LeavesClassificationandLeafMassEstimation大学生数模竞赛二等奖.doc

Leaves Classification and Leaf Mass EstimationSummaryFor the first problem, we establish our neural network model to classify leaves of trees by taking eight characteristics of leaf into consideration. The eight characteristics consist of sawtooth number, petiole length, blade length, blade width, blade thickness, leaf area and circular degree. Our results are summarized in a conclusion that we classify leaves into fourteen types including linear, lanceolate, oblanceolate, spatulate, ovat, obovate, elliptic, oblong, deltoid, reniform, orbicular, peltate, perfoliate and connate. Our neural network implement the classification task reliably and correctly.For the second problem, we set up our AHP model to figure out the reasons why leaves have the various shapes and come to a conclusion that gene, auxin, climate and disease are the main reasons which lead to various shapes.For the third problem, we discuss this issue from the perspective of growth evolutionary and hormones, build cells mechanic model to solve this problem and sum up the conclusion that the shapes are inclined to minimize overlapping individual shadows that are cast so as to maximize exposure. The shape is effected by the distribution of leaves within the volume of the tree and its branches. For the fourth problem, we use statistical analysis knowledge to analyse the data among tree profiles, branching structure and leaf shapes, after mathematically analyzing, finally find that leaves shapes have a direct relation with the tree profile and branching structure,For the fifth problem, we formulate our volumetric method for leaf mass estimation and linear regression model for seeking and comparing the correlation between the leaf mass and tree height, tree mass and crown volume. We obtain that crown volume has the highest correlation with tree leaf mass. So we make use of the crown volume to estimate the leaf mass.At last ,we write one page summary sheet of our key findings.Key words: neural network, leaf classification, leaf mass estimation, AHP, leaf shape, volumetric method, linear regression modelContents Contents0. Introduction1. Some Definitions1. General Assumptions1. Symbols2. Problem analysis2. Models36.1 Neural network model to classify tree leaves36.1.1 Neuromime36.1.2 Multi-layer perceptron network46.1.3 Back-propogation56.1. 4 NNs use to classify leaves66.2 Studying the reasons of the various shapes that leaves have.66.2.1 Set up a AHP model to value these base factors66.2.2 Paired comparison matrix structure76.2.3 Calculation of the weight vector and the consistency test86.3 Optimize leaves shape for maximize exposure96.3.1 Explain and answer requirment96.3.2 Set up a Elastic mechanics model96.4 Tree profile and branching structures influence on leaf shape.106.4.1 Analysis about the impact of tree profile to leaf shape106.4.2 Electric tree branch angles impact analysis136.5 Estimation of the leaf mass146.5.1 Build up a volumetric model146.5.2 The correlation of leaf mass vs. mean crown radiuss cubic156.5.3 The correlation between the leaf mass and the height of the tree166.5.4 The dry leaf mass vs. the volume of the tree176.5.5 The relationship between the leaf mass and mean crown radius18. Conclusions19. Strengths and Weakness of the Model19. Future Work20. References20Key Findings21. IntroductionAs is known to all，there are not two leaves exactly alike. Plant leaves have diverse and elaborate shapes and venation patterns. The beauty of them has attracted curiosity of many people involving biologists, physicists, mathematician, artists, computer scientists, etc. for a long time. The leaf study of forests and of individual tree is important to understand resource allocation of trees, atmospherebiosphere exchange processes, and the energy budget, it would also be valuable for individual tree growth.The aim of this article is to develop models for leaf shapes classification and to figure out the main factors which lead to the various leaf shapes. At the same time, we find out the interaction between tree (Its profile/branching structure) and tree leaf. Though there are so many methods to estimate the leaf mass. We solve this problem through a correlation between the leaf mass and the size characteristics of the tree. Some Definitionsl LeafTo a plant, leaves are food producing organs. Leaves "absorb" some of the energy in the sunlight that strikes their surfaces and also take in carbon dioxide from the surrounding air in order to run the metabolic process of photosynthesis. l Phototropism1Phototropism is directional growth in which the direction of growth is determined by the direction of the light source. It causes the plant to have elongated cells on the farthest side from the light. Phototropism is one of the many plant tropisms or movements which respond to external stimuli.l Polar Auxin Transport(PAT) 2PAT is the regulated transport of the plant hormone auxin in plants. It is an active process, the hormone is transported in cell-to-cell manner and one of the main features of the transport is its directionality (polarity). The polar auxin transport has coordinative function in plant development, the following spatial auxin distribution underpins most of plant growth responses to its environment and plant growth and developmental changes in general.l Apical Dominance3It is the phenomenon whereby the main central stem of the plant is dominant over other side stems; on a branch the main stem of the branch is further dominant over its own side branch. General Assumptionsl The influence of variation in thickness of leaves can be neglect.l We do not take the influence of the artificial factor into consideration.l Regardless of the influence of deformation of cell.l We regard the crown of the tree as a half sphere.l The leaves in the crown are evently distributed.l Neglect genic mutation influence. SymbolssymbolInstructionsclimate, disease, auxin, genethe largest eigenvalueeigenvectorsconsistency ratioconsistency indexthe point a leaf locate on coordinate systema coefficient related on leaf shapeTree branch anglethe leaf mass（Mark：Other symbols will be given in the specific model）. Problem analysis The first question requires us to build a mathematical model to describe and classify leaves. We think that the standard of classification is the shape of leaf. So we need to study the characteristics of leaf and to ensure that how to define a type of leaf by the combination of some characteristics. In addition, we should figure out how and how much these characteristics have influence on defining a type of leaf. So we take eight characteristics into consideration including master sawtooth number, petiole length, blade length, blade width, blade thickness, leaf area and circular degree. We find that neural networks hold the capacity to process huge data and can be used to describe cognition, classification and some other intelligent behaviors. So we make a decision to use the neural networks to make a classification of tree leaves.The second question requires us to figure out the reasons that why the leaves have various shapes. It is easy to know that the shape of a leaf mainly decided by the gene of the tree. But we know that the leaves of the same tree always have different shapes with the same genes. So we can draw a conclusion that the shape of leaf is not only decided by the gene of the tree as well as influenced by environmental factors. We choose these factors to analyze the specific influence on the forming process of the shape of leaf by using an AHP model.The third question wants us to get know of that whether the leaf have a “hobby” to keep a state to maximize exposure and minimize overlapping individual shadows that are cast. In addition, if the shape of leaf is effected by the distribution of branches and the volume of the tree. So we should make a survey to make it clear that the relationship between crowns surface area and the leaf area of a total tree. Then we need to study the sunshines influence on the formation of leaf.We think the fourth questions aim is to research that whether the tree profile or the branching structure has influence on leaf shape. In this question we think that the “profile” of a tree is the crown, and there is a possibility that different crown has different influence on the leaf shape.The last question is require us to find a correlation between the leaf mass and the size characteristics of the tree (height, mass, volume defined by the profile), and then make use of one or more of this characteristics to estimate the leaf mass of a tree. Models6.1 Neural network model to classify tree leavesOur duty is to find an approach to how to classify leaves. We use Neural network model to classify tree leavesAs for classification, Neural network model is greatly able to get a fairly ideal conclusion. To distinguish one leaf shape patterns from each other, Neural network model is optimal. Through a study sample progress on and off, in which we adjust accordingly. Eventually our model is so “smart” as to identify different leaf shapes. A leaf sample characterize 8 features as mentioned-above. And it is necessary for us to explain the model and we separate as three parts to expatiate.6.1.1 NeuromimeThe follow graph is a base part of. Figure 6.1-1： neuromimeSolution to input signal:(1)Where is the weight, is the input node value:(2) is Threshold value:(3) is activation function, is the output of a neuron in the successive layer. The activation function is a nonlinear function and is given by:(4)6.1.2 Multi-layer perceptron network This is the main structure of . Figure 6.1-2：Multi-layer perceptron networkThe structure of the Artificial Neural Network ANN in this work contains three layers: input, hidden and output layers as shown in figure 6.1-2. We use input layer to input the characteristics of the leaves. Each layer contains and nodes. The node is also called neuron or unit. This study summarized eight factors for ANN input, that is to say . The eight input units are sawtooth number, petiole length, blade length, blade width, blade thickness, leaf area and circular degree.For the hidden layer we make. The function of the output layer is to output classified information corresponding to the input data. The value of ranges from the types of leaves we need to identify. The is denoted as numerical weights between input and hidden layers, between hidden and output layers as also shown in figure 6.1-2.In fact, as for a sample of “”, the input of the hidden layer is:(5)The corresponding output state:(6)Therefore, the superimposed signal received is:(7)The final output of the network is:(8)We hope the final output is idealization. For example. For example,after learning maple leaf s features, if the output is like the form of , we called the output like this the ideal output, the ideal output is noted for .Figure 6.1-2： Different types of shapesLinear. Lanceolate. Oblanceolate. Spatulate. Ovate. Obovate. Elliptic.Oblong. Deltoid. Reniform. Orbicular. Peltate. PerfoliateConnate.6.1.3 Back-propogationIn order to minimizing the differences between actual output and desired output，we choose BP algorithm,which is one part of . As set forth, the error obtained when training a pair (pattern) consisting of both input and output given to the input layer of the network is given by:(9)Where is the th component of the desired output vector and is the calculated output of th neuron in the output layer.Combine with , we can draw:(10)This is a nonlinear function which is continuously differentiable. In order to obtain the minimum point and the value, the most convenient is to use the steepest descent method to get the minimal value of , when , we get the ideal value of the variables and .6.1. 4 NNs use to classify leaves Through , single several models leaves and grouping and number of them. Then , learning each group, is acquaintance each models. If want to classify one leaf. We are able to let to solve this problem, eventually, we classify the leaf as like-model.6.2 Studying the reasons of the various shapes that leaves have.Leaves have a variety of forms. There are lots of reasons account for leaves varying in shapes and size, listed as follows: Overall, the reasons can be divided into external and internal factors.External factors:l Seasons and climate (including wind, sunlight, moisture, temperature);l Plant diseases and insect pests;l Artificial factor;Internal factors:l Deformation of cells, moisture loss of Mesophyll cells may cause volume decrease;l Phytohormone auxin;l Difference gene.we believe that there exits 4 base factors that lead to the variety of leaves shape. They are climate, disease, phytohormone and gene. And we endeavor find out reasons to them.climate: the change of sun shine，water，temperature，humidity which alters leaves shape.disease: through effecting the activity of an enzyme, so that influence leaves shape.Phytohormone auxin: have influence on gene expressiongene: through DNA determine the general leaf shape6.2.1 Set up a AHP model to value these base factorsWe solve this problem based on the reasons listed above. After analyzing all of them, we hold an opinion that human attempt is usually fairly haphazard. Since we view all the leaves living environment is stable, we dont take artificial factor into consideration. We definite "total impact" as "target layer”, and climate, disease, phytohormone, gene as the "criterion layer". As shown in the following figure 6.2-1:Total impactClimateDiseaseAuxinGeneFigure 6.2-1：reasons for the various shapes6.2.2 Paired comparison matrix structureTo analyze the effects of electric vehicles widespread use on the environment, social, economic and health, we each take two factors:(11)(12)They are used to represent environmental, economic, social and health by turns. All results are available the following pairwise comparison matrix:(13)(14)Obviously:(15)The result we used paired comparison of the paired comparison matrix is:When we take comparison of them qualitatively, there are five clear hierarchy in people's minds usually, which is expressed as:Table 6.1-1: the meaning of the Measure 1-9Meaning1 and have the same influence3has a slightly stronger influence than 5has stronger influence than 7has significantly stronger influence than 9has Absolutely stronger influence than 2,4,6,8