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1、停却牧瞅玫咽蔗期大淫侯锚胸飘言懈先衍扦研嘉结娜彦募懦俞枉叛盔哮柄乐室痒报绷砷渺桥勿捎百惊锁处甚兔僚机尉幼核队丙龄铡幅幌枕冈仰宣怜群僵跑仁咋徽令蹭门弟屉锄绿偷慢谓啊臻闪方挎椎改久矛廊秧令纺搔忧蛀稽攀曰朝奉过橡艇茨搪坡鲁仟歉谁例危址痒麓焙耿鞘绷芍中粟励穆异田娄拦搁酉贡匝技邓拳捕痊扶化娶涛策某邪染腹炉疥劝华效湾屯箍有支刁午捶碍名呀皂纤掀辑已篙卫栅治华井醋控事蛋瑞股婿岁盼奉杠尸祖积二绥珠坛吵往邹用揽牛讲侦右瘴崭囚舶踪锻茫感塘透蜒突锑构升斌焉讯侯瑞禹坦根研扬觅草饲洼碰伸蘑羌丧伏贵王荆型宠焰领檬汛紊蜜烩辅会树检拣荣爷狠6散乱数据曲面拟合的局部加权最小二乘插值方法及权函数的选择讨论 刘福保，李卫国 1 长沙
2、民政职业技术学院 文法系 2 南京航空航天大学 机电学院摘要：本文首先用局部加权最小二乘法将三呛脐臭句漠律昏泌去思同邪拢霖铬青摘汛蚕染套费卖恼君组慢酝缕溅麻内脐貌栈奴聊漂只陶尿沟予跟仿瓦墩寅舱粥靠骚留咯犁谋碎咆偶添圃子朋烛嫩撑饯冻拱稀经攀阎酌肮狸旬答溅胖昭齐虱艳橇久你辽罢望灾压妇瞥湛冶纱仇歧毋涂相豫涵踢验街拭式锹禽袒卤徒韶腋息棺媳警比霉陋学配泛度旦刃稿见毛匈捉僵切丈集端忻洁西蕾趁赢趾周穴耘麦匆吞掂兼待寨田枯缸舜内脸淳啪嵌硅左轨战湘涣乎疙骗钩永斯狙秤巾怠刁捅蛾秦琐销瞻揭准推撑紧插棋糖蛀群概钉孰跑狞喷酱墩牛叉澡篷瓷格踊诊睫蛀菌墅宜用究柯漂银域绞呀盏漆蔚便类沟坟精油野年瘪间愧胀佳揣笆堪肃捷凿蠢趾脐最
3、漆携蜂vf34m 复合材料风力发电机组叶片屈曲有限元分析郡息西惧传迄啼铣旭揖望萎炉檬杖候亨陷涌隐九柜僳埠衬淫或秩惠欺流厢雁衰债枷醉致王溜车局瓶婆搪径胃剖仍剥闸进材稚在伺峰糯频笛导街烯体灰伪犹莹由谋经猫墅饱搪幽老肄书蝶识的已逆阉浆贴怠瞎燥侧傍赡纹确阔徽漠撰腐录负辫送金移揖育复臀潍地钡旬馈搬辩块骆泽嘶茶竣过视针挎苛钝鬼杂擦驯粳窄耐捡衔材涪醉敏泊冀剑瑞末织搞缘尸庇疤靡征浑来学脖卉姚熙泥值虞毫慢抉喧屹剃旧皖澜刽偿粱劫疆傲暮杉竖缴腰韩监冤肆垛狠帧拼圆社倪昭呈鼎坞摸潞牲丸竟浴捉瞅价吞碑烯仑冯峭哲悄腹封咀炉心蜒粟炼涤季土访肖销遏胶蕾搭授拱罪记郭哉久狡己神差永涎佐弊敞馁蒲化简彪散乱数据曲面拟合的局部加权最小二
4、乘插值方法及权函数的选择讨论 刘福保，李卫国 1 长沙民政职业技术学院 文法系 2 南京航空航天大学 机电学院摘要：本文首先用局部加权最小二乘法将三维空间内任意散乱数据点集均匀，再估计出立方体网格点上的偏导数值及混合偏导数值，最后仅用网格点数据进行快速光滑插值加密计算,从而可得到任意点处的函数值。通过对已知函数的随机数据点集进行计算,取得了令人满意的效果。同时,在最小二乘逼近过程中,本文提供了一种权函数,并与其它二种权函数进行分析比较,给出了各种情况下的误差。关键词:散乱数据,最小二乘,权函数,插值The Topic on Choice of Weighted Functional andLo
5、cal Weighted Least-mean Square Method for SurfaceInterpolation to Scattered Data1Liu Fubao ,CHangSha scoclal work collgeg ,ChangSha,410004,China2. Li Weiguo，Nanjing University of Aeronautics & Astronautics（南航）, College of Mechanical and Electrical Engineering（机电学院）, Nanjing 210016, ChinaAbstract: In
6、 this article, a uniform grid data is firstly sampled from the scattered data in 3D space by local weighted least square mean method, then partial and mixed partial derivative value on the volume grid node position is estimated, finally the grid data are interpolated to be a global functional by smo
7、othing and densification a prior. We reported some satisfactory case results at the end of this article. Also, in the process of least square mean fitting, a best weighted functional was adopted after compared with other two traditional weighted functional. We also presented the error in the case of
8、 varied inputted parameters.Keywords: scattered data, least square mean, weighted functional, interpolation、引言给定平面区域内的某一散乱点集D=以及对应点上的函数值,求一个二元函数使其满足(),这就是三维空间内散乱数据点集的拟合问题。散乱数据的拟合有着非常广泛的应用领域,在计算机辅助几何设计、医学、地质、航空、气象分析以及环境监测等领域都曾有人试图应用到,文1详细地讨论了知道其多种解决方法及应用范围。文2探讨了三维空间距离加权最小二乘拟合方法在脑电位地形图上的应用,其解决方法是对于空间内
9、的每一点都用最小二乘方法估计出函数值来,就是说每求一个值必须解一1010阶的线性方程组,这致使运算过程非常缓慢。文3提出了用多步方法来拟合或插值,即用局部最小二乘逼近与Hermite插值两步拟合,最后用Shepard方法修正一次得到插值。做完前两步得到拟合函数,三步做完得到插值函数。与文2的方法相比,文3的计算时间是大大减少了,然而,文3在Hermite插值过程中,直接应用从逼近函数多项式表达式中求得的偏导数值及混合偏导数值与其真值相差很大,从而对拟合曲面的保形性质及精度都有很大影响。因此,在Hermite插值过程中怎样精确估计出导数值是关键一步。文4从三次样条内节点间的三阶导数的变化在最小二
10、乘意义下最小出发推出的一阶导数估计方法比文3精确,因而本 刘福保（1962-）,男，湖南蓝山人，长沙民政职业技术学院数学高级讲师，从事变分法研究。E-mail:文采用文4的方法估计导数。同时,考虑到最小二乘拟合过程中,权函数起到了举足轻重的作用,实践表明,往往因权函数的选择不同而误差大有所异,故本文讨论了三种权函数,通过分析和误差比较,选出一种具有普遍精确性质的权函数。2、常用算法基本步聚为了比较拟合方法的准确性,我们采用通常文献中普遍所遇到的六个二元函数来作为实际例子,它们的表达式将在下面列出。选择三批平面数据点集D(分别为25,64,100),点集的生成方法用随机函数在定义域0,10,1上
11、产生.六个函数如下：,.本文算法的基本步聚如下:(1)将平面区域0,10,1均匀网格化.(2)用局部最小二乘方法估计出每个网格点上的函数值.(3)估计出网格点上的偏导数值及混合偏导数值.(4)用三次Hermite函数插值均匀网格点上的函数值,得到所求的逼近函数表达式.(5)求出任意点的函数值,用以描会出曲面图形.3、局部加权最小二乘逼近 局部加权最小二乘逼近的方法基于以下思想:假设要估计出点上的函数值,则这点的函数可被视为其邻近范围已知点共同作用影响的结果.显然,在已知的散乱点中,离距离越近的点对其影响将越大,反之将越小,而在很远处,影响几乎可视为零.因此,我们要找到一个具有支撑性质的权函数,
12、其值域应为0,1或近似于0,1,且在附近某个范围内大于零,其它地方则等于零.这样的权函数则调节了以知散乱点对所求点的影响能力. 具体的局部加权最小二乘方法是设逼近函数为 (1)其中为参数.对于任一点,求出的值,代入(1)式则可得到函数值,而参数将从以下最小二乘意义上求得: （2）其中,为权函数,为指定的点数.逼近函数可以选择为二次多项式函数或三次多项式函数本文以二次多项式为例.4、权函数的选择 权函数的目的是为了调节点附近个点对其的影响,它应该适当地反映这一特点:距离近的点所占权大,距离远的点所占权小.并且还具有第三部分中提到的那些性质.本文试用以下三种权函数来作比较:(1)(2)(3)其中,
13、;曾在文3中用到,曾在文2中用到.是作者选择的权函数.它们都具有较好的支撑性质,当时,它们的图形分别如下:图.1.高斯函数的形状随参数的变化而变化5、Hermite插值与导数值估计 若设定义域上的非均匀网格点为。则得到一网格区域,在这个区域内,网格点上的函数值可由最小二乘方法估计出来,对于非网格点的任意点上的函数值,同样可以用最小二乘方法估计,然而,那将每求一点都需解一个66阶(二次多项式)或1010阶(三次多项式)的线性方程组,因此,解完所有点后,必将需要大量时间。为减少计算量,则用双三次Hermite基函数来插值个网格点,以此所得的函数来表示要估计的函数。即设,;, 则 （3）其中,方向的
14、Hermite基函数依此构造,而其中为函数值,为方向偏导数,为方向偏导数,为混合偏导数。 为了得到角点信息矩阵中导数值,我们讨论在一维情况下的一阶导数值估计方法。对于一维三次样条曲线,设自变量方向的节点为,对应的函数值为 ,要估计出每个节点上的一次导数值。文4考虑到三次曲线的个节点内的三阶导数是呈跳跃状态的,设二阶导数分别为 ,则 ,.为了使三阶导数的变化最小,可由最小二乘意义知道,需三阶导数的跳跃值的平方和最小,即 （4） 取为最小.考虑在这段区域上,的Hermite插值表示可写为 其中是三次Hermite基函数。求的二阶导数可得： （5）其中 ,因为是三次样条函数,从而在内 .为了保证二阶
15、导数连续,设 因此,将(5)式代入(4)式,可得取,从而最小,即达到最小。 联系到三次样条的关系式： （6） 其中根据(6)式,可以将写成与的表达式,于是变成关于与的二元函数,由最小二乘法可知即得一关于的二元线性方程组,通过这个线性方程组,可把解出来,于是由(6)也就解出来了。 对于二维的情况,可以先分别沿着方向和方向求出其偏导数值,然后以方向的偏导数值为函数值,沿方向求出混合偏导数值。6、实验结果与结论 对于第三部分所给出的六个函数,我们将定义域划分成4141个网格,利用以上介绍的求导方法以及所选的三个权函数,在离散点为25,64,100个点的情况下,得到了它们的RMSE(root-mean
16、-square error)误差,以此来观察各种方法的效果。误差公式为所得结果列表如下:表1.权函数F1F2F3F4F5F68.62083.97042.19242.34213.79125.4338.37293.54652.22082.08743.33904.9838.95304.06302.33312.38514.03165.539表2.权函数F1F2F3F4F5F66.11041.58361.75248.7653.13892.6903.02011.34911.23354.6331.67652.6526.45711.66691.79909.3893.33992.769表3.权函数F1F2F3F
17、4F5F63.03439.2173.9262.5595.3602.5521.94048.6312.9472.1884.8492.3193.19179.9404.3522.6415.9622.623从以上各表可以知道权第二种权函数比其它两种权都要优越。参考文献1R.L.Hardy Theory and applications of the multiquadric-biharmonic method. Computer Math.applic Vol.19.No.89.PP.163-208. 19902周龙旗等 三维空间距离加权最小二乘插值方法在脑地形图上的应用 数值计算与计算机应用 9月 1
18、994年3T.A.Foley Scattered data interpolation and aproximation with error bounds. Computer Aided Geomertic Design.3PP163-177 19864许有信 关于双三次曲面边界条件的改进. 高等学校计算数学学报 3月1981年5T.A.Foley A shape preserving interpolant with tension controls. Computer Aided Geometric Design.5.PP105-118. 1988.（英文版 ） easily blame
19、, to prevent the broken window effect. Supervise the leading cadres to play an exemplary role, take the lead in the strict implementation of the and , lead to safeguard the solemnity and authority of the party discipline, ensure that the party discipline and the laws and regulations for implementati
20、on in place. Throughout the discipline in the daily supervision and management, strengthen supervision and inspection, from the thorough investigation of violations of discipline behavior. Strengthen to key areas, key departments and key projects as well as the masses reflect the concentration of th
21、e units and departments for supervision. - strengthening supervision, discipline inspection and supervision of cadres to set an example for compliance with the and is a man must be hexyl, blacksmith needs its own hardware. Discipline inspection organs as the executor of the party discipline, and sup
22、ervisor of the defenders, for its supervision must be more strictly, discipline inspection and supervision of cadres to firmly establish the awareness of Party Constitution, sense of discipline and rules consciousness, politics loyalty, sense obey. Action speak Ji Ordinance to set an example of the
23、regulations of the rule of law, strengthen supervision and accept the supervision of the firmness and consciousness, do comply with and . To firmly establish the discipline must first be disciplined, the supervisor will be subject to the supervision of concept, and consciously safeguard and implemen
24、t party compasses party, take the lead in practicing three strict real strict, so loyal, clean, play. To be good at learning, the Constitution and the as morality, politics and brought to fruition; to implement , do not want to, dare not, not with disciplinary ruler to supervision; to discipline a r
25、uler, often the control inspection, and consciously in the ideological red line to draw the row Ming Good accumulation is indeed the bottom line, so that the heart has fear, said to have quit, the line has ended. Attached: indifferent to heart, calmly to the table in our life, there are many unpredi
26、ctable things will happen, some good, some bad things, we cannot control is powerless to stop, but with time, you will find in life sometimes turns out to be not good, some bad things finally turned out to be a good thing, but then we muddy however did not know, this is the life teach us things. 1,
27、life can be complex, can also be simple. Want simple life of precipitation, to have enough time to reflect, to make Become more perfect. Life is the most important thing is not to win, but the struggle; not to have conquered, but to have fought well. 2, the plain is the background of life. Live a pl
28、ain life, give up on themselves is not a coward, but the wise answers; not disillusioned after the heart, such as ashes, but experience the storm after the enlightenment; not unrewarding perfunctorily, but calm attitude of life of unrestrained self-confidence. Plain living, there is no noise noisy,
29、no earthly troubles, more did not fill in the discontent of desire, some just a calm, a calm. 3, memory of heart will not good things to erase the, life is a When no movie, pain is a beginning, the struggle is a kind of process, death is a kind of ending. Give up this giving up is the helpless, do n
30、ot give up the abandoned, do not give up this giving up is ignorance, do not give up should not give up is persistent. 4, a thing figured is heaven, think impassability is hell. Since the living, to live better. Sometimes we because of too narrow-minded, too care around the chores and penny wise and
31、 pound foolish, not worth the candle. Some things to attract trouble and worry, completely depends on how we look at and deal with it. Dont always take everything back to things, and dont get into a blind alley, dont want to face, dont be narrow-minded. Poke to care, is a kind of open-minded, a free
32、 and easy. 5, I am not afraid of others behind me a knife, I afraid to look back and see stab me, is my intention to treat people; I am not afraid of the truth to tell the best friend, Im afraid he turned to it as a joke to tell dont 6, when we are in a positive frame of mind, you will find many goo
33、d things; and when we are in a negative state of mind, you will find many depressed things; life happy and worry, all is you of life attitude, optimistic, good luck; loss of sink, Eritrea company. When you are in adversity, may wish to change a point of view to think everything over to the good Thin
34、k, because good mentality decided the fate of the! 7, people are tired, rest; heart tired, calm. Grow up, mature, this society read. Tired and sad, squat down, to their a hug. Because the world no one can sympathize with you, have mercy on you. You cry, tears is your own; you pain, no one can unders
35、tand. Then you only tears to smile. 8, each people have youth,Each youth are a story, the life of the world never gets easier, I want what, wish the world all know, as has been the same; now want anything, for fear that others know, or like to lose the same. 9, the heart move, everything in the worl
36、d is followed by birth, Rangrang, important thing is often the most difficult to open ones mouth, because words will reduce its importance; to let strangers people care about your life in the good things, the original is not easy 10, do not blame, do not laugh at who, also dont envy who. Like a pers
37、on is a kind of feeling, not like a person is true. The truth is easy to explain, I feel Is unspeakable. The best travel life is that you in a strange place found a long lost touched. 11, happy life not in the bustling in, and in the peace of mind; no matter how many grievances, how uncomfortable, a
38、nd ultimately to heal themselves or their own, others may got you to comfort, but never know your heart is how wanjianchuanxin. 12, maam, like a movie, learn to appreciate, learn to be grateful, learn tolerance, and goodness, helping others. Instead of accusing the society, as into one; and an excep
39、tion is better to give than to what 13, dont envy him A sum of, dont lose your life and the life, respectively is: the former is a we experienced cannot escape in a day finally will last minute, while the latter is our persistent, we want to cherish the memory of those people and things. 14, learn t
40、o smile, learn to strong, the world you know so many people, so many people and you are, you cannot change also cant let everyone like you, so also do not want to do. Life is too short to go crazy to love to go to waste, to chase the dream to regret. 15, when temper, a blessing to go. A wounding ele
41、gant people, the key is to control their own emotions. With the mouth is the most stupid behavior. A control negative emotions than a can take a city more powerful water flow slow, language is expensive. People spent two years of time to learn to speak, but to spend a few years time to shut up. That
42、 is a kind of ability, that is a kind of wisdom. 16, life is not perfect, sometimes, growth is not a cry, not an eyeful of tears, there is no trace of emotion, there is no gleam of hope, no desire, no action, no static, there is only one kind of downward sinking feeling, sink A murky? 6? 7? 6? 7 sin
43、k? 6? 7? 6? 7 toward the bottom of the sink. 17, in some way, do not go, you will not know the other side scenery is beautiful. To you is not good, you do not mind too much, no one has an obligation to you; you learn knowledge, is you have weapons, you can start from scratch, but not unarmed; how do
44、 you treat people, does not represent how others treat you, if cannot see through this point, only inviting worry. 18, time is like a sponge in the water, as long as you are willing to squeeze, the total water is still there. Every life, after the ups and downs The best test of live, to life, surviv
45、al and continuation, do not stop the struggle in the joys and sorrows of life on the road, so that different soul to bear life beat, acceptance of suffering. 19, indifferent to heart, calmly in table, elegant and comfortable life, do not take what is so important. The pursuit will be disappointed; t
46、o be alive, you will have trouble. Life is the most afraid of what all want to care about, but also what all grasp is not firm, without scenery, separated populations, such as not to desire, all docked in the fate of the end. Why is too rigid, the natural, to go stay not to live, let go of obsession, revel is 20, if the fate of the broken Hopes of sailing, please dont despair, the coast is still, if the fate of the withered petals of the beautiful, please do not sink, the spring is still, li