"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 ， 图2 图2 中的灰度直方图应具有什么样的分布特点。 命题1 <math id="M82"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>λ</mi></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>α</mi></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718584&type= 79.24800110 6.18066692 ，其中 <math id="M83"><mi>λ</mi><mo>=</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>-</mo></mrow></msub><mo>/</mo><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>-</mo></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718586&type= 28.27866554 4.82600021 。 证明 <math display="block" id="M84"><mtable><mtr><mtd><mi>S</mi><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><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mo>-</mo></mrow></msub><mo>+</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></munderover></mstyle><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>o</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup></mrow><mo>=</mo></mtd></mtr><mtr><mtd><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 open="[" close="]" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></munderover></mstyle><mrow><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mo>-</mo></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></mfrac><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></munderover></mstyle><mrow><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mo>-</mo></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></mfrac><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup></mrow></mrow></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718604&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718603&type= 75.69200134 22.77533340 将 式（10） 式（10） 进行变形可以得出： <math display="block" id="M85"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mo>=</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>α</mi></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></munderover></mstyle><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup></mrow><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>λ</mi></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></munderover></mstyle><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718606&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718605&type= 59.52066422 8.89000034 . 将 式（7） 式（7） 和 式（8） 式（8） 代入到 式（11） 式（11） 中，得： <math id="M86"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mo>=</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>α</mi></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>λ</mi></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718608&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718607&type= 55.54133606 4.74133301 从而有： <math id="M87"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>λ</mi></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfenced><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>α</mi></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718610&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718609&type= 80.60266876 6.18066692 。 命题2 当 <math id="M88"><mi>λ</mi><mo>></mo><mn mathvariant="normal">1</mn><mo>/</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>-</mo><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718672&type= 28.95599937 3.55599999 时， <math id="M89"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 是关于 <math id="M90"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718674&type= 1.77800000 2.79399991 的单调递增函数。 证明： <math id="M91"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 关于 <math id="M92"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718674&type= 1.77800000 2.79399991 的偏导数为： <math display="block" id="M93"><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>λ</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718597&type= 32.93532944 10.24466610 . 令表达 式（12） 式（12） 大于0，可得关于 <math id="M94"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718674&type= 1.77800000 2.79399991 的不等式解： <math id="M95"><mi>λ</mi><mo>></mo><mn mathvariant="normal">1</mn><mo>/</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>-</mo><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718672&type= 28.95599937 3.55599999 。根据函数导数和函数单调性的关系，可知当 <math id="M96"><mi>λ</mi><mo>></mo><mn mathvariant="normal">1</mn><mo>/</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>-</mo><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718672&type= 28.95599937 3.55599999 时， <math id="M97"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 是关于 <math id="M98"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718674&type= 1.77800000 2.79399991 的单调递增函数。 命题3 <math id="M99"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 是关于 <math id="M100"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718701&type= 10.32933331 3.21733332 的单调递增函数。 证明： <math id="M101"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 关于 <math id="M102"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718701&type= 10.32933331 3.21733332 的偏导数为 <math id="M103"><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>λ</mi></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718636&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718635&type= 19.47333336 4.74133301 ，当求最大指数Renyi熵时，熵参数需满足 <math id="M104"><mi>α</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718638&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718637&type= 8.63599968 2.79399991 ［ 14 14 ］ ，又因为 <math id="M105"><mn mathvariant="normal">0</mn><mo><</mo><mi>λ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718665&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718664&type= 15.15533447 2.79399991 ，从而 <math id="M106"><mn mathvariant="normal">0</mn><mo><</mo><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>λ</mi></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718642&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718641&type= 32.85066605 4.74133301 。根据函数导数和函数单调性的关系，可知 <math id="M107"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 是关于 <math id="M108"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718701&type= 10.32933331 3.21733332 的单调递增函数。 命题4 <math id="M109"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 是关于 <math id="M110"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718703&type= 10.32933331 3.21733332 的单调递减函数。 证明： <math id="M111"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 关于 <math id="M112"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718703&type= 10.32933331 3.21733332 的偏导数为 <math id="M113"><mo>-</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>α</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718663&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718632&type= 6.18066692 3.04800010 ，因为 <math id="M114"><mn mathvariant="normal">0</mn><mo><</mo><mi>λ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718665&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718664&type= 15.15533447 2.79399991 ，从而 <math id="M115"><mo>-</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>α</mi></mrow></msup><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718666&type= 12.86933327 3.04800010 。根据函数导数和函数单调性的关系，可知 <math id="M116"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 是关于 <math id="M117"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718703&type= 10.32933331 3.21733332 的单调递减函数。 命题1到命题4表明：在 <math id="M118"><mi>λ</mi><mo>></mo><mn mathvariant="normal">1</mn><mo>/</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>-</mo><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718672&type= 28.95599937 3.55599999 的前提下， <math id="M119"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718674&type= 1.77800000 2.79399991 值和 <math id="M120"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718701&type= 10.32933331 3.21733332 值越大，而 <math id="M121"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718703&type= 10.32933331 3.21733332 值越小将使得 <math id="M122"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 取更大值。结合 <math id="M123"><mi>λ</mi><mo>=</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>-</mo></mrow></msub><mo>/</mo><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>-</mo></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718682&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718681&type= 28.27866554 4.82600021 以及 <math id="M124"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718701&type= 10.32933331 3.21733332 和 <math id="M125"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718703&type= 10.32933331 3.21733332 的相对大小关系，容易进一步推断出：相对更大的 <math id="M126"><msub><mrow><mi>μ</mi></mrow><mrow><mo>-</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718698&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718697&type= 3.80999994 4.57200003 或相对更小的 <math id="M127"><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718700&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718699&type= 3.89466691 4.57200003 ，以及相对更大的 <math id="M128"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718701&type= 10.32933331 3.21733332 或相对更小的 <math id="M129"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718703&type= 10.32933331 3.21733332 ，将使得 <math id="M130"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 取更大值。因为 <math id="M131"><msub><mrow><mi>μ</mi></mrow><mrow><mo>-</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718698&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718697&type= 3.80999994 4.57200003 和 <math id="M132"><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718700&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718699&type= 3.89466691 4.57200003 分别代表直方图左部和右部的总频数，这意味着由相对较少的像素构成的右部直方图对应的 <math id="M133"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718701&type= 10.32933331 3.21733332 应尽可能大，而由相对较多的像素构成的左部直方图对应的 <math id="M134"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>-</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718703&type= 10.32933331 3.21733332 应尽可能小，这将使 <math id="M135"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 取更大值。通过上述分析，具有如下分布特点的灰度直方图将有助于 <math id="M136"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718707&type= 24.46866608 3.21733332 取更大值：左部直方图应包含相对多的像素，且左部直方图在更窄的灰度级范围内应尽可能集中分布；右部直方图应包含相对少的像素，且右部直方图在更广的灰度级范围内应尽可能分散分布。 基于指数Renyi熵差的上述性质，同时结合第2节平稳小波变换的多尺度乘积效应，可知：为了寻找出相对恰当的 <math id="M137"><msub><mrow><mi>ω</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718750&type= 3.64066648 3.72533321 值，使得MPT图像 <math id="M138"><msub><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718765&type= 3.47133350 3.72533321 既保留必要的边缘信息，同时抑制噪声或随机细节，可以采用最大化MPT图像 <math id="M139"><msub><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718765&type= 3.47133350 3.72533321 的灰度直方图的指数Renyi熵差，其形式化描述为： <math display="block" id="M140"><mfenced separators="|"><mrow><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>l</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></mrow></mfenced><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></munder><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0,255</mn></mrow></mfenced></mrow></munder><mfenced separators="|"><mrow><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mo>+</mo></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718716&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718715&type= 69.08799744 6.51933336 . 一旦确定了某一方向上参与MPT变换的多尺度乘积层数 <math id="M141"><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718718&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718717&type= 3.64066648 3.97933316 ，就可以结合 式（2） 式（2） 计算出对应方向的MPT图像 <math id="M142"><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718771&type= 3.89466691 3.97933316 。为了综合利用3个单一方向的MPT图像提供的边缘信息，再将3幅单一方向MPT图像进行等权重线性融合，以构造出一幅融合图像。为方便起见，将融合图像记为 <math id="M143"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719411&type= 3.80999994 3.04800010 （参考 图4 图4 ）。 1;2; 此外，为了比较不同小波函数对评估合理 <math id="M152"><msub><mrow><mi>ω</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718750&type= 3.64066648 3.72533321 值的影响，这里对比分析了Haar，Sym4，Db4，Fk4和Coif4小波相应的MPT图像 <math id="M153"><msub><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718765&type= 3.47133350 3.72533321 。从 图3 图3 中可以观察到：一方面，受限于小波函数的自身数学特性，小波函数的选取将直接影响到多尺度乘积层数 <math id="M154"><msub><mrow><mi>ω</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718750&type= 3.64066648 3.72533321 的自动评估；另一方面，当不同小波函数的多尺度乘积层数 <math id="M155"><msub><mrow><mi>ω</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718750&type= 3.64066648 3.72533321 取到相同数值时，小波函数之间的性质差异仍将导致MPT图像 <math id="M156"><msub><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718765&type= 3.47133350 3.72533321 存在较大区别。总体而言，Haar小波对应的多尺度乘积效应在抑制噪声和凸显边缘方面表现更佳。 4 基于指数Renyi熵构造阈值选取的目标函数 1 4.1 目标函数的构造 2 对于输入的灰度图像 <math id="M157"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719514&type= 2.45533323 3.64066648 ，按第3节介绍的方法可以计算出每个方向对应的最佳分解层数 <math id="M158"><msub><mrow><mi>ω</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718750&type= 3.64066648 3.72533321 和MPT图像 <math id="M159"><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718771&type= 3.89466691 3.97933316 ，并获得最终的融合MPT图像 <math id="M160"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719411&type= 3.80999994 3.04800010 。假设灰度图像 <math id="M161"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719514&type= 2.45533323 3.64066648 的灰度取值范围为 <math id="M162"><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><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></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719312&type= 18.79599953 3.97933316 ，用该灰度范围内的一个灰度值 <math id="M163"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719492&type= 1.26999998 2.79399991 对灰度图像 <math id="M164"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719514&type= 2.45533323 3.64066648 进行阈值化可得一幅二值图像 <math id="M165"><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719384&type= 2.62466669 3.72533321 ，然后从图像 <math id="M166"><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719384&type= 2.62466669 3.72533321 中提取出一幅内外轮廓图像 <math id="M167"><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719396&type= 2.28600001 3.72533321 ，再用图像 <math id="M168"><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719396&type= 2.28600001 3.72533321 中值为1的像素对图像 <math id="M169"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719411&type= 3.80999994 3.04800010 进行取样，并以取样所得像素构造对应的灰度直方图 <math id="M170"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 。 图4 图4 给出了上述处理步骤的一个直观示意。 EREM方法采用如 式（14） 式（14） 所示的目标函数选取最终阈值 <math id="M171"><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719536&type= 2.37066650 3.04800010 ： <math display="block" id="M172"><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><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></mfenced></mrow></munder><mfenced separators="|"><mrow><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718879&type= 40.89400101 6.68866634 ， 其中： <math id="M173"><msub><mrow><mi>t</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=45718884&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718883&type= 4.31799984 3.72533321 和 <math id="M174"><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=45718886&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718888&type= 4.65666676 3.72533321 分别表示灰度图像 <math id="M175"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719514&type= 2.45533323 3.64066648 的最小灰度值和最大灰度值； <math id="M176"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719254&type= 13.97000027 4.06400013 表示基于灰度直方图 <math id="M177"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 计算出的指数Renyi熵（Exponential Renyi Entropy，ERE）。 4.2 目标函数的合理性 2 从上述选取最终阈值的目标函数可以看出，EREM方法和传统阈值方法的主要不同之处在于：传统阈值方法的目标函数大多构建于原始灰度图像或者原始灰度图像及其均值滤波图像之上，而EREM方法的目标函数构建于融合MPT图像 <math id="M178"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719411&type= 3.80999994 3.04800010 上。无论原始图像是无峰、单峰、双峰或者多峰直方图模式，融合MPT图像 <math id="M179"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719411&type= 3.80999994 3.04800010 具有统一的单峰右偏直方图模式，这为EREM方法统一处理无峰、单峰、双峰或者多峰直方图模式下图像的自动阈值选择问题奠定了基础。 下面进一步分析按 公式（14） 公式（14） 定义的目标函数计算最终阈值 <math id="M180"><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719536&type= 2.37066650 3.04800010 的合理性。有两种方法计算灰度直方图 <math id="M181"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 对应的指数Renyi熵 <math id="M182"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719254&type= 13.97000027 4.06400013 。一种方法是对灰度直方图 <math id="M183"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 做整体运算以直接计算出指数Renyi熵 <math id="M184"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719254&type= 13.97000027 4.06400013 。这种方法的效率更高，在算法实现时将采用该方法。另一种方法是利用 式（13） 式（13） 计算出的灰度 <math id="M185"><msubsup><mrow><mi>l</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719216&type= 2.62466669 3.97933316 将灰度直方图 <math id="M186"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 划分为左右两个部分，并采用第3节计算 <math id="M187"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718975&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718972&type= 2.20133328 2.79399991 一样的方式计算指数Renyi熵 <math id="M188"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719254&type= 13.97000027 4.06400013 。这种方法可以建立起一个涉及4个自变量的多元函数，便于从理论上分析阈值取值的合理性。根据第二种方法，可以计算出： <math display="block" id="M189"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>α</mi></mrow></msubsup><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>-</mo></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>α</mi></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>+</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718993&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45718961&type= 63.83866882 4.74133301 ， 其中： <math id="M190"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>+</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719174&type= 11.59933376 3.30200005 是灰度直方图 <math id="M191"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 中大于灰度 <math id="M192"><msubsup><mrow><mi>l</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719216&type= 2.62466669 3.97933316 的右部对应的指数Renyi熵， <math id="M193"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>-</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719177&type= 11.59933376 3.30200005 是灰度直方图 <math id="M194"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 中小于灰度 <math id="M195"><msubsup><mrow><mi>l</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719216&type= 2.62466669 3.97933316 的左部对应的指数Renyi熵，而 <math id="M196"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>t</mi><mo>,</mo></mrow></msub></mrow><mrow><mo>-</mo></mrow></msub><mo>/</mo><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>-</mo></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>+</mo></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719028&type= 35.13666534 4.82600021 ，其中 <math id="M197"><msub><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>t</mi><mo>,</mo></mrow></msub></mrow><mrow><mo>-</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719019&type= 5.92666674 4.57200003 为灰度直方图 <math id="M198"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 左部对应的总频数， <math id="M199"><msub><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>t</mi><mo>,</mo></mrow></msub></mrow><mrow><mo>+</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719034&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719033&type= 6.01133299 4.57200003 为灰度直方图 <math id="M200"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 右部对应的总频数。 命题5 令 <math id="M201"><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>=</mo><mn mathvariant="normal">1</mn><mo>/</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>+</mo></mrow></msub><mo>-</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>-</mo></mrow></msub></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719038&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719037&type= 32.51200104 4.65666676 ，则当 <math id="M202"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>∈</mo><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719092&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719108&type= 16.00200081 4.48733330 时，指数Renyi熵 <math id="M203"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719254&type= 13.97000027 4.06400013 关于 <math id="M204"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719251&type= 2.53999996 3.72533321 单调递增；当 <math id="M205"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>∈</mo><mfenced separators="|"><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>,</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719248&type= 16.00200081 4.48733330 时，指数Renyi熵 <math id="M206"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719254&type= 13.97000027 4.06400013 关于 <math id="M207"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719251&type= 2.53999996 3.72533321 单调递减。 证明： <math id="M208"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719254&type= 13.97000027 4.06400013 关于 <math id="M209"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719251&type= 2.53999996 3.72533321 的偏导数为： <math display="block" id="M210"><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>-</mo></mrow></msub></mrow></msup></mrow><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></msubsup></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>+</mo></mrow></msub></mrow></msup></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719067&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719051&type= 35.13666534 10.32933331 . 当 <math id="M211"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>∈</mo><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719069&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719068&type= 15.66333294 4.48733330 时， <math id="M212"><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>-</mo></mrow></msub></mrow></msup></mrow><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></msubsup></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>+</mo></mrow></msub></mrow></msup></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow></mfrac><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719070&type= 40.47066879 10.32933331 ，根据函数导数和函数单调性的关系，可知指数Renyi熵 <math id="M213"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719254&type= 13.97000027 4.06400013 在区间 <math id="M214"><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719074&type= 9.65200043 4.48733330 内关于 <math id="M215"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719251&type= 2.53999996 3.72533321 单调递增；当 <math id="M216"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>∈</mo><mfenced separators="|"><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>,</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719084&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719083&type= 16.00200081 4.48733330 时， <math id="M217"><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>-</mo></mrow></msub></mrow></msup></mrow><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></msubsup></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>+</mo></mrow></msub></mrow></msup></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></msup></mrow></mfrac><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719078&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719085&type= 41.82533264 10.32933331 ，根据函数导数和函数单调性的关系，可知指数Renyi熵 <math id="M218"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719081&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719086&type= 13.97000027 4.06400013 在区间 <math id="M219"><mfenced separators="|"><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>,</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719157&type= 9.65200043 4.48733330 内关于 <math id="M220"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719251&type= 2.53999996 3.72533321 单调递减。 命题5指出了指数Renyi熵 <math id="M221"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719254&type= 13.97000027 4.06400013 的两个重要性质，性质1：在 <math id="M222"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>∈</mo><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719092&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719108&type= 16.00200081 4.48733330 时，更小的 <math id="M223"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719251&type= 2.53999996 3.72533321 值对应于更小的指数Renyi熵 <math id="M224"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719254&type= 13.97000027 4.06400013 ；性质2：在 <math id="M225"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>∈</mo><mfenced separators="|"><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>,</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719248&type= 16.00200081 4.48733330 时，更小的 <math id="M226"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719251&type= 2.53999996 3.72533321 值对应于更大的指数Renyi熵 <math id="M227"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719254&type= 13.97000027 4.06400013 。据此有两种可能的候选策略来选择相对更合理的阈值：策略1是根据性质1在单调递增区间 <math id="M228"><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719131&type= 9.65200043 4.48733330 内寻找相对更小的指数Renyi熵 <math id="M229"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719254&type= 13.97000027 4.06400013 ；策略2是根据性质2在单调递减区间 <math id="M230"><mfenced separators="|"><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>,</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719157&type= 9.65200043 4.48733330 内寻找相对更大的指数Renyi熵 <math id="M231"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719254&type= 13.97000027 4.06400013 。下面分析策略2更可取，即利用 式（14） 式（14） 选择最终阈值 <math id="M232"><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719536&type= 2.37066650 3.04800010 的合理性。 一方面，结合最大化指数Renyi熵差的性质， <math id="M233"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719251&type= 2.53999996 3.72533321 位于单调递减区间 <math id="M234"><mfenced separators="|"><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>,</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719157&type= 9.65200043 4.48733330 内的机会更大。最大化指数Renyi熵差有利于产生较大的 <math id="M235"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>+</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719174&type= 11.59933376 3.30200005 和较小的 <math id="M236"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>-</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719177&type= 11.59933376 3.30200005 ，这有利于当内外轮廓图像 <math id="M237"><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719396&type= 2.28600001 3.72533321 中值为1的像素相对更多的位于融合图像 <math id="M238"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719411&type= 3.80999994 3.04800010 中的边缘区域时产生较大的 <math id="M239"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>+</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719174&type= 11.59933376 3.30200005 值和较小的 <math id="M240"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>-</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719177&type= 11.59933376 3.30200005 值。因为 <math id="M241"><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>=</mo><mn mathvariant="normal">1</mn><mo>/</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>+</mo></mrow></msub><mo>-</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>-</mo></mrow></msub></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719170&type= 32.51200104 4.65666676 ，当 <math id="M242"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>+</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719174&type= 11.59933376 3.30200005 越大或 <math id="M243"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi><mo>,</mo><mo>-</mo></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719177&type= 11.59933376 3.30200005 越小时， <math id="M244"><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719181&type= 2.87866688 3.97933316 将取较小值，因此单调递减区间 <math id="M245"><mfenced separators="|"><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>,</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719157&type= 9.65200043 4.48733330 将越宽， <math id="M246"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719251&type= 2.53999996 3.72533321 位于区间 <math id="M247"><mfenced separators="|"><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>,</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719157&type= 9.65200043 4.48733330 内的机会将越大。 另一方面，考虑到灰度直方图 <math id="M248"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 是通过内外轮廓图像 <math id="M249"><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719396&type= 2.28600001 3.72533321 中值为1的像素对融合图像 <math id="M250"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719411&type= 3.80999994 3.04800010 进行取样构造而来，这样灰度直方图 <math id="M251"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 将涉及融合图像 <math id="M252"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719411&type= 3.80999994 3.04800010 中的边缘像素区域和非边缘像素区域。在灰度直方图 <math id="M253"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 中，小于灰度 <math id="M254"><msubsup><mrow><mi>l</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719216&type= 2.62466669 3.97933316 的左部主要涉及融合图像 <math id="M255"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719411&type= 3.80999994 3.04800010 中的非边缘像素，而大于灰度 <math id="M256"><msubsup><mrow><mi>l</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719216&type= 2.62466669 3.97933316 的右部主要涉及融合图像 <math id="M257"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719411&type= 3.80999994 3.04800010 中的边缘像素，因此 <math id="M258"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719251&type= 2.53999996 3.72533321 值将反映内外轮廓图像 <math id="M259"><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719396&type= 2.28600001 3.72533321 中值为1的像素位于融合图像 <math id="M260"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719411&type= 3.80999994 3.04800010 的非边缘区域和边缘区域的比例。对内外轮廓图像 <math id="M261"><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719396&type= 2.28600001 3.72533321 而言，相对更小的 <math id="M262"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719251&type= 2.53999996 3.72533321 值意味着非边缘区域占比更小，结合命题5在 <math id="M263"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>∈</mo><mfenced separators="|"><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>,</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719248&type= 16.00200081 4.48733330 时的性质，更小的 <math id="M264"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719251&type= 2.53999996 3.72533321 值更容易使指数Renyi熵 <math id="M265"><msup><mrow><mi>e</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mi>S</mi></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719254&type= 13.97000027 4.06400013 取较大值。 5 算法步骤 1 算法名称：EREM 输入：待分割的灰度图像 <math id="M266"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719514&type= 2.45533323 3.64066648 输出：分割阈值 <math id="M267"><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719536&type= 2.37066650 3.04800010 和分割结果图像 <math id="M268"><msubsup><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719544&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719527&type= 3.04800010 3.97933316 步骤1：结合 式（2） 式（2） 和 式（13） 式（13） 计算水平、垂直和对角3个方向对应的MPT图像，并将这3幅MPT图像进行等权重线性融合得到图像 <math id="M269"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719411&type= 3.80999994 3.04800010 ； 步骤2：用 <math id="M270"><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719469&type= 4.91066647 3.64066648 记录后续循环处理过程中从直方图 <math id="M271"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 中计算的ERE值，用 <math id="M272"><msubsup><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719477&type= 4.91066647 3.80999994 记录后续循环处理过程中最大的 <math id="M273"><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719469&type= 4.91066647 3.64066648 值，用 <math id="M274"><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719536&type= 2.37066650 3.04800010 记录对应于 <math id="M275"><msubsup><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719477&type= 4.91066647 3.80999994 的灰度值。 <math id="M276"><msubsup><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719477&type= 4.91066647 3.80999994 和 <math id="M277"><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719536&type= 2.37066650 3.04800010 的初始值均为0。对于待分割图像 <math id="M278"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719514&type= 2.45533323 3.64066648 中出现的每一个灰度值 <math id="M279"><mi>t</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><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></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719312&type= 18.79599953 3.97933316 ，按从小到大的顺序重复执行下面的步骤3到步骤6； 步骤3：用灰度值 <math id="M280"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719492&type= 1.26999998 2.79399991 阈值化待分割图像 <math id="M281"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719514&type= 2.45533323 3.64066648 以产生一幅二值图像 <math id="M282"><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719384&type= 2.62466669 3.72533321 ； 步骤4：基于二值图像 <math id="M283"><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719384&type= 2.62466669 3.72533321 构建出一幅和 <math id="M284"><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719384&type= 2.62466669 3.72533321 大小相等的内外轮廓图像 <math id="M285"><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719396&type= 2.28600001 3.72533321 。在初始化二值图像 <math id="M286"><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719396&type= 2.28600001 3.72533321 中所有像素值为1后，先构建内部轮廓：在遍历 <math id="M287"><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719384&type= 2.62466669 3.72533321 的每个像素时，当某个像素值等于1且该像素的4邻域值也都为1，那么修改 <math id="M288"><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719396&type= 2.28600001 3.72533321 中和该像素位置对应的像素值为0；再构建外部轮廓：先将 <math id="M289"><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719384&type= 2.62466669 3.72533321 取反得 <math id="M290"><mover><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mo>~</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719399&type= 3.04800010 5.58799982 ，然后采用构建内部轮廓一样的方法来遍历 <math id="M291"><mover><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mo>~</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719399&type= 3.04800010 5.58799982 并相应修改 <math id="M292"><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719396&type= 2.28600001 3.72533321 ； 步骤5：用内外轮廓图像 <math id="M293"><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719396&type= 2.28600001 3.72533321 中值为1的像素对融合图像 <math id="M294"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719411&type= 3.80999994 3.04800010 进行采样，并将采样所得像素构造对应的灰度直方图 <math id="M295"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 （直方图的灰度值规范化到［0，255］），再从灰度直方图 <math id="M296"><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719425&type= 3.80999994 3.72533321 中计算出对应的ERE值并存储到 <math id="M297"><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719469&type= 4.91066647 3.64066648 中； 步骤6：判断 <math id="M298"><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719469&type= 4.91066647 3.64066648 和 <math id="M299"><msubsup><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719477&type= 4.91066647 3.80999994 的大小关系：如果 <math id="M300"><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub><mo>></mo><msubsup><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719476&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719462&type= 14.56266594 3.80999994 ，则令 <math id="M301"><msubsup><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719477&type= 4.91066647 3.80999994 等于 <math id="M302"><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719469&type= 4.91066647 3.64066648 ，并令 <math id="M303"><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719536&type= 2.37066650 3.04800010 等于 <math id="M304"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719492&type= 1.26999998 2.79399991 ； 步骤7：以最终计算得到的阈值 <math id="M305"><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719536&type= 2.37066650 3.04800010 对图像 <math id="M306"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719514&type= 2.45533323 3.64066648 进行阈值化得到分割结果图像 <math id="M307"><msubsup><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719544&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719527&type= 3.04800010 3.97933316 ，最后输出分割阈值 <math id="M308"><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719536&type= 2.37066650 3.04800010 和分割结果图像 <math id="M309"><msubsup><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719544&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719527&type= 3.04800010 3.97933316 。 6 实验结果与讨论 1 6.1 实验环境、参与比较的方法、定量评估指标 2 用于测试实验的主要硬件参数如下：Intel Core i7-12700H 2.7 GHz中央处理器，16 GB DDR5内存，操作系统为64位Windows 11，集成开发平台是Matlab R2021a。测试图像集包括16幅合成图像和50幅真实世界图像，它们的灰度直方图呈现为无峰、单峰、双峰或者多峰模式。每幅测试图像的分割参考图像均利用Adobe Photoshop软件手工产生。测试图像集和分割参考图像集可以访问 https：//share.weiyun.com/NaYeXTAW https://share.weiyun.com/NaYeXTAW 获得。 提出的EREM方法与8种新近提出的分割方法进行了比较，它们分别是基于信息冗余度的自适应Tsallis熵阈值方法（Adaptive Tsallis Entropy，ATE） ［ 25 25 ］ 、基于类间方差和多类平均值的无参数阈值方法（Parameter Free，PF） ［ 26 26 ］ 、结合Otsu方法的熵加权阈值方法（Entropy Weighted，EW） ［ 27 27 ］ 、基于直方图形状的斜率差分分布阈值方法（Slope Difference Distribution，SDD） ［ 5 5 ］ 、鲁棒自稀疏模糊均值聚类分割方法（Robust Self Sparse Fuzzy Clustering Algorithm，RSSFCA） ［ 28 28 ］ 、模糊子空间聚类分割方法（Fuzzy Subspace Clustering，FSC） ［ 29 29 ］ 、基于自适应距离正则化水平集方法（Adaptive Distance Regularization Level Set，ADRLS） ［ 30 30 ］ 以及全局和局部模糊图像拟合的活动轮廓分割方法（Global and Local Fuzzy Image Fitting，GLFIF） ［ 31 31 ］ 。注意到EREM方法带有一个指数Renyi熵参数，在后续的6.2小节到6.4小节中，指数Renyi熵参数 <math id="M310"><mi>α</mi><mo>=</mo><mn mathvariant="normal">0.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719755&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719753&type= 12.53066730 2.79399991 ；在6.5小节进一步分析了不同指数Renyi熵参数 <math id="M311"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719773&type= 1.94733346 2.79399991 对分割精度的影响。 采用马修斯相关系数（Matthews Correlation Coefficient，MCC） ［ 32 32 ］ 来评估以上9个分割方法的分割精度。和Precision，Recall，Specificity，Dice，Jaccard（也称为Intersection Over Union）等指标相比，MCC是一个更为鲁棒的指标。只有当真阳性、真阴性、假阳性和假阴性这4种情况下都获得好的分割结果，MCC指标值才会高，并且MCC指标值正比于阳性元素和阴性元素各自的元素数量，从而能更稳健地应对目标和背景大小比例失衡的情况。特别地，当分割结果图像和分割参考图像保持一致时，MCC指标值将等于1；当分割结果图像和分割参考图像完全相反时，MCC指标值将等于-1。 6.2 在合成图像上的比较实验 2 为了验证EREM方法的分割适应性，一方面通过调整目标和背景的大小比例，使得16幅合成图像既包括目标和背景大小比例失衡的情况（参考 图5 图5 ），也包括大小比例相对均衡的情况（参考 图6 图6 ）；另一方面在合成图像上依次添加高斯噪声、极值噪声、伽玛噪声、瑞利噪声、贝塔噪声和均匀噪声，采用目标和背景被不同噪声退化的方式，以形成具有无峰、单峰、双峰或者多峰直方图模式的合成图像（参考 图5 图5 ~ 图7 图7 ）。在 图5 图5 和 图6 图6 的每幅子图中，合成图像对应的灰度直方图用绿色区域表示，EREM方法计算阈值时所得目标函数曲线用红色曲线表示，红色虚线及旁边的数字则标识了EREM方法选择的阈值（彩图见期刊电子版）。 图7 图7 展示了EREM，ATE，PF，EW和SDD方法在4幅代表性合成图像上的阈值以及最大化MCC指标时对应的最优分割阈值，其中，第1列所展示的测试图像的灰度直方图模式分别为无峰、单峰、双峰和多峰，在每幅子图中，左列为合成图像，右列绿色区域是合成图像对应的灰度直方图，红色曲线是EREM方法计算阈值时所得目标函数曲线（彩图见期刊电子版）。4幅代表性合成图像的选取综合了图像的目标图案、直方图模式、目标与背景的大小比例以及灰度分布等多种要素，其中，第1张为无峰模式的合成图像，目标与背景的大小比例相对均衡（目标和背景的大小比例约等于1∶2），但随机纹理构成的背景以及直方图的无峰模式给分割增加了难度。第2张为单峰模式的合成图像，目标与背景的比例严重失衡（目标和背景的大小比例约等于1∶110）和直方图的单峰模式增加了一定的分割难度。第3张为双峰模式的合成图像，该图像在施加噪声后进行了高斯点扩散处理，这导致目标的边缘变得相对模糊，但直方图的双峰模式使得分割难度相对降低。第4张为多峰模式的合成图像，直方图中的多峰样式以及复杂的噪声背景使得分割难度相对增加。 通过对比5种阈值分割方法在 图7 图7 中的阈值选取结果，可以观察到：（1）在 图7 图7 （a）中，EREM方法所得阈值与最优分割阈值一致（最优分割阈值通过最大化MCC值来自动确定），ATE，EW和PF方法所得阈值偏离最优分割阈值的程度较大，且三种方法所得阈值相对集中在［125，130］的灰度值区域内，SDD方法所得阈值则与最优分割阈值相差124个灰度值；（2）在 图7 图7 （b）中，EREM方法所得阈值与最优分割阈值一致，ATE，PF、EW和SDD方法所得阈值相较最优分割阈值均存在不同程度的偏离；（3）在 图7 图7 （c）中，5种阈值分割方法均存在不同程度的偏离，而ATE方法所得阈值更接近最优分割阈值，仅相差3个灰度值；（4）在 图7 图7 （d）中，EREM方法所得阈值与最优分割阈值一致，ATE，SDD和EW方法所得阈值偏离最优分割阈值的程度较大，且三种方法所得阈值相对集中在［115，125］的灰度值区域内，PF方法所得阈值则与最优分割阈值相差87个灰度值；（5）相较其他4种分割方法，ATE方法所得阈值在 图7 图7 （c）中较为接近最优分割阈值，但其分割结果不稳定，在 图7 图7 （a）、 图7 图7 （b）和 图7 图7 （d）中ATE方法所得阈值和最优分割阈值存在较大偏离。 图8 图8 展示了9个分割方法在 图7 图7 中4幅合成图像上的定性分割结果比较， 表1 表1 则给出了9个分割方法在16幅合成图像上的MCC值。通过统计分析可知，ATE，PF，EW，SDD，RSSFCA，FSC，ADRLS，GLFIF和EREM方法的平均MCC值分别为0.594 3，0.588 0，0.701 9，0.581 9，0.512 0，0.591 7，0.497 3，0.427 7和0.990 8，且EREM方法相比于分割精度第2的EW方法，平均MCC值提高了41.2%。结合 图8 图8 的分割结果和 表1 表1 的数据可以进一步观察到：（1）由于受到随机纹理的干扰，对于第1幅无峰模式的合成图像，只有EREM和FSC方法将目标从纹理背景中分割出来， 表1 表1 的第13行数据也印证了这种定性的视觉判断；（2）在第2幅单峰模式的合成图像中，目标和背景的大小比例失衡，从 图8 图8 的第2行和 表1 表1 的第14行结果可以看出，EREM，ATE，EW，SDD，FSC和ADRLS方法不同程度的将目标从随机噪声构成的背景中分割出来，其中EREM方法输出的分割结果图像和分割参考图像完全一致，而其余3个方法均未能获得较好的分割结果；（3）对于第3幅双峰模式的合成图像，SDD方法较为明显地将一些背景像素误分割为目标，FSC和ADRLS方法的分割结果存在目标细节信息丢失，目标形状变形等问题（参考FSC和ADRLS方法中的“五角星”目标和“问号”目标），而其他6个方法都获得较好的分割结果；④在 图8 图8 的第4行多峰模式的合成图像中，只有EREM方法较为成功地将目标从随机噪声构成的背景中分割出来。EREM方法虽然也存在少量像素被误分割，但结合 表1 表1 的最后1行数据来看，EREM方法的MCC值相比其他8种方法更接近于1。 Test images ATE ［ 25 25 ］ PF ［ 26 26 ］ EW ［ 27 27 ］ SDD ［ 5 5 ］ RSSFCA ［ 28 28 ］ FSC ［ 29 29 ］ ADRLS ［ 30 30 ］ GLFIF ［ 31 31 ］ EREM Fig.5 Fig.5 （a） 0.342 3 0.042 9 0.728 8 0.242 2 0.038 3 0.027 4 0.000 0 0.000 0 1.000 0 Fig.5 Fig.5 （b） 0.007 3 1.000 0 0.004 3 1.000 0 0.053 5 0.030 6 0.916 4 0.000 0 1.000 0 Fig.5 Fig.5 （c） 0.372 0 0.084 8 0.592 6 0.120 1 0.045 5 0.024 0 0.019 8 0.000 0 0.980 6 Fig.5 Fig.5 （d） 0.276 1 0.063 0 0.644 8 0.146 7 0.041 5 0.017 4 0.000 0 0.000 0 1.000 0 Fig.5 Fig.5 （e） 0.339 8 0.092 4 0.339 5 0.082 1 0.043 3 0.008 7 0.000 0 0.000 0 0.980 6 Fig.5 Fig.5 （f） 0.049 3 0.039 5 0.039 5 0.068 1 0.037 2 0.048 8 0.000 0 0.000 0 1.000 0 Fig.6 Fig.6 （a） 0.983 2 0.991 9 0.985 3 0.954 0 0.992 0 0.946 7 0.963 8 0.936 4 0.992 8 Fig.6 Fig.6 （b） 0.983 6 0.991 0 0.983 6 0.980 8 0.992 0 0.947 4 0.963 5 0.944 9 0.990 0 Fig.6 Fig.6 （c） 0.991 6 0.992 7 0.972 0 0.863 4 0.991 0 0.935 8 0.962 6 0.946 1 0.993 5 Fig.6 Fig.6 （d） 0.994 4 0.993 6 0.979 5 0.967 6 0.994 5 0.934 3 0.965 9 0.942 9 0.994 5 Fig.6 Fig.6 （e） 0.308 9 0.990 8 0.985 0 0.774 0 0.985 0 0.965 7 0.963 6 0.892 0 0.973 9 Fig.6 Fig.6 （f） 0.978 0 0.922 5 0.972 4 0.936 9 0.856 0 0.944 5 0.000 0 0.901 3 0.978 0 Fig.7 Fig.7 （a） 0.650 4 0.633 3 0.643 5 0.341 8 0.638 5 0.895 1 0.000 0 0.396 8 0.994 4 Fig.7 Fig.7 （b） 0.734 8 0.136 3 0.900 8 0.473 9 0.104 0 0.962 6 0.940 6 0.000 0 1.000 0 Fig.7 Fig.7 （c） 0.990 7 0.988 9 0.970 0 0.864 5 0.981 1 0.904 6 0.908 5 0.883 2 0.980 1 Fig.7 Fig.7 （d） 0.506 6 0.443 9 0.488 5 0.493 5 0.398 7 0.873 0 0.352 5 0.000 0 0.994 4 6.3 在真实世界图像上的比较实验 2 具有不同灰度直方图模式的50幅真实世界图像被用于进一步检验9个分割方法的分割适应能力。这些真实世界图像的灰度直方图可以用一种或多种分布的组合来近似，且直方图呈现无峰、单峰、双峰或多峰模式。这50幅测试图像被分为4组，其中编号1~6，7~18，19~35和36~50的测试图像分别对应无峰、单峰、双峰和多峰模式，测试图像采集自不同应用领域，包括材料无损检测、舰船目标监测、行人视频监控、生物细胞分析、激光熔覆控制等，涉及的成像方法包括超声成像、红外热成像、涡流成像、光学显微镜成像、光学CCD成像等。 作为代表性示例， 图9 图9 展示了EREM，ATE，PF，EW和SDD方法在4幅真实世界图像上的阈值选取以及最大化MCC指标时对应的最优分割阈值，4幅真实世界图像编号依次为4,18,23和43，每幅子图中绿色区域是相应的灰度直方图，红色曲线是EREM方法计算阈值时所得目标函数曲线（彩图见期刊电子版）。这4幅测试图像的灰度直方图分别为无峰、单峰、双峰和多峰模式。其中，第1幅测试图像采集自生物细胞分析，其中复杂的环境背景对分割目标细胞形成一定的干扰。第2幅测试图像采集自行人视频监控，该图中目标和背景的大小比例失衡增加了分割的难度。第3幅测试图像是材料显微镜图像，该图存在一定的竖状条纹光斑干扰。第4幅测试图像为红外热成像图像，该图中稍暗的灌木丛背景为分割目标人物增加了分割的难度。 对比5种阈值分割方法在 图9 图9 中的阈值选取结果，从中可以观察到：（1）在 图9 图9 （a）中，EREM方法所得阈值与最优分割阈值一致，SDD方法所得阈值与最优分割阈值相差5个灰度值，而PF，EW和ATE方法所得阈值则偏离最优分割阈值的程度均较大，且三种方法所得阈值相对集中在［115，130］的灰度值区域内；（2）在 图9 图9 （b）中，EREM方法所得阈值与最优分割阈值一致，ATE和EW方法所得阈值相较最优分割阈值存在较小偏离，且两种方法所得阈值相对集中在［205，215］的灰度值区域内，PF和SDD方法所得阈值则与最优分割阈值分别相差35和65个灰度值；（3）在 图9 图9 （c）中，EREM方法所得阈值与最优分割阈值一致，ATE，PF和EW方法相较最优分割阈值均存在不同程度的偏离，且PF和EW方法所得阈值相对集中在［120，125］的灰度值区域内，SDD方法所得阈值则与最优分割阈值相差152个灰度值；（4）在 图9 图9 （d）中，5种分割方法所得阈值较最优分割阈值均存在不同程度的偏离，与ATE，PF，EW和SDD方法相比，EREM方法所得阈值偏离最优分割阈值相对较小，与最优分割阈值仅差6个灰度值。 图10 图10 给出了9个分割方法在这4幅真实世界图像上的分割结果。从中可以看出，只有EREM方法相对成功地从这4幅真实世界图像中分隔开目标和背景，其他8个分割方法大多存在明显的误分割。 图11 图11 （彩图见期刊电子版）展示了9个分割方法在50幅真实世界图像上的MCC值，编号1到6的测试图像对应无峰模式（红色点），编号7到18的测试图像对应单峰模式（黄色点），编号19到35的测试图像对应双峰模式（蓝色点），编号36到50的测试图像对应多峰模式（绿色点）。通过统计分析可知，ATE，PF，EW，SDD，RSSFCA，FSC，ADRLS，GLFIF和EREM方法的平均MCC值分别为0.809 3，0.782 4，0.802 9，0.579 9，0.576 5，0.723 4，0.518 5，0.608 2和0.977 7，且EREM方法相比于分割精度第2的ATE方法，平均MCC值提高了20.8%。从 图11 图11 中可以进一步观察到，EREM方法的MCC值都集中在0.95以上，且各散点的波动幅度较小，这表明该方法能够获得精确度更高的分割结果，且对无峰、单峰、双峰和多峰模式的图像具有更强的分割适应性。相对而言，ATE和EW方法在不同模式下的MCC值都在0到1之间散布，但MCC值大部分主要集中在0.6以上，这表明这两种方法在总体上具有较弱的分割适应性。PF和FSC方法在双峰模式下的MCC值主要集中在0.8以上，但在无峰、单峰和多峰模式下的MCC值散布在0到1之间，这表明PF和FSC方法相对ATE和EW方法更适合于双峰模式的测试图像，但不适用于无峰、单峰和多峰模式的测试图像。对SDD，RSSFCA，ADRLS和GLFIF方法而言，它们仅个别测试图像下具有相对较高的MCC值，其余测试图像均在0到1之间散布，这表明这4个方法均不适合无峰、单峰、双峰和多峰模式的测试图像。 6.4 计算效率的比较 2 在相同软硬件参数下，同一个分割方法所需的CPU运行时间一般会存在轻微差别。为了降低不同时间点上CPU的浮动影响，可以让每个方法在同一幅测试图像上连续运行10次，并取10次运行所需CPU时间的平均值作为该方法在该图像上的CPU耗时。 表2 表2 给出了9个分割方法在16幅合成图像和50幅真实世界图像上CPU耗时的均值，从中可以观察到，尽管提出的EREM方法在计算效率方面不及EW和SDD这两个自动阈值分割方法，但是与ATE和PF两个自动阈值分割方法的计算效率在同一个数量级。另外，EREM方法比FSC，RSSFCA，GLFIF和ADRLS这4个非阈值分割方法具有相对计算效率优势。 2 Methods CPU time on synthetic images CPU time on real-world images Mean Mean EW ［ 27 27 ］ 0.000 031 0.000 042 SDD ［ 5 5 ］ 0.001 340 0.002 117 ATE ［ 25 25 ］ 0.022 277 0.024 564 PF ［ 26 26 ］ 0.020 265 0.041 370 EREM 0.061 223 0.070 134 FSC ［ 29 29 ］ 0.329 234 0.714 120 RSSFCA ［ 28 28 ］ 1.811 577 3.798 606 GLFIF ［ 31 31 ］ 2.600 437 2.702 787 ADRLS ［ 30 30 ］ 7.070 475 12.368 462 s 6.5 熵参数 <math id="M312"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719773&type= 1.94733346 2.79399991 对分割精度的影响 2 EREM方法中有一个熵参数 <math id="M313"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719773&type= 1.94733346 2.79399991 。为了测试熵参数 <math id="M314"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719773&type= 1.94733346 2.79399991 对分割精度的影响，需要改变熵参数 <math id="M315"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719773&type= 1.94733346 2.79399991 的取值并统计不同取值下在16幅合成图像和50幅真实世界图像上的分割精度。为此，让熵参数 <math id="M316"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719773&type= 1.94733346 2.79399991 在［0.1，0.9］区间内按步长 <math id="M317"><mi>η</mi><mo>=</mo><mn mathvariant="normal">0.025</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719707&type= 16.08666801 3.64066648 进行变化，然后统计相应66幅测试图像的平均MCC值。 图12 图12 给出了熵参数 <math id="M318"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719773&type= 1.94733346 2.79399991 在不同取值时的平均MCC值，从中不难发现，在［0.3，0.9］区间内平均MCC值波动相对较小，且在［0.375，0.75］区间内MCC更容易获得较高值。实际上，在熵参数 <math id="M319"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719773&type= 1.94733346 2.79399991 位于［0.375，0.75］区间内时，各点相对于 <math id="M320"><mi>α</mi><mo>=</mo><mn mathvariant="normal">0.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719755&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719753&type= 12.53066730 2.79399991 时的MCC绝对差值均小于0.0012。这些结果表明在［0.375，0.75］这一较为广阔的区域内，EREM方法对熵参数 <math id="M321"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45719773&type= 1.94733346 2.79399991 的变化不敏感，且能获得相对鲁棒的分割结果。 7 结 论 1 为解决灰度直方图呈现出无峰、单峰、双峰或者多峰模式时的自动阈值选取问题，提出了一种平稳小波域多尺度乘积下的指数Renyi熵自动阈值选择方法来确定最优分割阈值。通过与ATE方法、PF方法、EW方法、SDD方法、RSSFCA方法、FSC方法、ADRLS方法和GLFIF方法进行对比实验，发现EREM方法虽然在计算效率方面不及EW方法、SDD方法、ATE方法和PF方法，但具有更稳健的分割适应能力，并能获得精确度更高的分割结果图像，在合成测试图像集和真实世界图像集上的平均MCC值分别达到了0.990 8和0.977 7。EREM方法的稳健分割适应能力主要得益于：（1）在灰度图像的直方图呈现不同模式时，EREM方法利用平稳小波变换的多尺度乘积效应将不同的直方图模式转化为统一的单峰右偏灰度直方图；（2）EREM方法通过最大化指数Renyi熵差自适应确定平稳小波分解层数；（3）EREM方法对熵参数变化不敏感。 参考文献 吴一全 ， 孟天亮 ， 吴诗婳 . 图像阈值分割方法研究进展20年（1994—2014） ［J］. 数据采集与处理 ， 2015 ， 30 （ 1 ）： 1 - 23 . doi: 10.16337/j.1004-9037.2015.01.001 http://dx.doi.org/10.16337/j.1004-9037.2015.01.001 图像阈值分割方法研究进展20年（1994—2014） WU Y Q ， MENG T L ， WU S H . Research progress of image thresholding methods in recent 20 years （1994-2014） ［J］. Journal of Data Acquisition & Processing ， 2015 ， 30 （ 1 ）： 1 - 23 . （in Chinese） . doi: 10.16337/j.1004-9037.2015.01.001 http://dx.doi.org/10.16337/j.1004-9037.2015.01.001 Research progress of image thresholding methods in recent 20 years （1994-2014） 袁小翠 ， 吴禄慎 ， 陈华伟 . 基于Otsu方法的钢轨图像分割 ［J］. 光学 精密工程 ， 2016 ， 24 （ 7 ）： 1772 - 1781 . doi: 10.3788/ope.20162407.1772 http://dx.doi.org/10.3788/ope.20162407.1772 基于Otsu方法的钢轨图像分割 YUAN X C ， WU L S ， CHEN H W . Rail image segmentation based on Otsu threshold method ［J］. Opt. Precision Eng. ， 2016 ， 24 （ 7 ）： 1772 - 1781 . （in Chinese） . doi: 10.3788/ope.20162407.1772 http://dx.doi.org/10.3788/ope.20162407.1772 Rail image segmentation based on Otsu threshold method 姜鑫 ， 陈武雄 ， 聂海涛 ， 等 . 航空遥感影像的实时舰船目标检测 ［J］. 光学 精密工程 ， 2020 ， 28 （ 10 ）： 2360 - 2369 . doi: 10.37188/ope.20202810.2360 http://dx.doi.org/10.37188/ope.20202810.2360 航空遥感影像的实时舰船目标检测 JIANG X ， CHEN W X ， NIE H T ， et al . Real-time ship target detection based on aerial remote sensing images ［J］. Opt. Precision Eng. ， 2020 ， 28 （ 10 ）： 2360 - 2369 . （in Chinese） . doi: 10.37188/ope.20202810.2360 http://dx.doi.org/10.37188/ope.20202810.2360 Real-time ship target detection based on aerial remote sensing images NIE F ， ZHANG P ， LI J ， et al . A novel generalized entropy and its application in image thresholding ［J］. Signal Processing ， 2017 ， 134 ： 23 - 34 . doi: 10.1016/j.sigpro.2016.11.004 http://dx.doi.org/10.1016/j.sigpro.2016.11.004 A novel generalized entropy and its application in image thresholding WANG Z Z ， XIONG J J ， YANG Y M ， et al . A flexible and robust threshold selection method ［J］. IEEE Transactions on Circuits and Systems for Video Technology ， 2017 ， 28 （ 9 ）： 2220 - 2232 . doi: 10.1109/tcsvt.2017.2719122 http://dx.doi.org/10.1109/tcsvt.2017.2719122 A flexible and robust threshold selection method YAZID H ， BASAH S N ， RAHIM S A ， et al . Performance analysis of entropy thresholding for successful image segmentation ［J］. Multimedia Tools and Applications ， 2022 ， 81 （ 5 ）： 6433 - 6450 . doi: 10.1007/s11042-021-11813-z http://dx.doi.org/10.1007/s11042-021-11813-z Performance analysis of entropy thresholding for successful image segmentation 罗希平 ， 田捷 . 用最大熵原则作多阈值选择的条件迭代算法 ［J］. 软件学报 ， 2000 ， 11 （ 3 ）： 379 - 385 . doi: 10.1038/sj.cr.7290056 http://dx.doi.org/10.1038/sj.cr.7290056 用最大熵原则作多阈值选择的条件迭代算法 LUO X P ， TIAN J . The ICM algorithm for multi-level threshold selection by maximum entropy criterion ［J］. Journal of Software ， 2000 ， 11 （ 3 ）： 379 - 385 . （in Chinese） . doi: 10.1038/sj.cr.7290056 http://dx.doi.org/10.1038/sj.cr.7290056 The ICM algorithm for multi-level threshold selection by maximum entropy criterion CHANG C I ， DU Y ， WANG J ， et al . Survey and comparative analysis of entropy and relative entropy thresholding techniques ［J］. IEE Proceedings-Vision， Image， and Signal Processing ， 2006 ， 153 （ 6 ）： 837 . doi: 10.1049/ip-vis:20050032 http://dx.doi.org/10.1049/ip-vis:20050032 Survey and comparative analysis of entropy and relative entropy thresholding techniques Pun T . A new method for grey-level picture thresholding using the entropy of the histogram ［J］. Signal Processing ， 1980 ， 2 （ 3 ）： 223 - 237 . doi: 10.1016/0165-1684(80)90020-1 http://dx.doi.org/10.1016/0165-1684(80)90020-1 A new method for grey-level picture thresholding using the entropy of the histogram KAPUR JN ， SAHOO PK ， WONG AK . A new method for gray-level picture thresholding using the entropy of the histogram ［J］. Computer Vision， Graphics， and Image Processing ， 1985 ， 29 （ 3 ）： 273 - 285 . doi: 10.1016/0734-189x(85)90125-2 http://dx.doi.org/10.1016/0734-189x(85)90125-2 A new method for gray-level picture thresholding using the entropy of the histogram ABUTALEB AS . Automatic thresholding of gray-level pictures using two-dimensional entropy ［J］. Computer Vision， Graphics， and Image Processing ， 1989 ， 47 （ 1 ）： 22 - 32 . doi: 10.1016/0734-189x(89)90051-0 http://dx.doi.org/10.1016/0734-189x(89)90051-0 Automatic thresholding of gray-level pictures using two-dimensional entropy WONG A K C ， SAHOO P K . A gray-level threshold selection method based on maximum entropy principle ［J］. IEEE Transactions on Systems， Man， and Cybernetics ， 1989 ， 19 （ 4 ）： 866 - 871 . doi: 10.1109/21.35351 http://dx.doi.org/10.1109/21.35351 A gray-level threshold selection method based on maximum entropy principle DE ALBUQUERQUE MP ， ESQUEF IA ， MELLO AG . Image thresholding using tsallis entropy ［J］. Pattern Recognition Letters ， 2004 ， 25 （ 9 ）： 1059 - 1065 . doi: 10.1016/j.patrec.2004.03.003 http://dx.doi.org/10.1016/j.patrec.2004.03.003 Image thresholding using tsallis entropy SAHOO P ， WILKINS C ， YEAGER J . Threshold selection using Renyi’s entropy ［J］. Pattern Recognition ， 1997 ， 30 （ 1 ）： 71 - 84 . doi: 10.1016/s0031-3203(96)00065-9 http://dx.doi.org/10.1016/s0031-3203(96)00065-9 Threshold selection using Renyi’s entropy Wang S ， Chung FL . Note on the equivalence relationship between Renyi-entropy based and Tsallis-entropy based image thresholding ［J］. Pattern Recognition Letters ， 2005 ， 26 （ 14 ）： 2309 - 2312 . doi: 10.1016/j.patrec.2005.03.027 http://dx.doi.org/10.1016/j.patrec.2005.03.027 Note on the equivalence relationship between Renyi-entropy based and Tsallis-entropy based image thresholding SAHOO PK ， ARORA G . A thresholding method based on two-dimensional Renyi’s entropy ［J］. Pattern Recognition ， 2004 ， 37 （ 6 ）： 1149 - 1161 . doi: 10.1016/j.patcog.2003.10.008 http://dx.doi.org/10.1016/j.patcog.2003.10.008 A thresholding method based on two-dimensional Renyi’s entropy SAHOO S ， SAHOO S S ， KUMAR S ， et al . Optimized Entropy Based Image Segmentation ［C］. 2020 International Conference on Computer Science， Engineering and Applications （ICCSEA） . 13 - 14 ， 2020， Gunupur， India. IEEE ， 2020： 1 - 6 . doi: 10.1109/iccsea49143.2020.9132959 http://dx.doi.org/10.1109/iccsea49143.2020.9132959 Optimized Entropy Based Image Segmentation ABDEL-KHALEK S ， ISHAK AB ， OMER OA ， et al . A two-dimensional image segmentation method based on genetic algorithm and entropy ［J］. Optik ， 2017 ， 131 ： 414 - 422 . doi: 10.1016/j.ijleo.2016.11.039 http://dx.doi.org/10.1016/j.ijleo.2016.11.039 A two-dimensional image segmentation method based on genetic algorithm and entropy 刘松涛 ， 刘振兴 ， 姜康辉 . 基于模糊Renyi熵和区域增长的图像目标分割方法 ［J］. 系统工程与电子技术 ， 2018 ， 40 （ 8 ）： 1693 - 1701 . doi: 10.3969/j.issn.1001-506X.2018.08.04 http://dx.doi.org/10.3969/j.issn.1001-506X.2018.08.04 基于模糊Renyi熵和区域增长的图像目标分割方法 LIU S T ， LIU Z X ， JIANG K H . Image target segmentation method based on fuzzy Renyi entropy and region growing ［J］. Systems Engineering and Electronics ， 2018 ， 40 （ 8 ）： 1693 - 1701 . （in Chinese） . doi: 10.3969/j.issn.1001-506X.2018.08.04 http://dx.doi.org/10.3969/j.issn.1001-506X.2018.08.04 Image target segmentation method based on fuzzy Renyi entropy and region growing WANG Y ， XU X . An improved level set method to image segmentation tation based on saliency ［J］. Journal of Information Processing Systems ， 2019 ， 15 （ 1 ）： 7 - 21 . An improved level set method to image segmentation tation based on saliency HE C M ， WANG X B ， DENG L Z ， et al . Image Threshold Segmentation Based on GLLE Histogram ［C］. 2019 International Conference on Internet of Things （iThings） and IEEE Green Computing and Communications （GreenCom） and IEEE Cyber， Physical and Social Computing （CPSCom） and IEEE Smart Data （SmartData） . 14 - 17 ， 2019， Atlanta， GA， USA. IEEE ， 2019： 410 - 415 . doi: 10.1109/ithings/greencom/cpscom/smartdata.2019.00088 http://dx.doi.org/10.1109/ithings/greencom/cpscom/smartdata.2019.00088 Image Threshold Segmentation Based on GLLE Histogram LEI B ， FAN J . Adaptive granulation Renyi rough entropy image thresholding method with nested optimization ［J］. Expert Systems With Applications ， 2022 ， 203 ： 117378 . doi: 10.1016/j.eswa.2022.117378 http://dx.doi.org/10.1016/j.eswa.2022.117378 Adaptive granulation Renyi rough entropy image thresholding method with nested optimization NASON G P ， SILVERMAN B W . The Stationary Wavelet Transform and Some Statistical Applications ［M］. Wavelets and Statistics . New York， NY ： Springer New York ， 1995 ： 281 - 299 . doi: 10.1007/978-1-4612-2544-7_17 http://dx.doi.org/10.1007/978-1-4612-2544-7_17 The Stationary Wavelet Transform and Some Statistical Applications 薛婷 ， 钟麦英 . 基于SWT与等价空间的LDTV系统故障检测 ［J］. 自动化学报 ， 2017 ， 43 （ 11 ）： 1920 - 1930 . doi: 10.16383/j.aas.2017.c160479 http://dx.doi.org/10.16383/j.aas.2017.c160479 基于SWT与等价空间的LDTV系统故障检测 XUE T ， ZHONG M Y . SWT and parity space based fault detection for linear discrete time-varying systems ［J］. Acta Automatica Sinica ， 2017 ， 43 （ 11 ）： 1920 - 1930 . （in Chinese） . doi: 10.16383/j.aas.2017.c160479 http://dx.doi.org/10.16383/j.aas.2017.c160479 SWT and parity space based fault detection for linear discrete time-varying systems RAMÍREZ-REYES A ， HERNÁNDEZ-MONTOYA A ， HERRERA-CORRAL G ， et al . Determining the entropic index q of tsallis entropy in images through redundancy ［J］. Entropy ， 2016 ， 18 （ 8 ）： 299 . doi: 10.3390/e18080299 http://dx.doi.org/10.3390/e18080299 Determining the entropic index q of tsallis entropy in images through redundancy CAO X H ， LI T H ， LI H L ， et al . A robust parameter-free thresholding method for image segmentation ［J］. IEEE Access ， 2018 ， 7 ： 3448 - 3458 . doi: 10.1109/access.2018.2889013 http://dx.doi.org/10.1109/access.2018.2889013 A robust parameter-free thresholding method for image segmentation TRUONG M T N ， KIM S . Automatic image thresholding using Otsu's method and entropy weighting scheme for surface defect detection ［J］. Soft Computing ， 2018 ， 22 （ 13 ）： 4197 - 4203 . doi: 10.1007/s00500-017-2709-1 http://dx.doi.org/10.1007/s00500-017-2709-1 Automatic image thresholding using Otsu's method and entropy weighting scheme for surface defect detection JIA X H ， LEI T ， DU X G ， et al . Robust self-sparse fuzzy clustering for image segmentation ［J］. IEEE Access ， 2020 ， 8 ： 146182 - 146195 . doi: 10.1109/access.2020.3015270 http://dx.doi.org/10.1109/access.2020.3015270 Robust self-sparse fuzzy clustering for image segmentation WEI T ， WANG X ， LI X ， et al . Fuzzy subspace clustering noisy image segmentation algorithm with adaptive local variance & non-local information and mean membership linking ［J］. Engineering Applications of Artificial Intelligence ， 2022 ， 110 ： 104672 . doi: 10.1016/j.engappai.2022.104672 http://dx.doi.org/10.1016/j.engappai.2022.104672 Fuzzy subspace clustering noisy image segmentation algorithm with adaptive local variance & non-local information and mean membership linking XIAO H G ， ZHANG B L ， LIU R H ， et al . Accurate image segmentation based on adaptive distance regularization level set method ［J］. International Journal of Wavelets， Multiresolution and Information Processing ， 2022 ， 20 （ 6 ）： 2250033 . doi: 10.1142/s0219691322500333 http://dx.doi.org/10.1142/s0219691322500333 Accurate image segmentation based on adaptive distance regularization level set method FANG J ， LIU H ， LIU J ， et al . Fuzzy region-based active contour driven by global and local fitting energy for image segmentation ［J］. Applied Soft Computing ， 2021 ， 100 ： 106982 . doi: 10.1016/j.asoc.2020.106982 http://dx.doi.org/10.1016/j.asoc.2020.106982 Fuzzy region-based active contour driven by global and local fitting energy for image segmentation CHICCO D ， JURMAN G . The advantages of the Matthews correlation coefficient （MCC） over F1 score and accuracy in binary classification evaluation ［J］. BMC Genomics ， 2020 ， 21 （ 1 ）： 6 . doi: 10.1186/s12864-019-6413-7 http://dx.doi.org/10.1186/s12864-019-6413-7 The advantages of the Matthews correlation coefficient （MCC） over F1 score and accuracy in binary classification evaluation 10.37188/OPE.20233112.1841 TP391 A 1004-924X（2023）12-1841-18 邹耀斌,孟祥丹,孙水发等.平稳小波域多尺度乘积下的指数Renyi熵自动阈值选择方法[J].光学精密工程,2023,31(12):1841-1858. 邹耀斌,孟祥丹,孙水发等.平稳小波域多尺度乘积下的指数Renyi熵自动阈值选择方法[J].光学精密工程,2023,31(12):1841-1858. ZOU Yaobin,MENG Xiangdan,SUN Shuifa,et al.Automatic threshold selection method using exponential Renyi entropy under multi-scale product in stationary wavelet domain[J].Optics and Precision Engineering,2023,31(12):1841-1858. ZOU Yaobin,MENG Xiangdan,SUN Shuifa,et al.Automatic threshold selection method using exponential Renyi entropy under multi-scale product in stationary wavelet domain[J].Optics and Precision Engineering,2023,31(12):1841-1858. 国家自然科学基金资助项目（No. 61871258） 2022-07-18 2022-09-22 2023-06-25 2023-07-17 39074252 光学精密工程 12 31
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