Learning Affine Invariant Pattern Recognition Using High-Order NeuralNetworksMohammed Sadiq Al-RawiComputer Science Dept., Jordan UniversityAmman, Jordan rawi@.jo.jo/faculties/index.htmAbstractIn this work, high order neural networks have been successfully used to recognize affine transformed images.Affine invariants are important features used to classify images of objects deteriorated by affine transformations. The dataset used in training and testing contains two classes, the star image and the wave image. The proposedhigh order neural network is inspired from the system ofaffine moment invariants, but the implementation is totally different and the performance is much higher. More than 27700 images are generated using affine transformations via four degrees of freedom and only 96images have been used in training and 27616 images have been used in testing. Some of the testing images areseriously deteriorated by the affine transform and they are hard to recognize even for the human observer yet therecognition accuracy is 97%. A white Gaussian random noise has also been added to all images under investigation and the proposed method is superior compared to previous methods. All testing experiments are performed using a set that is disjoint with that used intraining. The complexity of computing is of the order ofaffine moment invariants which is very fast compared toother techniques like cross-weighted moments and multi-scale auto-convolution.Keyword: Affine transform; invariant pattern recognit-ion; high order neural networks; object recognition;moment invariants.1. IntroductionObject recognition where shape of an object may be changed according to the imaging geometry is importantin computer vision and pattern recognition applications. The imaging geometry is represented by a projective transformation may be approximated by a simpler transformation called affine transformation. The complexity of shape generation of affine transformationsstands in the middle between rigid and projective transformations, yet they produce variant images of the same object that re not easy to recognize. Twoapproaches are used, mainly, to extract features that are invariant to affine transformations local and global. The local is based on detecting vertices and edges of the shape [1][2][3][4], while the whole shape that is placedon a black background is used to extract the invariant features in the global approach [5][6][7]. Both approaches, however, are based on well established mathematical theories and algorithms to ensure the invariance property of the extracted features or extractedregions of interest. Features that depends on moments like affine moment invariants [6] are very sensitive tonoise, cross weighted moments [4] are less sensitive to noise but they are very computationally expensive, and ina recent work [7] multi-scale autoconvolution have been used to generate affine invariant features that are more resistant to noise and faster than cross-weighted moments [4] but slower than affine moment invariants [6].When it comes to learning with neural networks, all previous approaches researched this problem using features that posses invariance [8][9][10][11][12], which means that they are linearly separable and a simple distance function or k-nearest neighbor algorithm can solve the problem easily. Using affine invariant features as input to neural networks would not be a good choice since the later is recommended as a powerful nonlinear classifier and the extracted features are already separable. Aside from this, there is another problem related to image recognition via neural networks which is the size of input image that is considered larger than the networks ability. Therefore, techniques like coarse coding [8] have beenproposed to manipulate this large size. Those techniques, however, were not good solutions due to the fact that the image representation is degraded in a way that accurate recognition would be impossible. In [8][9][10][11][12], high order neural networks up to order three have beenadopted and invariant features to only image translation and rotation have been used as input to those networks,their implementation (faced by the very large search24space) lacks the blind denonlinearity that the high order neural network should perform and non of them tried to train a network via affine invariants.However, a neural system that is capable of learning affine invariant shape recognition without feature extraction is still not available. In this work, the developed neural network affine invariant recognition does not depend on any mathematical foundation that supports the invariance property of some extracted features, rather than this the invariance property is learned via the proposed high order neural network. It is important to mention that previous neural based learning invariance is limited to small size images with very limited geometric variations that included translation, and rotation at most [12] and the most undesired issue in a generalized learning system which is feature extraction.2.High Order Neural Networks and AffineInvariantsGeneral learning and recognition systems are attracting more and more attention due to their capability to accommodate to various changes and the role they will play the automated intelligent systems of the near future. It would be of a great importance that a system learns invariance without a strong mathematical foundation to achieve invariance property. Neural networks and their variations founded a strong base to learning theories [13][14][15][16], they can be used as nonlinear classifiers without having prior knowledge of the data nonlinearity and how different data related to different classes are generated. There are many neural networks limitations related to the size of input image, neural networks topologies, and learning strategies [14].Learning invariance recognition with neural networks that can search all the parameters and their changes is impractical with respect to nowadays computers and learning strategies, therefore, it is necessary to assume many hypothesis to ease up the parameters search process, i.e., the network is a feedforward, interconnected, has no intraconnections, fixed number of neurons and layers, very few input neurons at the input layer, etc.. These limitations might however, trap the neural network in a local minimum without relaxing and converging into a solution [14].High order neural networks contain multiplication nodes in addition to summation nodes therefore they have the drawback of many local minima. Regularization techniques (using some optimization procedure) are important to reduce the topology of the network [10] high order neural networks proved as an efficient classifier however. The proposed system is as discussed next.Affine Moment InvariantsTo shorten many of the problems related to the design and topology of neural networks, we may make use of some existing system(s) or phenomena(s). Affine moment invariants [6] can work as such a system that inspires the design of a neural network that is applied to affine invariant shape recognition. Let's take a look at the definition of geometric moments and the affine features extracted from them. For an image f, geometric moments [17] are defined as follows:1111(,)p qpqm x y f x y dx dy−−=∫∫, (1) where 0,1,2,...p q+=is called the moment order. Geometric moments as given in (1) are not invariant to any transform and to make moments invariant to translation we should use central moments given as follows:1111()()(,)p qpq c cx x y y f x y dxdyµ−−=−−∫∫, (2) should be used instead of geometric moments where (,)c cx y is the center of mass that may be found usingthe zero and first order moments given as1000/cx m m=and0100/cy m m=, and for a digital image all integrations are replaced with summations. Many invarinats can be derived from (2) (see [6] for details), the simplest invariant is derived from the second order moments given by:2420021100()/Iµµµµ=−. (3) One can obtain higher order moment invariants too as described in [5]. It must be noted that the credit goes to Hu [18] for deriving invariant moments and their implementation in pattern recognition, despite the fact that his original work on affine invariants contained errors that were corrected lately in [6].The Moment Computation NetworkThe affine moment invariants shown above can be transformed into a moment computation neural network then to a multilayer network computation system, see Fig.1 for the moment computation network. The layer that generates geometric moments is the first hidden layer.Fig. 1: The moment computation network that computes moments up to the third order resulting in ten geometric moments, and all activation functions are linear. The weight from the input (,)f x y to the neuron indexedpqm is p qx y such that ,{1,1}x y∈−.Central moments are linear combinations of different geometric moments, so an extra summation layer after the first hidden layer might be needed to simulate the computation of central moments. The high order neural networks are adopted because there are multiplicationsm m m m m m m m mmΣ25between different moments in the affine moment invariants [6]. Therefore, it is adequate to implement the high order (multiplication) units after the first hidden layer. When the design is accomplished the, it is easy to see that the final neural network is an eight layer high-order neural network with many shortcut connections and other connections.Simplification of the High Order Neural NetworkInstead of the complicated eight layers high-order neural network, it might be more appropriate to try the below simplified topology:1- The input layer is fully feedforward connected and passed to the first hidden layer which is consisted of ten summation neurons each has a linear activation function. The weights between the input and the first hidden layer are fixed and their values are adopted from geometric moment monomials. Further improvement which is also inspired from the nominator of affine moment invariants as shown in (3) is accomplished by including and additional four units, these are, 234{1/,1/,1/,1/}m m m m where mis given by(1111(,)m f x y dxdy −−=∫∫, (4)for an image f of size M N ×. It is obvious that there will be a total of fourteen units at the first hidden layer.2- The outputs of the first hidden layer are passed into a unit that performs normalization so that the fourteen outputs have zero mean and unit variance. This normalization unit is considered to be an essential part of the first hidden layer.3- The high order layer comes after the first hidden layer, where each connection to each unit at this layer have a fixed weight with value one and the activation function of each unit is linear. Because there are fourteen outputs from the previous layer, the order of this high-order layer is fourteen which yields 105 units (14*(14+1)/2). As an example of the proposed high order units, suppose we have three outputs from the first hidden layer 123{,,}m m m with their signals passed to the high order layer, in this case, we may calculate up to the third order yielding (3*4/2)= 6 high order units, they are given by the set 222123121323{, , , , , }m m m m m m m m m . Notethat the square of the inputs as a high order unit (e.g.21m ) was not adopted in previous works and it is included here by inspiration of the moment computation network.4- A direct shortcut connection from the first hidden layer (fourteen units) to the third hidden summation layer is added, therefore, each unit at the third summation layer receives 119 incoming connections from previous layers (105 connections come from the second hidden high order layer and 14 connections come from the first hidden layer). The activation function at each neuron of the third hidden summation layer is a hyperbolic-tangent. The number of neurons at this layer which is called N 23 is left variable and can be adjusted as needed.5- All signals output from the third hidden layer arepassed to the fourth output layer that has one output neuron with a hyperbolic-tangent activation function. We may increase the number of output units if we have more than two categories to train and test.According to the above, the eight layers high order neural network is reduced into a five layers high order neural network.3. Experimental ResultsThis is the conclusion. The major contributions of this In the below experiments, we may deploy the simplified five-layer high order neural networks inspired from moment invariants in training and testing as well as comparisons with other methods. The classes to be used in training and testing of all neural networks in this paper are only two; the star and the wave images see Fig. 2.For each class, many affine transformed images have been generated according to the following affinetransform expression 1112x a x a y′=+ and 2122y a x a y ′=+, and the affine transformed image is ([])([])TTaff f x y f x y ′′=A ……(5)where the affine transformation matrix is given by11122122a a a a =⎡⎤⎢⎥⎣⎦A , (6) then to ensure the inverse of the transform, the constraint ()0≠A determinant should be valid. To obtain many transformed images, the following values are chosen for each of the coefficients [,],ij a a a a step δδ∈−−+,...,, for all ,1,2i j =. In each train/test experiment, the step value is chosen in a way that the training and testing sets are disjoint, i.e., there is no chance that the affine transformation matrix A is repeated. Moreover, it is the intention to complicate the task such that ()ij a absolute of the training set is less than ()ij a absolute of the testing set. Therefore, the testingimages have higher variations and many of them are larger in size than those used in training. In all cases, if 00m of the affine transformed image is less than 200(indicating a very distorted image that looks like a distorted line) then this image is rejected from training and testing. As an example of how the images are generated, let us consider the case for using {1,0.5,...,1}j i a ∈−−, which means: 11{1,0.5,...,1}a =−,Fig. 2: The star and the wave images are the originalimages used to generate the affine transformed versions. Each of them is a 256*256 sized image.2612{1,0.5, (1)a =−,21{1,0.5, (1)a =−, and22{1,0.5,...,1}a =−. Since we have two original images the star and the wave,there will be a total of 2*[(1-(-1))/0.5+1]^4=2*(5^4)=1250 possible generated images ofwhich only 992 images are eligible image samples due toexcluding ()0=A determinant parameters and the power less than 200 exclusion principle. Some of the affine transformed images are shown in Fig. 3.Two sets of training and testing images have beengenerated from the star and wave images, they are referred as Experiment1 and Experiment2, see Table 1 for the affine parameters used and the number of generated images. All experiments are performed using MATLAB simulation environment.Advanced features are used for comparisons with the proposed method; these are features of the affine moment invariants [6] and the multi-scale autoconvolution. As for the multi-scale autoconvolution [7], a total of 29 featurevalues have been calculated for each image then they are used as inputs to a multi-layer neural network. The affine moment invariants are calculated according to [6] which yield a feature vector of 4 values. Due to the largedynamic numerical range of affine moment invariants, each feature vector is normalized by taking the absoluteof logarithm based 2 of absolute of the affine momentvalue. Then, each logarithmically normalized vector is statistically normalized to have zero mean and unit variance one. The statistically normalized affine moment invariants are used as inputs to a multilayer neuralnetwork.Table 1: Parameters used to generate the affinetransformed image datasetExperiment NameAffine Parameters Used for Generating Training Images, Total no. of Generated ImagesAffine ParametersUsed for Generating Testing Images, Total no. of GeneratedImagesExperiment1{1,0,1}ij a ∈−, 96{2,0.6, (2)ij a ∈−−, 4540Experiment2{1,0,1}ij a ∈−, 96{3,0.6, (3)ij a ∈−−,27616More recognition experiments have been carried out after adding a white Gaussian random noise with variances {0.4, 0.7, 1,2,4,6,10} and zero mean to all the training and the testing images. Then, a simple smoothing filter is applied to all images to eliminate the noise effect followed by image thresholding prior to the learning process. Since all images are binary (a white object on a black background), the shapes will have a degraded edge and the interior will not be affected by that noise except with some holes inside. Other noise distributions (i.e., salt and pepper) would have similar and less deterioration effects on the images. See Fig. 4 for an image with added noise.The proposed high order-neural network achieved an excellent success as listed in Tables 2 & 3. They areFig. 3: Some affine transformed star and wave images, they are displayed as they appear compared to the size of original star and wave images as shown in Fig. 2. The sizes of images as they appear left to right, top to bottom are (684*513), (513*793), (155*1021), (513*546), (700*513), (513*769), and (155*307).a)b)Fig. 4: a) The wave image with added noise of variance 10, b) The image in (a) is filtered with a 7x7 average filter, then converted into a binary image with no further processing is applied to the image shown in (b).27trained with backpropagation of error using the scaled conjugate gradient learning algorithm [15], and the number of hidden neurons is N23=5 , N23=20 and/or N23=50. Averages of 20 trails have been performed for each training/testing experiment.All high order neural network experiments converged in less than 100 epochs and all errors occurred in the wave image category while all star images are classified correctly. In both the proposed high order neural network and the multi-scale autoconvolution N23=5 is adequate for the network to converge and increasing this number of neurons does not affect the result. Normalizing the 29 feature vector of the multi-scale auto convolution to have zero mean and unit variance did not improve the recognition accuracy.Table 2: Recognition accuracy results obtained using high order neural networks proposed in this work in comparison to other approaches on dataset of Experiment1. A white Gaussian noise is added with the variances shown.Noise Variance Proposed HighOrder NeuralNetworkMulti-ScaleAutoconvoluti-on [7]AffineMomentInvariants[6]No noise 98% 100% 91%0.4 98% 99% 60%0.7 98% 99% 60%1.0 98% 98% 59%2.0 98% 98% 58%4.0 98% 95% 57%6.0 96% 89% 55%10.0 92% 85% 55%Table 3: Recognition accuracy results obtained using high order neural networks proposed in this work in comparison to other approaches on dataset of Experiment2. A white Gaussian noise is added with the variances shown.Noise Variance Proposed HighOrder NeuralNetworkMulti-ScaleAutoconvoluti-on [7]AffineMomentInvariants[6]No noise 97% 97% 84%0.4 97% 97% 55%0.7 97% 97% 55%1.0 97% 97% 54%2.0 97% 97% 53%4.0 97% 93% 51%6.0 95% 87% 47%10.0 90% 82% 45%It is worth mentioning that the affine moment invariants needs a very large number of epochs to converge and the trained neural network never reaches the error goal when the images contain noise, in other words the convergence error is always high. For example, one experiment is leftto something near 100,000 epochs and the convergence error is something near 0.5. Performance comparison of the proposed method to previous methods is shown graphically in Fig.5 & 6, and the computational complexity is shown in Fig. 7.0.20.40.60.811.200.40.7124610Noise VarianceRecognitionAccuracyFig. 5: Using Experimen1 data set the recognition accuracy comparison for the proposed high order neural network (HONN), and multi-scale auto convolution (MSA), and affine moment invariants (AMI).0.20.40.60.811.200.40.7124610Noise VarianceRecognitionAccuracyFig. 6: Using Experimen2 data set the Recognition accuracy comparison for the proposed high order neural network (HONN), and multi-scale auto convolution (MSA), and affine moment invariants (AMI).4.ConclusionsAll previous works used as inputs to neural networks some features that are invariant to the imaging transformation and linearly separable which is a trivial task for neural networks. It is the first time that a neural network learns the affine invariant pattern recognition by presenting the whole image as input to the network and without feature extraction and this is the main contribution of this work. The image dataset used in this work is obtained by varying the four affine transformation parameters to obtain a large number of affine transformed images that are hard to recognize even for the human observer. Using a high order neural network of five hidden layers, the recognition accuracy ismore than 97% when using only 96 images in training and 27616 images in testing. To make experiments harder, a white Gaussian random noise have been added to all images (those used in training as well as testing) and the recognition accuracy is better than the bestuptodate known method which is the multi-scaleautoconvolution [7]. The time complexity ofimplementing the proposed high order neural network is the same as that of affine moment invariants [6] which is2()O N , and it is faster than that of cross-weightedmoments [4] which is 4()O N and multi-scale autoconvolution [7] which is 22(log )O N N . Theproposed high order neural network makes use of the general neural networks learning strategies without having such a strong neither mathematical nor algorithmic background as that of affine moment invariants and multi-scale autoconvolution. 024681012321282243241651260870480896992M i l l i o n sImage Size N=(32 to 992) for an NxN imageA p p r o x i m a t e C o m p u t a t i o n s n e e d e dFig.7: Computational complexity comparison between the time needed for the proposed high order neuralnetwork (HONN) and multi-scale auto convolution (MSA).5. References [1] Y. Lamdan, J. T. Schwartz, and H. J. Wolfson. Affine invariant model-based object recognition. IEEE Trans. Robot. Automat ., 6: 578–589, 1990. [2] S. Kurogi, Competitive neural network for affine invariant pattern recognition. In Proc. Int. Joint Conf. Neural Networks, 181–184, 1993. [3] G. Bebis, M. Georgiopoulos, N. da Vitoria Lobo, and M. Shah. Learning affine transformations. Pattern Recognition, 32: 1783–1799, 1999.[4] Z. Yang and F. Cohen. Cross-weighted moments andaffine invariants for image registration and matching. IEEE Trans. Pattern Analysis and Machine Intelligence, 21(8): 804-814, 1999. [5] G. Taubin, and D.B. Cooper. Object recognitionbased on moment (or algebraic) invariants. In Geometric invariance in computer vision, J L Mundy and A Zisserman(eds.), MIT Press, Cambridge, Mass, 375-497, 1992.[6] J. Flusser, and T. Suk. Pattern recognition by affinemoment invariants. Pattern Recognition, 26(1): 167-174, 1993.[7] E. Rahtu, M. Salo, and J. Heikkila. Affine invariant pattern recognition using multiscale autoconvolution. IEEE Trans. Pattern Analysis and Machine Intelligence, 27(6): 908-918, 2005. [8] L. Spirkovska, and M.B. Reid. Coarse-coded higher-order neural networks for PSRI object recognition. IEEE Trans. Neural Networks, 4(2): 276 – 283, 1993.[9] C.L. Giles, and T. Maxwell. Learning invariance andgeneralization in high-order neural networks. AppliedOptics, 26(23): 4972-4978, 1987.[10] G. Thimm, and E. Fiesler. High-order and multilayer perceptron initialization. IEEE Trans. Neural Networks, 8(2): 249-259, 1997.[11] S.J. Perantonis, and P.J.G. Lisboa. Translation,rotation, and scale invariant pattern recognition by high-order neural networks and moment classifiers. IEEE Trans. Neural Networks, 3(2): 241-251, 1992. [12] Z. He, and M.Y. Siyal. Improvement on higher-order neural networks for invariant object recognition.International Journal of Computer Vision, 10(1): 49-55, 1999.[13] D. E. Rumelhart, G.E. Hinton, and R.J. Williams. Learning internal representations by error propagation. In D. E. Rumelhart and J. L. McClelland, eds. Parallel Data Processing, Cambridge, MA: The M.I.T. Press, 1: 318-362, 1986.[14] O. Duda E. Hart, and G. Stork. Pattern classification.Wiley-Interscience; 2nd edition, 2000. [15] M.F. Miller. A scaled conjugate gradient algorithmfor fast supervised learning. Neural Networks, 6(4): 525-533, 1993. [16] J. Elman. Finding structure in time. CognitiveScience, 14: 179-211, 1990.[17] C. Teh, and R. Chin. On image analysis by the method of moments. IEEE Trans. Pattern Analysis and Machine Intelligence, 10: 291-310, 1988.[18] M.K. Hu. Visual pattern recognition by momentinvariants. IRE Trans. Information Theory, 8: 179-187, 1962.。
机器学习专业词汇中英⽂对照activation 激活值activation function 激活函数additive noise 加性噪声autoencoder ⾃编码器Autoencoders ⾃编码算法average firing rate 平均激活率average sum-of-squares error 均⽅差backpropagation 后向传播basis 基basis feature vectors 特征基向量batch gradient ascent 批量梯度上升法Bayesian regularization method 贝叶斯规则化⽅法Bernoulli random variable 伯努利随机变量bias term 偏置项binary classfication ⼆元分类class labels 类型标记concatenation 级联conjugate gradient 共轭梯度contiguous groups 联通区域convex optimization software 凸优化软件convolution 卷积cost function 代价函数covariance matrix 协⽅差矩阵DC component 直流分量decorrelation 去相关degeneracy 退化demensionality reduction 降维derivative 导函数diagonal 对⾓线diffusion of gradients 梯度的弥散eigenvalue 特征值eigenvector 特征向量error term 残差feature matrix 特征矩阵feature standardization 特征标准化feedforward architectures 前馈结构算法feedforward neural network 前馈神经⽹络feedforward pass 前馈传导fine-tuned 微调first-order feature ⼀阶特征forward pass 前向传导forward propagation 前向传播Gaussian prior ⾼斯先验概率generative model ⽣成模型gradient descent 梯度下降Greedy layer-wise training 逐层贪婪训练⽅法grouping matrix 分组矩阵Hadamard product 阿达马乘积Hessian matrix Hessian 矩阵hidden layer 隐含层hidden units 隐藏神经元Hierarchical grouping 层次型分组higher-order features 更⾼阶特征highly non-convex optimization problem ⾼度⾮凸的优化问题histogram 直⽅图hyperbolic tangent 双曲正切函数hypothesis 估值，假设identity activation function 恒等激励函数IID 独⽴同分布illumination 照明inactive 抑制independent component analysis 独⽴成份分析input domains 输⼊域input layer 输⼊层intensity 亮度/灰度intercept term 截距KL divergence 相对熵KL divergence KL分散度k-Means K-均值learning rate 学习速率least squares 最⼩⼆乘法linear correspondence 线性响应linear superposition 线性叠加line-search algorithm 线搜索算法local mean subtraction 局部均值消减local optima 局部最优解logistic regression 逻辑回归loss function 损失函数low-pass filtering 低通滤波magnitude 幅值MAP 极⼤后验估计maximum likelihood estimation 极⼤似然估计mean 平均值MFCC Mel 倒频系数multi-class classification 多元分类neural networks 神经⽹络neuron 神经元Newton’s method ⽜顿法non-convex function ⾮凸函数non-linear feature ⾮线性特征norm 范式norm bounded 有界范数norm constrained 范数约束normalization 归⼀化numerical roundoff errors 数值舍⼊误差numerically checking 数值检验numerically reliable 数值计算上稳定object detection 物体检测objective function ⽬标函数off-by-one error 缺位错误orthogonalization 正交化output layer 输出层overall cost function 总体代价函数over-complete basis 超完备基over-fitting 过拟合parts of objects ⽬标的部件part-whole decompostion 部分-整体分解PCA 主元分析penalty term 惩罚因⼦per-example mean subtraction 逐样本均值消减pooling 池化pretrain 预训练principal components analysis 主成份分析quadratic constraints ⼆次约束RBMs 受限Boltzman机reconstruction based models 基于重构的模型reconstruction cost 重建代价reconstruction term 重构项redundant 冗余reflection matrix 反射矩阵regularization 正则化regularization term 正则化项rescaling 缩放robust 鲁棒性run ⾏程second-order feature ⼆阶特征sigmoid activation function S型激励函数significant digits 有效数字singular value 奇异值singular vector 奇异向量smoothed L1 penalty 平滑的L1范数惩罚Smoothed topographic L1 sparsity penalty 平滑地形L1稀疏惩罚函数smoothing 平滑Softmax Regresson Softmax回归sorted in decreasing order 降序排列source features 源特征sparse autoencoder 消减归⼀化Sparsity 稀疏性sparsity parameter 稀疏性参数sparsity penalty 稀疏惩罚square function 平⽅函数squared-error ⽅差stationary 平稳性（不变性）stationary stochastic process 平稳随机过程step-size 步长值supervised learning 监督学习symmetric positive semi-definite matrix 对称半正定矩阵symmetry breaking 对称失效tanh function 双曲正切函数the average activation 平均活跃度the derivative checking method 梯度验证⽅法the empirical distribution 经验分布函数the energy function 能量函数the Lagrange dual 拉格朗⽇对偶函数the log likelihood 对数似然函数the pixel intensity value 像素灰度值the rate of convergence 收敛速度topographic cost term 拓扑代价项topographic ordered 拓扑秩序transformation 变换translation invariant 平移不变性trivial answer 平凡解under-complete basis 不完备基unrolling 组合扩展unsupervised learning ⽆监督学习variance ⽅差vecotrized implementation 向量化实现vectorization ⽮量化visual cortex 视觉⽪层weight decay 权重衰减weighted average 加权平均值whitening ⽩化zero-mean 均值为零Letter AAccumulated error backpropagation 累积误差逆传播Activation Function 激活函数Adaptive Resonance Theory/ART ⾃适应谐振理论Addictive model 加性学习Adversarial Networks 对抗⽹络Affine Layer 仿射层Affinity matrix 亲和矩阵Agent 代理 / 智能体Algorithm 算法Alpha-beta pruning α-β剪枝Anomaly detection 异常检测Approximation 近似Area Under ROC Curve／AUC Roc 曲线下⾯积Artificial General Intelligence/AGI 通⽤⼈⼯智能Artificial Intelligence/AI ⼈⼯智能Association analysis 关联分析Attention mechanism 注意⼒机制Attribute conditional independence assumption 属性条件独⽴性假设Attribute space 属性空间Attribute value 属性值Autoencoder ⾃编码器Automatic speech recognition ⾃动语⾳识别Automatic summarization ⾃动摘要Average gradient 平均梯度Average-Pooling 平均池化Letter BBackpropagation Through Time 通过时间的反向传播Backpropagation/BP 反向传播Base learner 基学习器Base learning algorithm 基学习算法Batch Normalization/BN 批量归⼀化Bayes decision rule 贝叶斯判定准则Bayes Model Averaging／BMA 贝叶斯模型平均Bayes optimal classifier 贝叶斯最优分类器Bayesian decision theory 贝叶斯决策论Bayesian network 贝叶斯⽹络Between-class scatter matrix 类间散度矩阵Bias 偏置 / 偏差Bias-variance decomposition 偏差-⽅差分解Bias-Variance Dilemma 偏差 – ⽅差困境Bi-directional Long-Short Term Memory/Bi-LSTM 双向长短期记忆Binary classification ⼆分类Binomial test ⼆项检验Bi-partition ⼆分法Boltzmann machine 玻尔兹曼机Bootstrap sampling ⾃助采样法／可重复采样／有放回采样Bootstrapping ⾃助法Break-Event Point／BEP 平衡点Letter CCalibration 校准Cascade-Correlation 级联相关Categorical attribute 离散属性Class-conditional probability 类条件概率Classification and regression tree/CART 分类与回归树Classifier 分类器Class-imbalance 类别不平衡Closed -form 闭式Cluster 簇/类/集群Cluster analysis 聚类分析Clustering 聚类Clustering ensemble 聚类集成Co-adapting 共适应Coding matrix 编码矩阵COLT 国际学习理论会议Committee-based learning 基于委员会的学习Competitive learning 竞争型学习Component learner 组件学习器Comprehensibility 可解释性Computation Cost 计算成本Computational Linguistics 计算语⾔学Computer vision 计算机视觉Concept drift 概念漂移Concept Learning System /CLS 概念学习系统Conditional entropy 条件熵Conditional mutual information 条件互信息Conditional Probability Table／CPT 条件概率表Conditional random field/CRF 条件随机场Conditional risk 条件风险Confidence 置信度Confusion matrix 混淆矩阵Connection weight 连接权Connectionism 连结主义Consistency ⼀致性／相合性Contingency table 列联表Continuous attribute 连续属性Convergence 收敛Conversational agent 会话智能体Convex quadratic programming 凸⼆次规划Convexity 凸性Convolutional neural network/CNN 卷积神经⽹络Co-occurrence 同现Correlation coefficient 相关系数Cosine similarity 余弦相似度Cost curve 成本曲线Cost Function 成本函数Cost matrix 成本矩阵Cost-sensitive 成本敏感Cross entropy 交叉熵Cross validation 交叉验证Crowdsourcing 众包Curse of dimensionality 维数灾难Cut point 截断点Cutting plane algorithm 割平⾯法Letter DData mining 数据挖掘Data set 数据集Decision Boundary 决策边界Decision stump 决策树桩Decision tree 决策树／判定树Deduction 演绎Deep Belief Network 深度信念⽹络Deep Convolutional Generative Adversarial Network/DCGAN 深度卷积⽣成对抗⽹络Deep learning 深度学习Deep neural network/DNN 深度神经⽹络Deep Q-Learning 深度 Q 学习Deep Q-Network 深度 Q ⽹络Density estimation 密度估计Density-based clustering 密度聚类Differentiable neural computer 可微分神经计算机Dimensionality reduction algorithm 降维算法Directed edge 有向边Disagreement measure 不合度量Discriminative model 判别模型Discriminator 判别器Distance measure 距离度量Distance metric learning 距离度量学习Distribution 分布Divergence 散度Diversity measure 多样性度量／差异性度量Domain adaption 领域⾃适应Downsampling 下采样D-separation （Directed separation）有向分离Dual problem 对偶问题Dummy node 哑结点Dynamic Fusion 动态融合Dynamic programming 动态规划Letter EEigenvalue decomposition 特征值分解Embedding 嵌⼊Emotional analysis 情绪分析Empirical conditional entropy 经验条件熵Empirical entropy 经验熵Empirical error 经验误差Empirical risk 经验风险End-to-End 端到端Energy-based model 基于能量的模型Ensemble learning 集成学习Ensemble pruning 集成修剪Error Correcting Output Codes／ECOC 纠错输出码Error rate 错误率Error-ambiguity decomposition 误差-分歧分解Euclidean distance 欧⽒距离Evolutionary computation 演化计算Expectation-Maximization 期望最⼤化Expected loss 期望损失Exploding Gradient Problem 梯度爆炸问题Exponential loss function 指数损失函数Extreme Learning Machine/ELM 超限学习机Letter FFactorization 因⼦分解False negative 假负类False positive 假正类False Positive Rate/FPR 假正例率Feature engineering 特征⼯程Feature selection 特征选择Feature vector 特征向量Featured Learning 特征学习Feedforward Neural Networks/FNN 前馈神经⽹络Fine-tuning 微调Flipping output 翻转法Fluctuation 震荡Forward stagewise algorithm 前向分步算法Frequentist 频率主义学派Full-rank matrix 满秩矩阵Functional neuron 功能神经元Letter GGain ratio 增益率Game theory 博弈论Gaussian kernel function ⾼斯核函数Gaussian Mixture Model ⾼斯混合模型General Problem Solving 通⽤问题求解Generalization 泛化Generalization error 泛化误差Generalization error bound 泛化误差上界Generalized Lagrange function ⼴义拉格朗⽇函数Generalized linear model ⼴义线性模型Generalized Rayleigh quotient ⼴义瑞利商Generative Adversarial Networks/GAN ⽣成对抗⽹络Generative Model ⽣成模型Generator ⽣成器Genetic Algorithm/GA 遗传算法Gibbs sampling 吉布斯采样Gini index 基尼指数Global minimum 全局最⼩Global Optimization 全局优化Gradient boosting 梯度提升Gradient Descent 梯度下降Graph theory 图论Ground-truth 真相／真实Letter HHard margin 硬间隔Hard voting 硬投票Harmonic mean 调和平均Hesse matrix 海塞矩阵Hidden dynamic model 隐动态模型Hidden layer 隐藏层Hidden Markov Model/HMM 隐马尔可夫模型Hierarchical clustering 层次聚类Hilbert space 希尔伯特空间Hinge loss function 合页损失函数Hold-out 留出法Homogeneous 同质Hybrid computing 混合计算Hyperparameter 超参数Hypothesis 假设Hypothesis test 假设验证Letter IICML 国际机器学习会议Improved iterative scaling/IIS 改进的迭代尺度法Incremental learning 增量学习Independent and identically distributed/i.i.d. 独⽴同分布Independent Component Analysis/ICA 独⽴成分分析Indicator function 指⽰函数Individual learner 个体学习器Induction 归纳Inductive bias 归纳偏好Inductive learning 归纳学习Inductive Logic Programming／ILP 归纳逻辑程序设计Information entropy 信息熵Information gain 信息增益Input layer 输⼊层Insensitive loss 不敏感损失Inter-cluster similarity 簇间相似度International Conference for Machine Learning/ICML 国际机器学习⼤会Intra-cluster similarity 簇内相似度Intrinsic value 固有值Isometric Mapping/Isomap 等度量映射Isotonic regression 等分回归Iterative Dichotomiser 迭代⼆分器Letter KKernel method 核⽅法Kernel trick 核技巧Kernelized Linear Discriminant Analysis／KLDA 核线性判别分析K-fold cross validation k 折交叉验证／k 倍交叉验证K-Means Clustering K – 均值聚类K-Nearest Neighbours Algorithm/KNN K近邻算法Knowledge base 知识库Knowledge Representation 知识表征Letter LLabel space 标记空间Lagrange duality 拉格朗⽇对偶性Lagrange multiplier 拉格朗⽇乘⼦Laplace smoothing 拉普拉斯平滑Laplacian correction 拉普拉斯修正Latent Dirichlet Allocation 隐狄利克雷分布Latent semantic analysis 潜在语义分析Latent variable 隐变量Lazy learning 懒惰学习Learner 学习器Learning by analogy 类⽐学习Learning rate 学习率Learning Vector Quantization/LVQ 学习向量量化Least squares regression tree 最⼩⼆乘回归树Leave-One-Out/LOO 留⼀法linear chain conditional random field 线性链条件随机场Linear Discriminant Analysis／LDA 线性判别分析Linear model 线性模型Linear Regression 线性回归Link function 联系函数Local Markov property 局部马尔可夫性Local minimum 局部最⼩Log likelihood 对数似然Log odds／logit 对数⼏率Logistic Regression Logistic 回归Log-likelihood 对数似然Log-linear regression 对数线性回归Long-Short Term Memory/LSTM 长短期记忆Loss function 损失函数Letter MMachine translation/MT 机器翻译Macron-P 宏查准率Macron-R 宏查全率Majority voting 绝对多数投票法Manifold assumption 流形假设Manifold learning 流形学习Margin theory 间隔理论Marginal distribution 边际分布Marginal independence 边际独⽴性Marginalization 边际化Markov Chain Monte Carlo/MCMC 马尔可夫链蒙特卡罗⽅法Markov Random Field 马尔可夫随机场Maximal clique 最⼤团Maximum Likelihood Estimation/MLE 极⼤似然估计／极⼤似然法Maximum margin 最⼤间隔Maximum weighted spanning tree 最⼤带权⽣成树Max-Pooling 最⼤池化Mean squared error 均⽅误差Meta-learner 元学习器Metric learning 度量学习Micro-P 微查准率Micro-R 微查全率Minimal Description Length/MDL 最⼩描述长度Minimax game 极⼩极⼤博弈Misclassification cost 误分类成本Mixture of experts 混合专家Momentum 动量Moral graph 道德图／端正图Multi-class classification 多分类Multi-document summarization 多⽂档摘要Multi-layer feedforward neural networks 多层前馈神经⽹络Multilayer Perceptron/MLP 多层感知器Multimodal learning 多模态学习Multiple Dimensional Scaling 多维缩放Multiple linear regression 多元线性回归Multi-response Linear Regression ／MLR 多响应线性回归Mutual information 互信息Letter NNaive bayes 朴素贝叶斯Naive Bayes Classifier 朴素贝叶斯分类器Named entity recognition 命名实体识别Nash equilibrium 纳什均衡Natural language generation/NLG ⾃然语⾔⽣成Natural language processing ⾃然语⾔处理Negative class 负类Negative correlation 负相关法Negative Log Likelihood 负对数似然Neighbourhood Component Analysis/NCA 近邻成分分析Neural Machine Translation 神经机器翻译Neural Turing Machine 神经图灵机Newton method ⽜顿法NIPS 国际神经信息处理系统会议No Free Lunch Theorem／NFL 没有免费的午餐定理Noise-contrastive estimation 噪⾳对⽐估计Nominal attribute 列名属性Non-convex optimization ⾮凸优化Nonlinear model ⾮线性模型Non-metric distance ⾮度量距离Non-negative matrix factorization ⾮负矩阵分解Non-ordinal attribute ⽆序属性Non-Saturating Game ⾮饱和博弈Norm 范数Normalization 归⼀化Nuclear norm 核范数Numerical attribute 数值属性Letter OObjective function ⽬标函数Oblique decision tree 斜决策树Occam’s razor 奥卡姆剃⼑Odds ⼏率Off-Policy 离策略One shot learning ⼀次性学习One-Dependent Estimator／ODE 独依赖估计On-Policy 在策略Ordinal attribute 有序属性Out-of-bag estimate 包外估计Output layer 输出层Output smearing 输出调制法Overfitting 过拟合／过配Oversampling 过采样Letter PPaired t-test 成对 t 检验Pairwise 成对型Pairwise Markov property 成对马尔可夫性Parameter 参数Parameter estimation 参数估计Parameter tuning 调参Parse tree 解析树Particle Swarm Optimization/PSO 粒⼦群优化算法Part-of-speech tagging 词性标注Perceptron 感知机Performance measure 性能度量Plug and Play Generative Network 即插即⽤⽣成⽹络Plurality voting 相对多数投票法Polarity detection 极性检测Polynomial kernel function 多项式核函数Pooling 池化Positive class 正类Positive definite matrix 正定矩阵Post-hoc test 后续检验Post-pruning 后剪枝potential function 势函数Precision 查准率／准确率Prepruning 预剪枝Principal component analysis/PCA 主成分分析Principle of multiple explanations 多释原则Prior 先验Probability Graphical Model 概率图模型Proximal Gradient Descent/PGD 近端梯度下降Pruning 剪枝Pseudo-label 伪标记Letter QQuantized Neural Network 量⼦化神经⽹络Quantum computer 量⼦计算机Quantum Computing 量⼦计算Quasi Newton method 拟⽜顿法Letter RRadial Basis Function／RBF 径向基函数Random Forest Algorithm 随机森林算法Random walk 随机漫步Recall 查全率／召回率Receiver Operating Characteristic/ROC 受试者⼯作特征Rectified Linear Unit/ReLU 线性修正单元Recurrent Neural Network 循环神经⽹络Recursive neural network 递归神经⽹络Reference model 参考模型Regression 回归Regularization 正则化Reinforcement learning/RL 强化学习Representation learning 表征学习Representer theorem 表⽰定理reproducing kernel Hilbert space/RKHS 再⽣核希尔伯特空间Re-sampling 重采样法Rescaling 再缩放Residual Mapping 残差映射Residual Network 残差⽹络Restricted Boltzmann Machine/RBM 受限玻尔兹曼机Restricted Isometry Property/RIP 限定等距性Re-weighting 重赋权法Robustness 稳健性/鲁棒性Root node 根结点Rule Engine 规则引擎Rule learning 规则学习Letter SSaddle point 鞍点Sample space 样本空间Sampling 采样Score function 评分函数Self-Driving ⾃动驾驶Self-Organizing Map／SOM ⾃组织映射Semi-naive Bayes classifiers 半朴素贝叶斯分类器Semi-Supervised Learning 半监督学习semi-Supervised Support Vector Machine 半监督⽀持向量机Sentiment analysis 情感分析Separating hyperplane 分离超平⾯Sigmoid function Sigmoid 函数Similarity measure 相似度度量Simulated annealing 模拟退⽕Simultaneous localization and mapping 同步定位与地图构建Singular Value Decomposition 奇异值分解Slack variables 松弛变量Smoothing 平滑Soft margin 软间隔Soft margin maximization 软间隔最⼤化Soft voting 软投票Sparse representation 稀疏表征Sparsity 稀疏性Specialization 特化Spectral Clustering 谱聚类Speech Recognition 语⾳识别Splitting variable 切分变量Squashing function 挤压函数Stability-plasticity dilemma 可塑性-稳定性困境Statistical learning 统计学习Status feature function 状态特征函Stochastic gradient descent 随机梯度下降Stratified sampling 分层采样Structural risk 结构风险Structural risk minimization/SRM 结构风险最⼩化Subspace ⼦空间Supervised learning 监督学习／有导师学习support vector expansion ⽀持向量展式Support Vector Machine/SVM ⽀持向量机Surrogat loss 替代损失Surrogate function 替代函数Symbolic learning 符号学习Symbolism 符号主义Synset 同义词集Letter TT-Distribution Stochastic Neighbour Embedding/t-SNE T – 分布随机近邻嵌⼊Tensor 张量Tensor Processing Units/TPU 张量处理单元The least square method 最⼩⼆乘法Threshold 阈值Threshold logic unit 阈值逻辑单元Threshold-moving 阈值移动Time Step 时间步骤Tokenization 标记化Training error 训练误差Training instance 训练⽰例／训练例Transductive learning 直推学习Transfer learning 迁移学习Treebank 树库Tria-by-error 试错法True negative 真负类True positive 真正类True Positive Rate/TPR 真正例率Turing Machine 图灵机Twice-learning ⼆次学习Letter UUnderfitting ⽋拟合／⽋配Undersampling ⽋采样Understandability 可理解性Unequal cost ⾮均等代价Unit-step function 单位阶跃函数Univariate decision tree 单变量决策树Unsupervised learning ⽆监督学习／⽆导师学习Unsupervised layer-wise training ⽆监督逐层训练Upsampling 上采样Letter VVanishing Gradient Problem 梯度消失问题Variational inference 变分推断VC Theory VC维理论Version space 版本空间Viterbi algorithm 维特⽐算法Von Neumann architecture 冯 · 诺伊曼架构Letter WWasserstein GAN/WGAN Wasserstein⽣成对抗⽹络Weak learner 弱学习器Weight 权重Weight sharing 权共享Weighted voting 加权投票法Within-class scatter matrix 类内散度矩阵Word embedding 词嵌⼊Word sense disambiguation 词义消歧Letter ZZero-data learning 零数据学习Zero-shot learning 零次学习Aapproximations近似值arbitrary随意的affine仿射的arbitrary任意的amino acid氨基酸amenable经得起检验的axiom公理，原则abstract提取architecture架构，体系结构；建造业absolute绝对的arsenal军⽕库assignment分配algebra线性代数asymptotically⽆症状的appropriate恰当的Bbias偏差brevity简短，简洁；短暂broader⼴泛briefly简短的batch批量Cconvergence 收敛，集中到⼀点convex凸的contours轮廓constraint约束constant常理commercial商务的complementarity补充coordinate ascent同等级上升clipping剪下物；剪报；修剪component分量；部件continuous连续的covariance协⽅差canonical正规的，正则的concave⾮凸的corresponds相符合；相当；通信corollary推论concrete具体的事物，实在的东西cross validation交叉验证correlation相互关系convention约定cluster⼀簇centroids 质⼼，形⼼converge收敛computationally计算(机)的calculus计算Dderive获得，取得dual⼆元的duality⼆元性；⼆象性；对偶性derivation求导；得到；起源denote预⽰，表⽰，是…的标志；意味着，[逻]指称divergence 散度；发散性dimension尺度，规格；维数dot⼩圆点distortion变形density概率密度函数discrete离散的discriminative有识别能⼒的diagonal对⾓dispersion分散，散开determinant决定因素disjoint不相交的Eencounter遇到ellipses椭圆equality等式extra额外的empirical经验；观察ennmerate例举，计数exceed超过，越出expectation期望efficient⽣效的endow赋予explicitly清楚的exponential family指数家族equivalently等价的Ffeasible可⾏的forary初次尝试finite有限的，限定的forgo摒弃，放弃fliter过滤frequentist最常发⽣的forward search前向式搜索formalize使定形Ggeneralized归纳的generalization概括，归纳；普遍化；判断（根据不⾜）guarantee保证；抵押品generate形成，产⽣geometric margins⼏何边界gap裂⼝generative⽣产的；有⽣产⼒的Hheuristic启发式的；启发法；启发程序hone怀恋；磨hyperplane超平⾯Linitial最初的implement执⾏intuitive凭直觉获知的incremental增加的intercept截距intuitious直觉instantiation例⼦indicator指⽰物，指⽰器interative重复的，迭代的integral积分identical相等的；完全相同的indicate表⽰，指出invariance不变性，恒定性impose把…强加于intermediate中间的interpretation解释，翻译Jjoint distribution联合概率Llieu替代logarithmic对数的，⽤对数表⽰的latent潜在的Leave-one-out cross validation留⼀法交叉验证Mmagnitude巨⼤mapping绘图，制图；映射matrix矩阵mutual相互的，共同的monotonically单调的minor较⼩的，次要的multinomial多项的multi-class classification⼆分类问题Nnasty讨厌的notation标志，注释naïve朴素的Oobtain得到oscillate摆动optimization problem最优化问题objective function⽬标函数optimal最理想的orthogonal(⽮量，矩阵等)正交的orientation⽅向ordinary普通的occasionally偶然的Ppartial derivative偏导数property性质proportional成⽐例的primal原始的，最初的permit允许pseudocode伪代码permissible可允许的polynomial多项式preliminary预备precision精度perturbation 不安，扰乱poist假定，设想positive semi-definite半正定的parentheses圆括号posterior probability后验概率plementarity补充pictorially图像的parameterize确定…的参数poisson distribution柏松分布pertinent相关的Qquadratic⼆次的quantity量，数量；分量query疑问的Rregularization使系统化；调整reoptimize重新优化restrict限制；限定；约束reminiscent回忆往事的；提醒的；使⼈联想…的（of）remark注意random variable随机变量respect考虑respectively各⾃的；分别的redundant过多的；冗余的Ssusceptible敏感的stochastic可能的；随机的symmetric对称的sophisticated复杂的spurious假的；伪造的subtract减去；减法器simultaneously同时发⽣地；同步地suffice满⾜scarce稀有的，难得的split分解，分离subset⼦集statistic统计量successive iteratious连续的迭代scale标度sort of有⼏分的squares平⽅Ttrajectory轨迹temporarily暂时的terminology专⽤名词tolerance容忍；公差thumb翻阅threshold阈，临界theorem定理tangent正弦Uunit-length vector单位向量Vvalid有效的，正确的variance⽅差variable变量；变元vocabulary词汇valued经估价的；宝贵的Wwrapper包装分类:。
仿射变换在初等几何解题中的应用…………摘要：仿射变换，即平行投影变换，是几何学中的一个重要变换，是从运动变换过渡到射影变换的桥梁.本文将从仿射变换的有关概念入手，了解仿射几何所研究的几何通过仿射变换的不变性质和不变的数量关系以及经过变形后的形状和位置关系，并讨论仿射变换在初等几何中的一些应用. 关键词：仿射变换；仿射不变性；初等几何Abstract:Affinetransformation,namelyparallelprojection,isanimpo rtanttransformationingeometry.Itisthebridgefromthemotionconvertingt otheprojectivetransformation.Thisarticlewillstartwiththeconceptofaf finetransform,tounderstandthegeometryofaffinegeometryresearchbyaffi netransformationinvariantpropertiesandconstantrelationshipbetweenth enumberafterthedeformedshapeandpositionalrelationship,anddiscusseds omeapplicationsofaffinetransformationinelementarygeometry.Keywords ：affinetransformation ；affineinvariance ；elementarygeometry1仿射变换的基本概念及相关性质1.1仿射变换的概念定义1.1[1]设同一平面内有n 条直线1a ，2a ，3a ，…n a ，1T ，2T ，3T ，…1-n T 顺次表示1a 到2a ，2a 到3a ，1-n a 到n a 的透视仿射，经过这一串平行射影，使1a 上的点与n a 上的点建立了一一对应，称为1a 到n a 的仿射或仿射变换如图1-1.T =1-n T 122T T T n ⋅⋅⋅⋅- ，T 称为1T ，2T ，3T ，…1-n T 按这个顺序的乘积.)(A T =1-n T 122T T T n ⋅⋅⋅⋅- )(A =1-n T )(22A T T n '⋅⋅⋅- =…=n A ，)(B T =n B 等图1-1定义1.2设A ，B ，C 为共线三点，这三点的简比()ABC 定义为下述有向线段的比：其中AC ，BC 是有向线段AC ，BC 的代数长，A ，B 叫基点，C 叫分点.当C 在A ，B 之间时，()ABC <0；当C 不在A ，B 之间时，()ABC >0；当C 与A 重合时，()ABC =0；当C 与B 重合时，()ABC 不存在.1.2 仿射变换的性质（1）仿射变换保持同素性：即仿射变换将点变成点，直线变成直线；（2）仿射变换保持结合性：即仿射变换保持点与直线的结合关系；（3）仿射变换将向量变成向量，且保持向量的线性关系v u λ=.定理1两条平行直线经仿射变换后仍变为两条平行直线.推论1两条相交直线经仿射变换后仍变成两相交直线.推论2共点的直线经仿射变换后仍变为共点直线.定理2两条平行线段之比是仿射不变量.推论一直线上两线段之比是仿射不变量.定理3两封闭图形（如三角形、平行四边形、椭圆等）面积之比是仿射不变量. 2 仿射变换与初等几何的相关联系从总体上看,高等几何对初等几何具有多方面的指导意义.在此,笔者择要阐述两种,以此说明高等几何对初等几何普遍指导意义[2].一是学习高等几何能深化对初等几何的认识和理解.几何学是一种研究在相应的变换群下图形保持不变的质和量的科学,射影群、仿射群、正交群所对应的是射影几何、仿射几何、欧氏几何,根据普遍性包含于特殊性的原理可知,射影几何包含于仿射几何包含于欧氏几何,这其中,射影几何内容最少,欧氏几何内容最丰富.不同的几何课程在内容上的侧重点不同,解析几何主要研究图形的性质,将空间几何结构代数化是其本质特征;欧氏几何主要研究整个空间的几何结构,它利用图形的直观形象启发人类的想象思维,从而促使人们不断探索发现图形间的关系与性质；高等几何尤其是其中的射影几何则包含、融合了上述两者的内容.也就是说,学习高等几何能使我们站得更高一些,看得更远一些,能进一步认清几种几何学间的关系,进一步开阔几何学的视野,从而更好地理解和把握初等几何的本质和精髓.二是学习高等几何能有效扩充初等几何的研究方法.从实用主义的角度看,数学与应用数学专业的学生或中学数学教师学好高等几何,一方面可扩展几何学的认知范畴,在更高的水准上搞好教学工作,另一方面可用高等几何的理念和观点来指导和反思初等几何的教学内容与研究方法,从而不断改进初等几何的教学方式,优化其研究手段和教学模式,切实提高中学几何的教学质量.3 仿射变换在初等几何解题中的应用根据仿射变换的性质可知，通过特殊仿射变换可将某些一般图形变为特殊图形，如可将任何三角形变成正三角形，平行四边形变为正方形或长方形，梯形变为等腰梯形或直角梯形.因此，对于一个仅涉及仿射性质的初等几何命题，如果能证明它在特殊图形中成立，则在仿射变换下，这个命题对于相应地一般图形也应成立.利用仿射变换可以解决许多初等几何问题，下面给出它在以下几个方面的应用.3.1 平行投影平行投影是仿射变换中最基本、最简单的一类.因此平行投影变换具有仿射变换中的一切性质.解这类题的关键是选定平行投影方向，应用平行线段之比是仿射不变量.例3.1P 是ABC ∆内任一点，连结AP 、BP 、CP 并延长分别交对边于D 、E 、F .求证：1=++CFPF BE PE AD PD . 图3-1证明如图3-1，分别沿AB 和AC 方向作平行投影.P →P '、P →P ''由仿射变换保简单比不变得：DC DP BD D P AD PD '''==，所以BC P P AD PD '''=， 同理BC C P BE PE ''=，BC BP CF PF '=， 所以1''''''=++=++BCBP BC C P BC P P CF PF BE PE AD PD . 例3.2一直线截三角形的边或其延长线,所得的顶点到分点和分点到顶点的有向线段的比的乘积等于﹣1，其逆也真.（梅涅劳斯定理）[3]分析如图3-2，本题要求证明当L 、M 、N 三点共线时，1-=⋅⋅NBAN MA CM LC BL 。
