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MARGINALIZED MAXIMUM A POSTERIORI HYPER-PARAMETER ESTIMATION FOR GLOBAL OPTICAL FLOW TECHNI

MARGINALIZED MAXIMUM A

POSTERIORI HYPER-PARAMETER

ESTIMATION FOR GLOBAL OPTICAL

FLOW TECHNIQUES

Kai Krajsek and Rudolf Mester

Visual Sensorics and Information Processing Lab,

Institute for Computer Science,

J.W.Goethe University,

Robert Mayer Str.2-4,60054Frankfurt am Main,Germany

Abstract.Global optical?ow estimation methods contain a regularization parameter(or prior and likelihood hyper-parameters if we consider the statistical point of view)which control the tradeoff between the different constraints on the optical?ow?eld.Although experiments(see e.g.Ng et al.[Ng and Solo(1997)])indicate the importance of the optimal choice of the hyper-parameters, only little attention has been focused on the optimal choice of these parameters in global motion estimation techniques in literature so far(the authors are only aware of one contribution[Ng and Solo(1997)]which attempts to estimate only the prior hyper-parameter whereas the likelihood hyper-parameter needs to be known).We adapt the marginalized maximum a posteriori(MMAP) estimator proposed in[Mohammad-Djafari(1995)]to simultaneously estimating hyper-parameters and optical?ow for global motion estimation techniques.Experiments demonstrate the performance of this optimization technique and show that the choice of the regularization parameter/hyper-parameters is an essential key-point in order to obtain precise motion estimates.

Keywords:Optical Flow,Bayesian motion estimation,Hyper-parameter estimation

PACS:https://www.wendangku.net/doc/d513697034.html,,02.50.Tt,07.05.Pj

INTRODUCTION

Motion estimation in image sequences is of crucial importance in computer vision as well as in image processing with a wide range of applications spanning from robot nav-igation over medical image analysis to video compressions.The motion of a single ob-ject,i.e.its displacement vector from frame to frame,which is inferred from brightness changes in the image sequence is denoted as the optical?ow vector.The set of all optical ?ow vectors is called the optical?ow?eld.Optical?ow estimation methods with a prior term that couples all optical?ow vectors are usually characterized as global methods.In this contribution,we develop a marginalized maximum a posteriori(MMAP)estimator for simultaneous estimation of hyper-parameters of the likelihood term as well as of the prior term and the optical?ow?eld.The optimal hyper-parameters can be estimated without any prior knowledge or assumption of the current optical?ow?eld,but due to the Bayesian framework,prior knowledge could also be incorporated,if needed.Lit-tle attention has been focused on the optimal choice of the regularization parameter in motion estimation techniques in literature so far(the authors are only aware of one con-

tribution [1]).Some authors even assume that the optimal choice of the regularization parameter is of minor importance [2].In contradiction to this assumption,obviously is the different value of the regularization parameter which has been proposed by different authors (0.5in [3],100in [4])and the dependency of the motion estimate on the regu-larization parameter (as demonstrated in [4,1]).But the proposal of a certain value of the regularization parameter is meaningless since its optimal value depends on image statistics,on the image noise statistics as well as on the statistics of the optical ?ow ?eld

[5].Experiments demonstrate the performance of this optimization technique and show that the choice of the regularization parameter is an essential key-point in order to obtain precise motion estimation.

DIFFERENTIAL MOTION ESTIMATION

The general principle behind all differential approaches to motion estimation is that the conservation of some local image characteristics throughout its temporal evolution is re?ected in terms of differential-geometric entities on the space-time signal s (x ),x =(x,y,t )T .In its simplest form,the assumed conservation of brightness along the motion trajectory through space-time leads to the well-known brightness constancy constraint equation (BCCE),where g denotes the gradient of the gray value signal s ,u h =(u x ,u y ,1)T the homogenous form of the direction of motion and u =(u x ,u y )T the optical ?ow ?eld

g T u h =0.(1)

Since it is fundamentally impossible to solve for u h by a single linear equation (aperture problem ),additional constraints have to be considered.Whereas local methods minimize an error function over a local area V ?A assuming a certain motion model in this neighborhood,global methods [3,6,7]estimate the optical ?ow ?eld by minimizing an error functional (or error function if u is considered on a discrete grid)over the whole region of interest in space-time.The necessary additional constraint is incorporated by a regularization term ρ(u )(ρdenotes an operator acting on u )imposing supplementary information on the solution,e.g.the optical ?ow ?eld should be smooth and should not vary abruptly [3].The regularization parameter λspeci?es the in?uence of the regularization term ρ(u )relative to the data term ψ g T u h ,(ψ=real positive function).The optical ?ow ?eld is estimated by minimizing

J (u )= A ψ

g T u h +λρ(u ) d x (2)

with respect to the optical ?ow ?eld u .Since the introduction of regularization in motion estimation by Horn and Schunk [3],a large class of regularization terms has been examined (for an overview we refer to [6]and to the references therein).But also new data terms have been proposed [7].Recently,a combination of global and local constraints (proposed by Bruhn et al.[2],which forms the combined local-global (CLG)method)has been exempli?ed to increase precision as well as robustness against noise.

BAYESIAN MOTION ESTIMATION

In a Bayesian formulation(see e.g.[8]),the optical?ow is estimated via a probability density function pdf which connects the observable signal or its gradient with the entity of interest,the optical?ow.In order to design such a pdf,we assume a regular grid in space-time considering only signal values and optical?ow vectors on the knots of the grid.Since N knots in space-time are isomorphic to the Euclidian space IR N,the signal and the optical?ow?eld can be expressed by a set of vectors s∈IR N and u∈IR2N.The approximated gradients w of the optical?ow components u as well as the approximated gradients g of the signal components s can be written in compact matrix vector equations w=Hu∈IR6N and g=Ps∈IR3N.The matrices P and H encode the approximation schemes of the gradient computation by?nite differences of values at neighborhood positions.In the Bayesian framework,not only the gradients g=(g(x1),g(x2),...,g(x N)),but also the estimated parameters u are considered as realizations of random vectors with corresponding pdfs p(u)and p(g),respectively. Prior knowledge about u is incorporated into the estimation framework via the prior pdf p(u).The maximum a posteriori(MAP)estimator infers the optical?ow?eld by maximizing the posterior pdf p(u|g)or minimizing its negative https://www.wendangku.net/doc/d513697034.html,ing Bayes’law,this leads to

?u=arg min

{?ln(p(g|u))?ln(p(u))}.(3)

The term in the bracket on the right side of equ.(3)is denoted as the objective function L. For exponential pdfs with partition functions Z L(α),Z p(β),energies J L(α)and J p(β) and corresponding hyper-parametersα,β,the objective function becomes

L=J L(α)+J p(β)+ln(Z L(α)Z p(β)).(4) After discussing the explicit form of the likelihood energy and the prior energy for the case of motion estimation,we develop a method for optimizing the hyper-parameters directly from the observable data.Since all variational methods are equivalent to a corresponding Bayesian formulation[9]and the regularization parameter corresponds to the ratio of the hyper-parametersα,β,our approach allows to optimize the regularization parameters of global optical?ow methods in general.

LIKELIHOOD FUNCTIONS AND PRIOR DISTRIBUTIONS FOR

MOTION ESTIMATION

In Bayesian estimation,the relation between the observed data g and the optical?ow ?eld u has to be established by the likelihood function p(g|u).The errorεtj in the temporal gradient componentsˇg tj at position j is here approximated as being identical independent noise which follows a Gaussian distribution,g tj=ˇg tj+εtj,whereas the spatial gradient components g sj:=(g xj,g yj)are assumed to be error free.The BCCE g T

u hj=εtj changes accordingly.We obtain the likelihood function p(g t|u,g s)by j

expressing each realization of the random variableεtj in the joint pdf p(εt)= N j=1p(εtj)

by the corresponding BCCE

p (g t |u ,g s )=1Z L (α)e ?α N j =1ψ1(||g T j u hj ||2),(5)

where ψ1(x )is a positive symmetric function that is ψ1(x )=x for Gaussian noise.If the optical ?ow ?eld can be assumed to be constant within spatial neighborhoods and the errors εtj within each of these regions are i.i.d Gaussian noise,the likelihood func-tion can be derived as an expression depending on the structure tensors C gj with the energy function J L =α N j =1u T hj C gj u hj .The robust version of the likelihood energy function reads J L =α N j =1ψ1 u T hj C gj u hj .The prior pdf encodes our prior informa-tion/assumption of the optical ?ow ?eld.The prior pdf corresponding to the smoothness assumption reads

p (u )=1Z p (β)e ?β N j =1ψ2(||w 2j ||2),(6)

where ψ2is again a positive symmetric function.

MARGINALIZED MAXIMUM A POSTERIORI

HYPER-PARAMETER ESTIMATION

The advantage of the Bayesian formulation of the motion estimation problem is that not only the optical ?ow but also hyper-parameters can be included into the estima-tion procedure.Firstly,introduced by MacKay [10]in the context of interpolation,the Bayesian hyper-parameter estimation techniques have been applied to different kind of problems.Other techniques for hyper-parameter estimation have been developed as well but as mentioned in the introduction,only one [1]of these has been derived for motion estimation so far.The drawback of the method of Ng et al.is its computational cost:a full search in parameter space is necessary to obtain the optimal regularization parame-ter.Furthermore,only one hyper-parameter can be estimated which corresponds to the prior hyper-parameter in our approach.Thus,it is necessary to know or estimate other-wise the hyper-parameter αwhich depends on the noise of the gradient ?eld [9].On the contrary,our approach allows the estimation of all hyper-parameters directly from the observable data.However,if the noise distribution is known,αcan be computed [9]and it is only necessary to estimate β.Based on the approach of Mohammad-Djafari [11],we derive a marginalized maximum a posteriori (MMAP)hyper-parameter estimator for the case of motion estimation which estimates hyper-parameters and the optical ?ow ?eld simultaneously.The main idea of our approach is to approximate the likelihood function p (g |u ,α)as well as the prior pdf p (u |β)by Gaussian distributions (if they are not already Gaussian)using a Taylor expansion with respect to u up to second order of the logarithm of the corresponding pdf.The joint pdf ?p (u ,g |α,β)of the gradient ?eld and the optical ?ow ?eld is then obtained by the product of the approximated prior pdf ?p (u |β)and the approximated likelihood function ?p (g |u ,α)

?p (u ,g |α,β)=?p (g |u ,α)?p (u |β),(7)

which is again a Gaussian function with respect to the optical?ow u and therefore analytically integrable.Furthermore,the partition function?Z(α,β)=?Z L(α)?Z p(β)of the joint pdf is the product of the partition functions of the Gaussian likelihood function and Gaussian prior pdf and thus also analytically tractable.In the current approximation,the likelihood energy J L as well as the prior energy J p are expanded around their individual minimum,which are not known a priori.In order to obtain an approximated energy of the joint pdf around its minimum,we exchange the energy by a Taylor series of the original,non approximated joint pdf,up to second order around its minimum with respect to the optical?ow?eld.The optical?ow?u maximizing the joint pdf is in fact the entity we are searching for and also not known in advance,but as shown below,can be estimated iteratively.Note that the joint pdf is proportional to the posterior pdf of the optical?ow?eld u and thus maximizing the joint pdf is equivalent to maximizing the joint pdf with respect to the optical?ow?eld.Integrating the resulting approximated joint pdf?p(u,g|α,β,?u)over u results in the likelihood function of the hyper-parameters

?p(g|α,β,?u)=

?p(u,g|α,β,?u)d u.(8)

If prior knowledge about the hyper-parameters are available,it can be encoded into the corresponding hyper-parameter prior pdfs p(α)and p(β).The resulting posterior pdf of the hyper-parameters yields

?p(α,β|g,?u)∝?p(g|α,β,?u)p(α)p(β).(9) The hyper-parameters are then estimated by minimizing the negative logarithm of the posterior pdf?p(α,β|g,?u)with respect toαandβfor the present realization of the gradi-ent?eld g.We now compute the concrete likelihood function of the hyper-parameters. Let Q(?u,α,β)denote the Hessian of the joint energy J(u,α,β)=J L(u,α)+J p(u,β) and A(?u,α),B(?u,β)the Hessians of the likelihood and prior energy,respectively,taken at the optical?ow?eld for which the joint pdf attains its maximum.The optical?ow ?eld which minimizes J(u,α,β)is in fact the MAP estimator of the optical?ow?eld for?xedα,β.With the notation?J=J(?u,α,β)and?Q=Q(?u,α,β)the approximated posterior energy reads

J(u,α,β)≈?J+1

(u??u)T?Q(u??u).(10)

Inserting the approximated posterior pdf in equ.(8)and integrating over u yields the likelihood function of the hyper-parameters

?p(g|α,β,?u)=

(2π)N

(α)?Z p(β)

exp

??J

.(11)

Since the computation of the determinant|Q(?u,α,β)|is not feasible for usual image sequence sizes,an approximation has to be performed.For computing|Q(?u,α,β)|,we neglect interactions between different pixels,i.e.the interaction matrix B(u,β)becomes a block diagonal matrix.Then the determinant of Q(?u,α,β)factorizes into the product

of determinants of Q j(?u,α,β)=A j+B j.The approximated objective function for the hyper-parameters then becomes

L(?u,α,β)∝?J+1

j=1

?Q j

+ln

Z L?Z p

.(12)

Since?u itself depends on the hyper-parametersα,βwe have to apply an iterative scheme for estimating u,αandβsimultaneously

u k+1=arg min

J(u,αk,βk)

(13)

αk+1=arg min

L(u k,α,βk)

βk+1=arg min

L(u k,αk,β)

Note that the?rst step in the iterative scheme is nothing but the usual optical?ow estimation for?xed hyper-parameters as used usually in global approaches.The second and third term in equ.(12)which distinguishes our objective function from others, enables the simultaneous hyper-parameter and motion estimation.

EXPERIMENTAL RESULTS

In this section,the performance of our MMAP estimator is shown.For the global motion estimation we choose the3D-linear-CLG estimator as described in[2].For the experiment we used three image sequences,together with their true optical?ow1:’Yosemite’(without clouds),’Diverging Tree’and’Of?ce’.An averaging volume of size5×5×5was applied with a Gaussian weighting function of widthσ=1.The derivatives occurring in the BCCE were designed according to[12]and are of size 5×5×5.The optical?ow u and the hyper-parametersαandβwere simultaneously estimated according to our iterative scheme(13).The?rst step was performed using the successive over-relaxation(SOR)method with a relaxation parameter ofγ=1.97.For the second and third step,we set the derivative of L(?u,α,β)with respect toαandβto zero and solve for the corresponding hyper-parameters to obtain a?x-point equation which was solved iteratively.For performance evaluation,the average angular error (AAE)[4]was computed.Figure1shows one image of the’Of?ce’sequence(left),the ground truth(middle)and the estimated optical?ow?eld(right).Figure2shows the computed AAE depending on the regularization parameterλ=α/βfor the three image sequences.

1The’Diverging Tree’sequence has been taken from Barron’s web-site,the’Yosemite’se-quence from"https://www.wendangku.net/doc/d513697034.html,/people/black/images.html"and the’Of?ce’sequence from "https://www.wendangku.net/doc/d513697034.html,/research/vision/".

Fig.1:Left:one image of the truth of the’Of?ce’sequence, the amplitude of the optical?ow is encoded in intensity/color with additional optical?ow vectors depicted at certain positions;right:estimated optical?ow using the3D-linear-CLG estimator.

Fig.2:Average angular error(AAE)vs.regularization parameterλ=α/βfor the image sequences:Diverging Tree(left),Of?ce(middle)and Yosemite(right).The solid lines denote the 3D-nonlinear-CLG estimates whereas the dash-dot lines denote the3D-linear-CLG estimates. The?lled circles denote the corresponding MMAP estimates.

The estimated regularization parameter delivers for all image sequences and all mo-tion estimation techniques a value that is quite close to the optimal value,i.e.the mini-mum of the corresponding curve.A constant regularization parameter for all sequences as well as for all motion estimation techniques would lead to erroneous results.Thus,λis a critical parameter which is properly estimated by our MMAP estimator.

SUMMARY AND CONCLUSION

In this contribution,we present a MMAP estimator for simultaneously estimating hyper-parameters and optical?ow directly from the observed signal without any prior knowl-edge of the optical?ow.Experiments show that our MMAP estimator delivers the op-timal hyper-parameters and show the need for optimizing the hyper-parameters to each image sequence for precise motion estimation.Since there are still free parameters left in global motion estimation techniques like the?lter size,our future work will focus on deriving optimization schemes for these still free parameters.Another task will be the incorporation of more complex prior pdfs like those proposed in[13]into our frame-work.

REFERENCES

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Frankfurt University(2006).

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International Conf.on Computer Vision,(ICCV93),Berlin,Germany,1993,pp.231–236.

8. E.Simoncelli,E.H.Adelson,and D.J.Heeger,“Probability Distribution of Optical Flow,”in

Proc.IEEE Conference on Computer Vision and Pattern Recognition,Hawaii,1991,pp.310–315.

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中考专题一次函数

O 时间距离图4 第8题 1000 2000 3000 x(km) 1000 2000 3000 y(元) y1 y2 o y x o y x o y x o y x 中考专题（一）一次函数一、选择题（2010哈尔滨）小明的爸爸早晨出去散步，从家走了20分到达距离家800米的公园，他在公园休息了10分，然后用30分原路返回家中，那么小明的爸爸离家的距离S与离家的时间t之间的函数关系图象大致是（）．（2010镇江）两直线1 : ,1 2 : 2 1 + = - =x y l x y l的交点坐标为（） A．（—2，3）B．（2，—3）C．（—2，—3）D．（2，3） (2010遵义)在“寻宝”游戏中,“寻宝”人找到了如图所标示的两个标志点A（2，3）、B（4，1）, A、B两点到“宝藏”点的距离都是10，则“宝藏”点的坐标是（） A．（1，0）Ｂ．（5，4）Ｃ．（1，0）或（5，4）Ｄ．（0，1）或（4，5） (2010玉溪) 王芳周末到新华书店购买资料。如图4，是她离家的距离与时间的函数图象.若黑点表示她家的位置，则王芳走的路线可能是（） A B C D （2010无锡）一次函数y kx b =+,当x的值减小1，y的值减小2；当x的值增加2时，则y值（）A．增加4 B．减小4 C．增加2 D．减小2 （2010连云港）某公司准备与汽车租凭公司签订租车合同，以每月用车路程x计算，甲汽车租凭公司每月收取的租赁费为y1元，乙汽车租凭公司每月收取的租赁费为y2元，若y1、y2与x之间的函数关系如图所示，其中x＝0对应的函数值为月固定租赁费，则下列判断错误．．的是（）A．当月用车路程为2000km时，两家汽车租赁公司租赁费用相同 B．当月用车路程为2300km时，租赁乙汽车租赁公车比较合算 C．除去月固定租赁费，甲租赁公司每公里收取的费用比乙租赁公司多 D．甲租赁公司平均每公里收到的费用比乙租赁公司少（2010珠海）在平面直角坐标系中，将点P（-2,3）沿x轴方向向右平移3个单位得到点Q，则点Q的坐标是（） A.(-2,6) B.(-2,0) C.(-5,3) D.(1,3) （2010温州）直线y=x+3与y轴的交点坐标是() A.(0，3) B.(0，1) C.(3，O) D.(1，0) （2010益阳）如图，火车匀速通过隧道（隧道长大于火车长）时，火车进入隧道的时间x与火车在隧道内的长度y之间的关系用图象描述大致是（） (A) (B) (C) (D) 火车隧道

初中数学一次函数的最值问题

初中数学一次函数的最值问题一次函数在自变量x允许取值范围（即全体实数）内，它是没有最大或最小值的。但是，如果给定了自变量的某一个取值范围（全体实数的一部分），那么y=kx+b 的最大值或最小值就有可能存在。一般地，有下面的结论：1、如果，那么有最大值或最小值（如图1）：当时，，；当时，，。图1 2、如果，那么有最小值或最大值（如图2）：当时，；当时，。图2

3、如果，那么有最大值或最小值（如图3）当时，；当，。图3 4、如果，那么既没有最大值也没有最小值。凡是用一次函数式来表达实际问题，求其最值时，都需要用到边界特性，像物质的运输与供应、生产任务的分配和订货、邮件的投递及空袋的调运等。下面是一道利用一次函数的最小值的决策问题，供参考：某送奶公司计划在三栋楼之间建一个奶站，三栋楼在同一条直线上，顺次为A楼，B楼，C楼，其中A楼与B楼之间的距离为40m，B楼与C楼之间的距离为60m，已知A楼每天有20人取奶，B楼每天有70人取奶，C楼每天有60人取奶，送奶公司提出两种建站方案：方案一：让每天所有取奶的人到奶站的距离总和最小；方案二：让每天A楼与C楼所有取奶的人到奶站的距离之和等于B楼所有取奶的人到奶站距离之和。（1）若按照方案一建站，取奶站应建在什么位置？

（2）若按照方案二建站，取奶站应建在什么位置？（3）在方案二的情况下，若A楼每天取奶的人数增加（增加的人数不超过22人），那么取奶站将离B楼越来越远，还是越来越近？请说明理由。解：（1）设取奶站建在距A楼xm处，所有取奶的人到奶站的距离总和为ym.。 ①当时， ∴当x=40时，y的最小值为4400。 ②当时，，此时y的值大于4400。因此按方案一建奶站，取奶站应建在B楼处。（2）设取奶站建在距A楼xm处。 ①当时，，解得（舍去）。 ②当时，解得x=80，因此按方案二建奶站，取奶站应建在距A楼80m处。

一次函数专题

九年级数学《一次函数》中考专项复习教学设计教学目标: 1. 知识与技能目标：使学生能系统掌握的一次函数相关中考考点，充分利用近三年的陕西中考一次函数试题，归类总结一次函数的出题方向，引导学生自觉掌握三种不同类型的考题的解题思路及规范书写。 2. 过程与方法：通过对近几年陕西中考一次函数原题的分析与归类，让学生总结一次函数的基本解题方法，形成解决此类问题的基本思路，提高学生解答中考原题的能力和技巧。 3. 情感态度与价值观：培养学生良好的合作、交流意识，发展学生合作探究的思想意识。教学重点：直击陕西中考原题，形成解答一次函数的知识架构，提升解答此类数学问题的能力。难点：归类运用解答一次函数的基本方法与思路。教学过程：一、课题引入：直击命题趋势破解（中考总复习《中考通鉴》P31页）：明确陕西中考对一次函数的考查情况与内容。关于一次函数的考查在选择和解答中各有一道试题，选择题注重考查关系式的确定（待定系数法和数形结合思想）、利用图象和性质把一次函数问题转化为方程和不等式的问题（函数性质），要求学生具有一定的作图能力和图像阅读能力。解答题一般在第21题会有一道一次函数的实际应用问题，常以文字、表格、图象的方式呈现，问题均为先确定函数表达式，再利用函数性质为依据，综合不等式知识确定方案，解决实际问题。要求学生有较强的图表阅读能力，能从图表中提取有效信息，准确找出相等关系，建立函数模型。解决这类题目的关键与方程应用题类似，仍是找等量关系，同时要注意函数关系式中自变量的实际意义. 板书课题：主题6函数与一次函数二、组织教学： 1.教师直言：近些年陕西中考对一次函数的考查重点。（结合P32—33知识解读梳理，引导学生复习知识点）

一些常用函数的曲线图及应用简说

1：正弦余弦曲线：更一般应用的正弦曲线公式为： A 为波幅（纵轴），ω 为（相位矢量）角频率=2PI/T，T为周期，t 为时间（横轴），θ 为相位（横轴左右）。周期函数：正余弦函数可用来表达周期函数。例如，正弦和余弦函数被用来描述简谐运动，还可描述很多自然现象，比如附着在弹簧上的物体的振动，挂在绳子上物体的小角度摆动。正弦和余弦函数是圆周运动一维投影。三角函数在一般周期函数的研究中极为有用。这些函数有作为图像的特征波模式，在描述循环现象比如声波或光波的时候很有用。每一个信号都可以记为不同频率的正弦和。谐波数目递增的方波的加法合成的动画。余弦函数的（通常是无限的）和；这是傅立叶分析的基础想法。例如，方波可以写为傅立叶级数：在动画中，可以看到只用少数的项就已经形成了非常准确的估计。如果明白了上书基本原理，也就不难理解我所用的浮动频率合成曲线的道理。

2：指数函数：形如y=ka x的函数，k为常系数，这里的a叫做“底数”，是不等于1 的任何正实数。指数函数按恒定速率翻倍，可以用来表达形象与刻画发展型的体系，比如金价2001年以来的牛市轨迹基本就是指数方程曲线。特例：应用到值x上的这个函数可写为exp(x)。还可以等价的写为e x，这里的e是数学常数，就是自然对数的底数，近似等于 2.718281828，还叫做欧拉数。即函数：定义于所有的a > 0，和所有的实数x。它叫做底数为a的指数函数。注意这个的定义依赖于先前确立的定义于所有实数上的函数的存在。注意上述等式对于a = e成立，因为指数函数可“在加法和乘法之间转换”，在下列“指数定律”的前三个和第五个中表述: 它们对所有正实数a与b和所有实数x与y都是有效的。

一次函数综合专题

2017年八年级秋季培优讲义一次函数综合专题一、知识要点 1．理解一次函数的代数表达式与图象之间的对应关系，掌握一次函数y ＝kx ＋b (k ≠0)的性质； 2．能较熟悉作出一次函数的图象； 3．结合图象体会一次函数k 、b 的取值和直线位置的关系，提高数形结合能力； 4．理解一次函数图象上的点与二元一次方程组解的关系，会用图象法解二元一次方程组； 5．从函数的观点明确一元一次方程、二元一次方程组、不等式之间的关系．二、基础能力测试 1．下列图象不能表示y 是x 的函数关系的是( ) 2．下列函数中，自变量x 的取值围是x ≥2的是( ) A ．y ＝2－x B ．y ＝1x －2 C ．y ＝4－x 2 D ．y ＝x ＋2·x －2 3．下面哪个点在函数y ＝1 2x ＋1的图象上( ) A ．y ＝2x －1 B ．y ＝x 3 C ．y ＝2x 2 D ．y ＝－2x ＋1 5．一次函数y ＝kx ＋b 的图象经过点(2，－1)和(0，3)，那么这个一次函数的解析式是( ) A ．y ＝－2x ＋3 B ．y ＝－3x ＋2 C ．y ＝3x －2 D ．y ＝1 2 x －3 6．若一次函数y ＝(3－k )x －k 的图象经过第二、三、四象限，则k 的取值围是( ) A ．k ＞3 B ．0＜k ≤3 C ．0≤k ＜3 D ．0＜k ＜3 7．已知一次函数的图象与直线y ＝－x ＋1平行，且过点(8，2)，那么这个一次函数的解析式为( ) A ．y ＝－x －2 B ．y ＝x －6 C ．y ＝－x ＋10 D ．y ＝－x －1 8．已知一次函数y ＝－x ＋a 与y ＝x ＋b 的图象相交于点(m ，8)，则－a －b ＝_____ 9．如果直线y ＝－2x ＋k 与两坐标轴所围成的三角形面积是9，则k 的值为_______． 10．已知两点A (－1，2)，B (2，3)，若x 轴上存在一点，能使得PA ＋PB 的值最小，则P 点的坐标为_____． 11．如图，A (0，1)，M (3，2)，N (4，4)．动点P 从点一出发，沿轴以每秒1个单位长的速度向上移动，且过点P 的直线l ：y ＝－x ＋b 也随之移动，设移动时间为t 秒． ⑴当t ＝3时，求l 的解析式； ⑵若点M ，N 位于l 的异侧，确定t 取值围； ⑶直接写出t 为何值时，点M 关于l 的对称点落在坐标轴上．

非线性回归分析(常见曲线及方程)

非线性回归分析回归分析中，当研究的因果关系只涉及因变量和一个自变量时，叫做一元回归分析；当研究的因果关系涉及因变量和两个或两个以上自变量时，叫做多元回归分析。此外，回归分析中，又依据描述自变量与因变量之间因果关系的函数表达式是线性的还是非线性的，分为线性回归分析和非线性回归分析。通常线性回归分析法是最基本的分析方法，遇到非线性回归问题可以借助数学手段化为线性回归问题处理两个现象变量之间的相关关系并非线性关系，而呈现某种非线性的曲线关系，如：双曲线、二次曲线、三次曲线、幂函数曲线、指数函数曲线(Gompertz)、S型曲线(Logistic) 对数曲线、指数曲线等，以这些变量之间的曲线相关关系，拟合相应的回归曲线，建立非线性回归方程，进行回归分析称为非线性回归分析常见非线性规划曲线 1.双曲线1b a y x =+ 2.二次曲线 3.三次曲线 4.幂函数曲线 5.指数函数曲线(Gompertz) 6.倒指数曲线y=a / e b x其中a>0， 7.S型曲线(Logistic) 1 e x y a b-= + 8.对数曲线y=a+b log x,x>0 9.指数曲线y=a e bx其中参数a>0 1．回归：（1）确定回归系数的命令 [beta，r，J]=nlinfit（x,y,’model’,beta0）（2）非线性回归命令：nlintool（x，y，’model’, beta0，alpha） 2．预测和预测误差估计： [Y，DELTA]=nlpredci（’model’, x，beta，r，J）求nlinfit 或lintool所得的回归函数在x处的预测值Y及预测值的显著性水平为1-alpha的置信区间Y，DELTA. 例2 观测物体降落的距离s与时间t的关系，得到数据如下表，求s 2 解： 1. 对将要拟合的非线性模型y=a/ e b x，建立M文件volum.m如下：

一次函数专项练习题

一次函数专项练习题题型一、点的坐标方法： x 轴上的点纵坐标为0，y 轴上的点横坐标为0；若两个点关于x 轴对称，则他们的横坐标相同，纵坐标互为相反数；若两个点关于y 轴对称，则它们的纵坐标相同，横坐标互为相反数；若两个点关于原点对称，则它们的横坐标互为相反数，纵坐标也互为相反数； 1、若点A （m,n ）在第二象限，则点（|m|,-n ）在第____象限； 2、若点P （2a-1,2-3b ）是第二象限的点，则a,b 的范围为______________________； 3、已知A （4，b ），B （a,-2），若A ，B 关于x 轴对称，则a=_______,b=_________;若A,B 关于y 轴对称，则a=_______,b=__________;若若A ， B 关于原点对称，则a=_______,b=_________； 4、若点M （1-x,1-y ）在第二象限，那么点N （1-x,y-1）关于原点的对称点在第______象限。题型二、关于点的距离的问题方法：点到x 轴的距离用纵坐标的绝对值表示，点到y 轴的距离用横坐标的绝对值表示；任意两点 (,),(,)A A B B A x y B x y 若AB ∥x 轴，则(,0),(,0)A B A x B x 的距离为 A B x x -；若AB ∥y 轴，则 (0,),(0,)A B A y B y 的距离为A B y y -；点(,)A A A x y 1、点B （2，-2）到x 轴的距离是_________；到y 轴的距离是____________； 2、点C （0，-5）到x 轴的距离是_________；到y 轴的距离是____________；到原点的距离是____________； 3、点D （a,b ）到x 轴的距离是_________；到y 轴的距离是____________；到原点的距离是____________； 4、已知点P （3,0），Q(-2,0),则PQ=__________,已知点110,,0,22M N ????- ? ???? ?,则MQ=________; ()()2,1,2,8E F --,则EF 两点之间的距离是__________;已知点G （2，-3）、H （3,4），则G 、H 两点之间的距离是_________； 5、两点（3，-4）、（5，a ）间的距离是2，则a 的值为__________； 6、已知点A （0,2）、B （-3，-2）、C （a,b ），若C 点在x 轴上，且∠ACB=90°，则C 点坐标为___________. 题型三、一次函数与正比例函数的识别方法：若y=kx+b(k,b 是常数，k ≠0)，那么y 叫做x 的一次函数，特别的，当b=0时，一次函数就成为y=kx(k 是常数，k ≠0)，这时，y 叫做x 的正比例函数，当k=0时，一次函数就成为若y=b ，这时，y 叫做常函数。 ☆A 与B 成正比例 A=kB(k ≠0) 1、当k_____________时，()2323y k x x =-++-是一次函数； 2、当m_____________时，()21345m y m x x +=-+-是一次函数； 3、当m_____________时，()21445m y m x x += -+-是一次函数；4、2y-3与3x+1成正比例，且x=2,y=12,则函数解析式为________________；题型四、函数图像及其性质方法： ☆一次函数y=kx+b （k≠0）中k 、b 的意义： k(称为斜率)表示直线y=kx+b （k≠0）的倾斜程度； b （称为截距）表示直线y=kx+b （k≠0）与y 轴交点的，也表示直线在y 轴上的。 ☆同一平面内，不重合的两直线 y=k 1x+b 1（k 1≠0）与 y=k 2x+b 2（k 2≠0）的位置关系：当时，两直线平行。当时，两直线垂直。当时，两直线相交。当时，两直线交于y 轴上同一点。 ☆特殊直线方程： X 轴 : 直线 Y 轴 : 直线与X 轴平行的直线与Y 轴平行的直线一、三象限角平分线二、四象限角平分线 1、对于函数y ＝5x+6，y 的值随x 值的减小而___________。 2、对于函数1223 y x =-, y 的值随x 值的________而增大。 3、一次函数 y=(6-3m)x ＋(2n －4)不经过第三象限，则m 、n 的范围是__。4、直线y=(6-3m)x ＋(2n －4)不经过第三象限，则m 、n 的范围是_________。 5、直线y=kx+b 经过第一、二、四象限，则直线y=-bx+k 经过第____象限。 6、无论m 为何值，直线y=x+2m 与直线y=-x+4的交点不可能在第______象限。 7、已知一次函数（1）当m 取何值时，y 随x 的增大而减小？（2）当m 取何值时，函数的图象过原点？题型五、待定系数法求解析式方法：依据两个独立的条件确定k,b 的值，即可求解出一次函数y=kx+b （k ≠0）的解析式。 ☆ 已知是直线或一次函数可以设y=kx+b （k ≠0）； ☆ 若点在直线上，则可以将点的坐标代入解析式构建方程。 1、若函数y=3x+b 经过点（2，-6），求函数的解析式。 2、直线y=kx+b 的图像经过A （3，4）和点B （2，7）， 4、一次函数的图像与y=2x-5平行且与x 轴交于点（-2,0）求解析式。6、已知直线y=kx+b 与直线y= -3x+7关于y 轴对称，求k 、b 的值。 7、已知直线y=kx+b 与直线y= -3x+7关于x 轴对称，求k 、b 的值。8、已知直线y=kx+b 与直线y= -3x+7关于原点对称，求k 、b 的值。 5、若一次函数y=kx+b 的自变量x 的取值范围是-2≤x ≤6，相应的函数值的范围是-11≤y ≤9，求此函数的解析式。题型六、平移方法：直线y=kx+b 与y 轴交点为（0，b ），直线平移则直线上的点（0，b ）也会同样的平移，平移不改变斜率k ，则将平移后的点代入解析式求出b 即可。直线y=kx+b 向左平移2向上平移3 <=> y=k(x+2)+b+3;（“左加右减，上加下减”）。 1. 直线y=5x-3向左平移2个单位得到直线。 2. 直线y=-x-2向右平移2个单位得到直线 3. 直线y=21x 向右平移2个单位得到直线 4. 直线y=22 3+-x 向左平移2个单位得到直线 5. 直线y=2x+1向上平移4个单位得到直线 6. 直线y=-3x+5向下平移6个单位得到直线

高中的常见函数图像及基本性质

常见函数性质汇总及简单评议对称变换常数函数 f (x )=b (b ∈R) 1）、y=a 和 x=a 的图像和走势 2）、图象及其性质：函数f (x )的图象是平行于x 轴或与x 轴重合（垂直于y 轴）的直线一次函数 f (x )=kx +b (k ≠0,b ∈R) 1)、两种常用的一次函数形式:斜截式—— 点斜式—— 2）、对斜截式而言，k 、b 的正负在直角坐标系中对应的图像走势： 3）、|k|越大，图象越陡；|k|越小，图象越平缓 4）、定义域：R 值域：R 单调性：当k>0时；当k<0时奇偶性：当b =0时，函数f (x )为奇函数；当b ≠0时，函数f (x )没有奇偶性；例题：y=f （x ）; y=g （x ）都有反函数，且f （x-1）和g -1 (x)函数的图像关于y=x 对称，若g （5）=2016，求）= 周期性：无 5）、一次函数与其它函数之间的练习 1、常用解题方法： b

反比例函数 f (x )= x k (k ≠0，k 值不相等永不相交；k 越大，离坐标轴越远) 图象及其性质：永不相交，渐趋平行；当k>0时，函数f (x )的图象分别在第一、第三象限；当k<0时，函数f (x )的图象分别在第二、第四象限；双曲线型曲线，x 轴与y 轴分别是曲线的两条渐近线；既是中心对成图形也是轴对称图形定义域：),0()0,(+∞-∞ 值域：),0()0,(+∞-∞ 单调性：当k> 0时；当k< 0时周期性：无奇偶性：奇函数反函数：原函数本身补充：1、反比例函数的性质 2、与曲线函数的联合运用（常考查有无交点、交点围城图行的面积）——入手点常有两个——⑴直接带入，利用二次函数判别式计算未知数的取值；⑵利用斜率，数形结合判断未知数取值（计算面积基本方法也基于此） 3、反函数变形（如右图） 1）、y=1/（x-2）和y=1/x-2的图像移动比较 2）、y=1/(-x)和y=-（1/x ）图像移动比较 3）、f (x )= d cx b ax ++ (c ≠0且 d ≠0)（补充一下分离常数）（对比标准反比例函数，总结各项内容）二次函数一般式：)0()(2 ≠++=a c bx ax x f 顶点式：)0()()(2 ≠+-=a h k x a x f 两根式：)0)()(()(21≠--=a x x x x a x f 图象及其性质：①图形为抛物线，对称轴为，顶点坐标为 ②当0>a 时，开口向上，有最低点当00时，函数图象与x 轴有两个交点（）；当<0时，函数图象与x 轴有一个交点（）；当=0时，函数图象与x 轴没有交点。 ④)0()(2 ≠++=a c bx ax x f 关系 )0()(2 ≠=a ax x f 定义域：R 值域：当0>a 时，值域为（）；当0 a 时；当0