MCMICM-Paper-ZH01 2012A

公众号 数学模型 微信号 MATHmodels A Close Look On Leaves 细看树叶 Bo Zhang Yi Zhang Tiankun Lu COMAP Mathematical Contest in Modeling February 13 2012 Shanghai Foreign Language School 翻译 周吕文 2 MATHmodels 关注 数学模型 公众号 获取更多信息 0若发现翻译问题 请邮件告知 谢谢 公众号 数学模型 微信号 MATHmodels Abstract We construct four models to study leaf classification relationships between我们构建了四个模型来分别研究叶子的分类 叶子形状 leaf shape and leaf distribution correlations between leaf shape and tree profile 与叶子分布之间的关系 叶子形状与树形轮廓之间的相关性 and total leaf mass of a tree 以及一棵树的树叶总质量 Model 1 deals with the classification of leaves We focus primarily on the most模型 1 处理叶子的分类 我们主要关注叶子最显着的特 conspicuous characteristic of leaves namely shape We create seven geometric征 即形状 我们构造了七个几何参数来量化形状 然后我 parameters to quantify the shape Then we select six common types of leaves to们选择了六种常见的叶子来构建一个数据库 通过计算样本 construct a database By calculating the deviation index of the parameters of a叶片与典型叶片的参数偏差指数 我们可以对叶子进行分类 sample leaf from those of typical leaves we can classify the leaf To illustrate this为了说明这个分类过程 我们使用枫叶作为测试用例 classification process we use a maple leaf as a test case Model 2 studies the relationship between leaf shape and leaf distribution First 模型 2 研究了叶子形状和叶子分布之间的关系 首先 we simplify a tree into an idealized model and then introduce the concept of solar我们将一棵树简化成一个理想模型 然后介绍太阳高度的概 altitude By analyzing the overlapping individual shadows through considering the念 通过考虑不同太阳高度下叶片长度和节间长度的关系 relationship between leaf length and internode length under different solar altitudes 分析重叠的叶片阴影 我们发现叶片形状和分布被优化 以 we find that the leaf shape and distribution are optimized to maximize sunlight根据太阳高度最大化日照 我们将模型应用于三种测试类型 exposure according to the solar altitude We apply the model to three test types的树木 of trees Model 3 discusses the possible association between tree profile and leaf shape 模型 3 讨论了树木轮廓和叶子形状之间可能的关联 根 Based on the similarity between the leaf veins and branch structure of trees we据叶脉与树枝结构的相似性 我们提出叶片形状是树形轮廓 propose that leaf shape is a two dimensional mimic of the tree profile Employing的二维近似 采用模型 1 的方法 我们设置了几个参数来刻 the method of Model 1 we set several parameters reflecting the general shape of化每棵树的大概形状 并将它们与其叶子的参数进行比较 each tree and compare them with those of its leaves With the help of statistical在统计工具的帮助下 我们展示了树形轮廓和叶形之间的粗 tools we demonstrate a rough association between tree profile and leaf shape 略关联 Model 4 estimates the total leaf mass of a tree given size characteristics Carbon模型 4 对给定尺寸特征的树 估计了其树叶总质量 引 dioxide CO2 sequestration rate and tree age are introduced to establish the link入二氧化碳固碳率和树龄来建立树叶质量和树木大小之间 between leaf mass and tree size Since a unit mass of a leaf sequesters CO2 at a的联系 由于单位质量的叶片以一个恒等的速率固碳 二氧 constant rate the CO2 sequestration rate has a quadratic relationship with the化碳固碳率与树龄之间是一个二次关系 并且树的大小呈现 age of the tree and the size the tree experiences logistic growth logistic 增长 公众号 数学模型 微信号 MATHmodels 翻译 周吕文关注 数学模型 公众号 获取更多信息第 1 页 共 33 页 Contents 1Introduction 引言3 2Breaking Down the Problem 问题的分解3 3Assumptions 假设4 4Nomenclatures 术语6 5Model One Leaf Classification 模型一 叶子的分类6 5 1Decisive Parameters 决定性参数 6 5 2Comparison 9 5 3Model Testing 12 5 4Conclusion 14 6Model Two Leaf Distribution and Leaf Shape15 6 1Introduction 15 6 2Idealized Leaf Distribution Model 15 6 3Analysis of Overlapping Areas 15 6 3 1Solar Altitude trending toward 90 15 6 3 2Solar Altitude trending toward 0 16 6 4Solar Altitude within normal range 17 6 5Model Testing 19 7Model Three Tree Profile and Leaf Shape20 7 1Introduction 20 7 2Hypothesis 20 7 3Comparison of Leaf Shape and Tree Contour 21 7 4Tree Profile Classification 21 7 5Parameters of the tree 22 7 6Conclusion 24 数学模型 2 MATHmodels 公众号 数学模型 微信号 MATHmodels 翻译 周吕文关注 数学模型 公众号 获取更多信息第 2 页 共 33 页 8Model Four Leaf Mass25 8 1Introduction 25 8 2Leaf Mass and Tree Age 25 8 2 1Leaf Mass and CO2 Sequestration 25 8 2 2CO2 Sequestration and Tree Age 26 8 2 3Leaf Mass and Tree Age 27 8 3Tree Age and Tree Size 27 8 4Leaf Mass and Tree Size 30 9Strengths and Weaknesses31 数学模型 2 MATHmodels 公众号 数学模型 微信号 MATHmodels 翻译 周吕文关注 数学模型 公众号 获取更多信息第 3 页 共 33 页 1Introduction 引言 Leaf which is responsible for photosynthesis and storage of food and water is a叶是植物非常重要的器官 它承担着光合作用 并为植 very important organ of a plant Leaf has so many different and interesting shapes物贮存水和养份 叶子具有许多不同而有趣的形状轮廓 这 and contours which make it so catching and conspicuous in the plant kingdom 也让它在大自然的植物王国中格外引人注目 几个世纪以来 For centuries people have always wondered why the leaf has so many splendid人们总是想知道为什么叶子有这么多精美的形状 难道这只 shapes Is this just a gift from the Almighty God Or is it the adjustment the leaf是全能的上帝带来的礼物 还是叶子在进化和适应过程中所 takes during its evolution and adaptation 做调整的结果 Another mystery of the leaf lies in the classification of different leaf types Can叶子的另一个不解之谜在于它繁多种类的区分 我们能 we do so in a more reliable and scientific way instead of judging subjectively We不能用一种更加科学可靠的方法来对树叶进行分类 而不是 have tried our best to build a model on a quantitative base to increase the accuracy凭借主观判断 因此我们在数值的定量基础上建立了模型 and efficiency of classifying different leaves 来提高树叶分类的准确率和有效性 The final mystery about the tree is how much the total leaf mass on a tree关于树叶还有一项终极难题 如何确定一棵树上树叶的 is Just like people always want to figure out how much hair a person has human总质量 就像人们总是想弄清楚一个人有多少头发一样 人 beings always have a fancy to find out a solution from the seemingly infinity and们总是喜欢从看似无穷无尽的事物中寻找答案 incalculable We are deeply convinced that the true glamour of scientific and mathematical我们一直深信 科学和数学模型的魔力在于解开大自然 models is to unveil the intriguing and the changeable of the nature Aiming to错综复杂和变幻莫测的面纱 以此为目标 我们将建立能适 do so we will generate models that can fit large quantities of data with highest应大量数据的模型 并且以最高的准确率在弹指之间解决问 accuracy within a quick flick 题 2Breaking Down the Problem 问题的分解 After carefully analyzing the problem we conclude four main sub problems to经过对这个问题的仔细分析 我们归纳出本文需要解决 tackle in our paper 的四个主要子问题 1 Classification the different types of leaves 1 对不同树叶进行分类 2 Relationship between the leaf distribution and leaf shape 2 树叶分布与叶形二者间的联系 3 Relationship between the tree profile and leaf shape 3 树的轮廓与叶形二者间的联系 4 Calculation of the total leaf mass on a tree 4 计算一棵树上的树叶总质量 数学模型 2 MATHmodels 公众号 数学模型 微信号 MATHmodels 翻译 周吕文关注 数学模型 公众号 获取更多信息第 4 页 共 33 页 To tackle the first problem we set a set of parameters to quantify the characters处理第一个问题时 我们设置了一组参数来量化叶形特征 从 of the leaf shape and use the leaf shape as the main standard for our classification而将叶形作为树叶分类过程的主要参照标准 process As for the second question we want to use the overlapping area that one leaf对于第二个问题 我们希望用一片树叶遮挡它正下方叶 casts on the leaf directly under it as a medium to associate the leaf distribution子的重叠面积来作为一个中间量来把树叶分布和叶形联系起 and leaf shape since the leaf shape will affect the overlapping and we assume the来 这是因为叶形会影响树叶间的重叠 同时我们假设树叶 leaf distribution will try to minimize the overlapping area 分布有令重叠面积最小的趋向 As for the third question we want to refer to the process we take when tackling对于第三个问题 我们希望能够参考第一个问题的解决 the first problem and also set some parameters for the tree profile After that we过程 同样设置一些描述树形的参数 然后 我们将对比叶 will compare their parameters and judge whether there is a relation between tree形 树形的两组参数 并判断树形轮廓和叶形之间是否存在 profile and leaf shape 着联系 To deal with the total mass of the leaves we want to use the age to link the为了计算树上树叶的总质量 我们用树龄把树的尺寸大 size of tree and the total weight of leaves of it because the tree size has an obvious小和树叶总质量连结起来 因为树的尺寸和树龄有明显的关 relationship with its age and the age will affect a tree s sequestration of carbon系 而树龄将反映了树的二氧化碳积累 净吸收 量 即而 dioxide which will reflect the weight of a tree s total leaves 反映出它的树叶总质量 3Assumptions 假设 1 The trees being studied are all individual open grown trees such as trees1 所分析的树木均为独立的 开阔成型的 open grown typically planted along streets in yards and in parks Our calculation does 树 最典型的就像我们街道边 庭院和公园里种植的 not apply to densely raised trees as in typical reforestation projects where树木 我们所做的计算不适用于密集分布的树 例如再 large numbers of trees are planted closely together 造林工程 把大量树密集地种植在一块区域内 2 Assume the shape of the leaves does not reflect special uses for the trees such2 假设树叶的形状与树的特殊应用无关 即不考虑叶形抵 as to resist extremely windy cold parched wet or dry conditions or to catch抗大风 严寒 炎热 潮湿或潮湿的环境 抑或为了捕 food 食 3 Assume the type of the leaf distribution to be discussed leaf length and3 假设将要讨论的树叶分布的类型 叶长及节间距离关 internode distance relation is only a reflection of the tree s natural tendency系 仅反映为树木对阳光的自然倾向 to sunlight 4 假设我们考虑的树形是地面上的部分 包括树干 树枝 4 Assume the tree profile we consider is the part above ground including the和树叶 数学模型 2 MATHmodels 公众号 数学模型 微信号 MATHmodels 翻译 周吕文关注 数学模型 公众号 获取更多信息第 5 页 共 33 页 trunk the branches and leaves 5 Assume all parts of leaf lay on a flat surface and the thickness or protrusion5 假设叶的所有部分展在同一平面上 即忽略厚度及叶脉 of veins are neglectable 的突出部分 6 Assume leaves are the only part of the tree that reacts in photosynthesis and6 假设叶是整棵树中唯一进行光合作用和呼吸作用的部 respiration so that the carbon dioxide sequestration of a tree is the sum of位 进而树的二氧化碳积累量就等于所有树叶积累的总 the sequestration of the leaves 和 7 Assume the sequestration of a tree or a leaf is the net amount of CO2 fixed7 假设树 叶的二氧化碳积累量是 固定 在该树上的二 in a tree which is the difference between the CO2 released in respiration and氧化碳净余量 即意味着光合作用中吸收的吸收量与呼 the CO2 absorbed in photosynthesis 吸作用的释放量之差 8 Assume the trees are in healthy mature and stable condition The trees of8 假设树木均处于健康 成熟和稳定的状态下 同种树木 the same species have same characteristics 有着相同的特征 数学模型 2 MATHmodels 公众号 数学模型 微信号 MATHmodels 翻译 周吕文关注 数学模型 公众号 获取更多信息第 6 页 共 33 页 4Nomenclatures 术语 SymbolMeaning表示 RRectangularity矩度 Aleafthe area of leaf叶子的面积 Arectanglethe area of minimum bounding rectangle最小外接矩形的面积 ARthe aspect ratio宽高比 Lshortthe length of shorter side 最小外接矩形 较短一边的边长 Llongthe length of longer side 最小外接矩形 较长一边的边长 Cthe circularity圆度 Rinthe radius of in circle内接圆半径 Rexthe radius of ex circle外接圆半径 FFForm factor形状因子 Pleafthe perimeter of leaf树叶的周长 ERAIthe edge regularity area index边缘面积平滑指数 BPAThe bounding polygon area外接多边形面积 ERPIthe edge regularity perimeter index边缘周长平滑指数 BPPthe bounding polygon perimeter外接多边形周长 PIiProportional index比例因子 IDthe Index of Deviation偏差指数 Lmajorthe length of major axis菱形长轴长 Aoverlappingthe overlapping area重叠面积 the solar altitude太阳高度角 Mleafthe total mass of leaves on a tree树叶的总质量 Mcarbon dioxidemass of carbon dioxide sequestered lbs 二氧化碳积累量 Asability to sequester carbon dioxide lbs g 二氧化碳积累能力 Athe age of the tree树龄 5Model One Leaf Classification 模型一 叶子的分类 5 1Decisive Parameters 决定性参数 In order to classify the shapes of the given leaf we want to set a number of为了将任意一片树叶按形状分到已知的某类当中 我们 数学模型 2 MATHmodels 公众号 数学模型 微信号 MATHmodels 翻译 周吕文关注 数学模型 公众号 获取更多信息第 7 页 共 33 页 parameters and establish a database for comparison After carefully and thoroughly希望设置一组参数 继而建立数据集用以对比 在仔细完整 analyzing the leaves we develop seven most significant parameters as shown below 地分析了树叶后 我们归纳出 7 个最重要的参数 如下 1 Rectangularity1 矩度 Rectangularity Firstly we define the ratio of the area of the leaf to the Minimum Bounding首先 我们将树叶面积与它的最小外接矩形面积定义为 Rectangle as the leaf s Rectangularity how much does the leaf resemble a该树叶的矩度 就是说这片树叶的形状跟矩形有多相似 rectangle refer to figure 1 The maximum possible value of this parameter 参看图1 这一参数的最大理论取值为 1 is 1 R Aleaf Arectangle R Aleaf Arectangle Aleafstands for the area of leaf Arectanglestands for the area of minimumAleaf代表树叶的面积 Arectangle代表其最小外接矩形 bounding rectangle 的面积 2 Aspect ratio2 宽高比 Aspect ratio After defining the rectangularity now we define the Aspect Ratio which在定义了矩度之后 现在我们定义宽高比来描述一片树 describes the proportional relationship between the width of and height of叶最小外接矩形宽 高的比例关系 作为另一个对叶形 a leaf s Minimum Bounding Rectangle as another key character to classify一般分类的关键特征 宽高比越大 树叶就越像正方形 general shape of a leaf The bigger this ratio is the more this leaf resembles 参看图1 这一参数的最大取值也是 1 a square refer to figure 1 The maximum possible value of this parameter is 1 AR Lshort Llong AR Lshort Llong Where AR stands for the aspect ratio Lshortstands for the length of shorter其中 AR 表示宽高比 Lshort表示 最小外接矩形 较 side Llongstands for the length of longer side 短一边的边长 Llong表示 最小外接矩形 较长一边的 边长 3 Circularity3 圆度 Circularity To evaluate how roundish a leaf is we consider that the respective radius of为了评价树叶有多圆 我们将考虑树叶内 外接圆半径 in circle and ex circle The ratio of the former to the latter which we define的相对值 定义前者 内接圆 与后者 外接圆 半径之 as Circularity may well reflect this characteristic The greater the ratio of比为树叶的圆度 来反映原型程度这一特征 圆度越大 Circularity is the closer the leaf is to a circle refer to figure 2 The of this表明叶形越接近圆形 参看图2 同样此参数最大取值 parameter is 1 C Rin Rex 为 1 C Rin Rex 数学模型 2 MATHmodels 公众号 数学模型 微信号 MATHmodels 翻译 周吕文关注 数学模型 公众号 获取更多信息第 8 页 共 33 页 Where C stands for the circularity Rinstands for the radius of in circle Rex其中 C 表示圆度 Rin代表内接圆半径 Rex代表外接 stands for the radius of ex circle 圆半径 4 Form factor4 形状因子 Form Factor Form Factor a famous shape description parameter is another essential in 形状因子这一著名的形状描述参数 是我们在树叶分类 dicator of leaf classification The maximum possible value of this parameter中另一重要的指示参数 它的最大值为 1 is 1 FF 4 Aleaf P2 leaf FF 4 Aleaf P2 leaf Where Pleafstands for the perimeter of leaf 其中 Pleaf表示树叶的周长 5 Edge regularity area index5 边缘面积平滑指数 Edge regularity area index Although the aspect ratio and the rectangularity of two leaves may resemble 有时尽管两片树叶的宽高比和矩度很相似 但它们的实 the contour or the exact shape of two leaves may vary much 际轮廓可能相差甚远 Thus In order to take the different contour of the leaf into consideration 因此 为了考虑不同的轮廓 我们将所有沿树叶轮廓 we join every convex dot along the contour and develop a specific parameter 的凸点连起来 定义一个特殊的参数 外接多边形面积 which we call Bounding Polygon Area The ratio between the leaf area and树叶面积与其外接多边形面积的比例是一个很好的数 this bounding polygon area is a good quantitative factor defined as Edge值因子 我们定义它为边缘面积平滑指数 Regularity Area Index The more this ratio is close to 1 the less jagged and the smoother this leaf s这个比例越接近于 1 叶形的锯齿状程度越低 轮廓 contour is refer to figure 3 The maximum possible value of this parameter越平滑 参看图3 此参数最大可能取值为 1 is 1 ERAI Aleaf BPA ERAI Aleaf BPA Where ERAI stands for the edge regularity area index BPA stands for其中 ERAI 表示边缘面积平滑指数 BPA 表示树叶的 bounding polygon area 外接多边形面积 6 Edge regularity perimeter index6 边缘周长平滑指数 Similarly we develop another parameter Bounding Polygon Perimeter 类似地 我们先指定另一个参数 外接多边形周长 即 the perimeter of the polygon when we join the convex dots of a leaf We所有沿树叶轮廓的凸点所连外接多边形的周长 于是 define the ratio of the convex dot perimeter and the perimeter of the leaf as我们将该外接多边形的周长与树叶的周长之比定义为 Edge Regularity Perimeter Index This time the smaller this ratio is 边缘周长平滑指数 要注意 这一次该比例越小 树叶 数学模型 2 MATHmodels 公众号 数学模型 微信号 MATHmodels 翻译 周吕文关注 数学模型 公众号 获取更多信息第 9 页 共 33 页 the more jagged and irregular the contour of the leaf is refer to figure 4 轮廓越不规则 参差不齐 参看图4 此参数的最大取 The maximum possible value of this parameter is 1 值仍是 1 ERPI BPP Pleaf ERPI BPP Pleaf Where ERPI stands for the edge regularity perimeter index BPP stands其中 ERPI 表示边缘周长平滑指数 BPP 表示外接 for the bounding polygon perimeter 多边形周长 7 Proportional index7 比例因子 Proportional index Since it is also highly critical to capture the spatial distribution of different鉴于树叶不同部位在竖直方向上的空间分布也同样重 portions of a leaf along its vertical axis we divide the minimum bounding要 我们将树叶的最小外接矩形水平地切成 4 块 然后 rectangle into four blocks horizontally and calculate the proportion of the分别计算某块特定区域面积与 4 块总面积之比 我们 leaf area in a particular region to the total leaf which we refer to as the称之为比例因子 Proportional Index PIi area of block i Aleaf PIi area of block i Aleaf Now we can develop a database of six most common leaves containing seven至此 我们可以针对 6 种最常见的树叶 根据以上 7 个 parameters discussed above 参数建立一个数据库 We develop a database of the six most common leaf types in North America Figure 5 using the seven parameters discussed above Table 1 gives the values of the parameters for each leaf type as measured from photos of leaves in 1 5 2Comparison When given a specific leaf we can calculate seven characteristics of it and当给定一个具体的树叶时 我们可以计算它的 7 个特征 compare them with our database by calculating the squared deviation of each of参数 然后将它们与我们建立的数据库进行比对 计算这 parameter of the given leaf from the corresponding standard parameter of each片树叶 7 个参数与某一标准树叶类 共 6 类 参数的方差 category We realize the fact that some of seven parameters are somehow more我们发现 这 7 个参数中有几个比其他的重要 因此 为了使 important than others So in an effort to make our model more accurate and模型更加准确可靠 我们引入偏差指数 Index of Deviation reliable we induce the conception of the Index of Deviation denoted as ID which的概念 用来表示 它是数据库和待分类树叶间方差乘上合 comes from the sum of the squared deviation between the database and the leaf 适权重后 对 7 个参数的求和 to be classified times the proper weight ID 7 i 1 IiWi 数学模型 2 MATHmodels 公众号 数学模型 微信号 MATHmodels 翻译 周吕文关注 数学模型 公众号 获取更多信息第 10 页 共 33 页 Figure 1 Aspect ratioFigure 2 CircularityFigure 3 Edge regularity area index Figure4 Edgeregularity perimeter index 123456 Figure 5 The six most common seen leaf types in North America 1 数学模型 2 MATHmodels 公众号 数学模型 微信号 MATHmodels 翻译 周吕文关注 数学模型 公众号 获取更多信息第 11 页 共 33 页 Table 1 Parameter values for the six leaf types 123456 Rectangularity0 66270 59020 62500 47720 48760 6576 Aspect Ratio0 86150 66000 18000 63830 47920 3111 Circularity0 81400 54320 45640 34540 31230 3311 Form Factor0 91390 62060 28230 24700 36620 4956 ER Area Index0 93220 87800 90910 85000 78800 8895 ER Perimeter Index0 87270 88890 93840 86020 82310 9903 PI10 06490 07690 11790 19090 12990 2920 PI20 29580 35550 22080 38920 36060 4187 PI30 34390 42430 41390 30470 41230 2677 PI40 29540 14330 24740 11520 09700 0220 where each Iiis the squared deviation except that其中 Ii为相应参数的方差 I7例外 I7 1 4 4 j 1 PI j PInew j 2 In order to decide the respective weight more scientifically we resort to the为了科学地确定各参数的权重 Wi 我们求助于层次分析 help of Analytical Hierarchy Process 2 法 2 First of all we build a seven by seven matrix reciprocal matrix by pair com 首先 我们建立一个 7 7 的对称倒数矩阵 或成对比较 parison 矩阵 来做比较 RARCFFERAIERPIPI Rectangularity11 311 41 21 21 7 Aspect Ratio3131221 3 Circularity11 311 41 21 21 7 Form Factor4141331 2 ER Area Index21 221 3111 4 ER Perimeter Index21 221 3111 4 Proportional Index7372441 数学模型 2 MATHmodels 公众号 数学模型 微信号