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  • 多维空间几何坐标系
    2018-08-17 18:25:00

    为什么叫齐次坐标系?

      齐次坐标系,英文名称Homogeneous coordinate system。也就是说Homogeneous国内翻译为“齐次”,查询“齐次”的解释,谷歌翻译Homogeneous是“同质”的意思,百度翻译结果是“均匀的;同性质的,同类的;由相同(或同类型)事物(或人)组成的;[数]齐性的,齐次的”。所以从名字上我们不能顾名思义,只能先理解齐次坐标系在来思考这个名字了。

      首先我们先从齐次性开始理解。

    齐次性定义

    在百度百科里的解释:

    一般地,在数学里面,如果一个函数的自变量乘以一个系数,那么这个函数将乘以这个系数的k次方,我们称这个函数为k次齐次函数,也就是:
    如果函数 f(v)满足
    f(ax)=a^k f(x),
    其中,x是输入变量,k是整数,a是非零的实数,则称f(x)是k次齐次函数。
     
    比如:一次齐次函数就是线性函数2.多项式函数 f(x,y)=x^2+y^2
    因为f(ax,ay)=a^2f(x,y),所以f(x,y)是2次齐次函数。
     
    齐次性在数学中描述的是函数的一个倍数的性质。
     
    齐次坐标系

    以下摘自维基百科对于齐次坐标系的描述:

    在数学里,齐次坐标(homogeneous coordinates),或投影坐标(projective coordinates)是指一个用于投影几何里的坐标系统,如同用于欧氏几何里的笛卡儿坐标一般。该词由奥古斯特·费迪南德·莫比乌斯于1827年在其著作《Der barycentrische Calcul》一书内引入[1][2]。齐次坐标可让包括无穷远点的点坐标以有限坐标表示。使用齐次坐标的公式通常会比用笛卡儿坐标表示更为简单,且更为对称。齐次坐标有着广泛的应用,包括电脑图形及3D电脑视觉。使用齐次坐标可让电脑进行仿射变换,并通常,其投影变换能简单地使用矩阵来表示。

    实投影平面可以看作是一个具有额外点的欧氏平面,这些点称之为无穷远点,并被认为是位于一条新的线上(该线称之为无穷远线)。每一个无穷远点对应至一个方向(由一条线之斜率给出),可非正式地定义为一个点自原点朝该方向移动之极限。在欧氏平面里的平行线可看成会在对应其共同方向之无穷远点上相交。给定欧氏平面上的一点 (x, y),对任意非零实数 Z,三元组 (xZ, yZ, Z) 即称之为该点的齐次坐标。依据定义,将齐次坐标内的数值乘上同一个非零实数,可得到同一点的另一组齐次坐标。例如,笛卡儿坐标上的点 (1,2) 在齐次坐标中即可标示成 (1,2,1) 或 (2,4,2)。原来的笛卡儿坐标可透过将前两个数值除以第三个数值取回。因此,与笛卡儿坐标不同,一个点可以有无限多个齐次坐标表示法。

    一条通过原点 (0, 0) 的线之方程可写作 nx + my = 0,其中 n 及 m 不能同时为 0。以参数表示,则能写成 x = mt, y = − nt。令 Z=1/t,则线上的点之笛卡儿坐标可写作 (m/Z, − n/Z)。在齐次坐标下,则写成 (m, − n, Z)。当 t 趋向无限大,亦即点远离原点时,Z 会趋近于 0,而该点的齐次坐标则会变成 (m, −n, 0)。因此,可定义 (m, −n, 0) 为对应 nx + my = 0 这条线之方向的无穷远点之齐次坐标。因为欧氏平面上的每条线都会与透过原点的某一条线平行,且因为平行线会有相同的无穷远点,欧氏平面每条线上的无穷远点都有其齐次坐标。

    概括来说:

    • 投影平面上的任何点都可以表示成一三元组 (X, Y, Z),称之为该点的'齐次坐标或投影坐标,其中 X、Y 及 Z 不全为 0。
    • 以齐次坐标表表示的点,若该坐标内的数值全乘上一相同非零实数,仍会表示该点。
    • 相反地,两个齐次坐标表示同一点,当且仅当其中一个齐次坐标可由另一个齐次坐标乘上一相同非零常数得取得。
    • 当 Z 不为 0,则该点表示欧氏平面上的该 (X/Z, Y/Z)。
    • 当 Z 为 0,则该点表示一无穷远点。
    • 注意,三元组 (0, 0, 0) 不表示任何点。原点表示为 (0, 0, 1)[3]。

      从上面的描述我们知道齐次坐标是用于投影几何里的坐标系统,和平时我们用的笛卡尔坐标系一样,是帮助我们理解宇宙的工具。但因为这是两种不同的坐标系,我们需要跳出笛卡尔坐标系,以更宏观的思维来理解,不然有些场景会让我们困惑。

      首先我们把下方的这个投影描述图印在脑海中。

     

      考虑一个点p,它的笛卡尔坐标是(x,y),齐次坐标是(x,y,1),齐次坐标比笛卡尔坐标多一个维度,按照现在书上和网络上的理解基本都是说笛卡尔坐标系就是齐次坐标系中w=1的那个平面,(x,y,1)是齐次坐标(kx,ky,k)表示的点在w=1上的映射。因为一开始我把这个投影想成了正交投影,所以总感觉不对。

      那为什么要引入齐次坐标系了?

      上面也提到了,主要是方便计算机图形学进行仿射几何变换。简单的理解就是可以使用矩阵同时描述旋转和平移,这样我们就可以使用矩阵相乘来表述物体的旋转、缩放和平移了,具体内容可参照《计算机图形学》等书籍或者网上的资料。

      如:https://oncemore.wang/blog/homogeneous/

     

    无穷远的点

      使用(∞,∞)?所以在笛卡尔坐标系中无穷远的点是没有定义的。但这个在齐次坐标系统中可以用w=0来表示无穷远的点,即任何(x,y,0)表示无穷远的点。

    两条平行线在无限远处相交

      笛卡尔坐标系中两条平行线没有交点,即使在三维空间也是,但在齐次坐标系中它们在无穷远点相交。

    引申的理解  

     1.坐标系理解

      在笛卡尔坐标系中有一个原点(0,0),对于这个点我以前没有过多的考虑,后来参考了很多资料,有一个说法,想象在宇宙中有一个绝对坐标系,对于我们现在使用的笛卡尔坐标系,其原点位于(a,b)点,当然同时也就还有无数的相同的坐标系,只不过它们的原点不同,对于笛卡尔坐标系中的点(x,y),它对于所有的笛卡尔坐标系都是相同的,有点多维宇宙的感觉,其中一个坐标系就是一个宇宙。我觉得这种思维也很有意思。

      2.向量和点

      关于向量和点,我觉得下面的这边文章说的很好,大家可以参考。

      https://blog.csdn.net/winbobob/article/details/38829001

     

      以下都是我根据这篇博客的得来的感悟。描述一个点比描述一个向量需要更多的信息。我们平时的描述一个点时,其实都忽略了一个信息,即参照点的信息,我们都是基于参照点描述一个点的位置,这个参照点就是原点。下面是我从上面博客截取的一段关于点和向量的一个解释:

    对于一个向量v以及基oabc,可以找到一组坐标(v1,v2,v3),使得v = v1 a + v2 b + v3 c          (1)

     而对于一个点p,则可以找到一组坐标(p1,p2,p3),使得 p – o = p1 a + p2 b + p3 c            (2),

    从上面对向量和点的表达,我们可以看出为了在坐标系中表示一个点(如p),我们把点的位置看作是对这个基的原点o所进行的一个位移,即一个向量——p – o(有的书中把这样的向量叫做位置向量——起始于坐标原点的特殊向量),我们在表达这个向量的同时用等价的方式表达出了点p:p = o + p1 a + p2 b + p3 c   (3) 

    (1)(3)是坐标系下表达一个向量和点的不同表达方式。这里可以看出,虽然都是用代数分量的形式表达向量和点,但表达一个点比一个向量需要额外的信息。如果我写出一个代数分量表达(1, 4, 7),谁知道它是个向量还是个点!

        我们现在把(1)(3)写成矩阵的形式:v = (v1 v2 v3 0) X (a b c o)

    p = (p1 p2 p3 1) X (a b c o),这里(a,b,c,o)是坐标基矩阵,右边的列向量分别是向量v和点p在基下的坐标。这样,向量和点在同一个基下就有了不同的表达:3D向量的第4个代数分量是0,而3D点的第4个代数分量是1。像这种这种用4个代数分量表示3D几何概念的方式是一种齐次坐标表示。

      

      上面的推导很完美,但还是有种只知其然的感觉。其实这里有一个非常重要的点没有指出来,就是变换对应矩阵的乘法。这里的变换有平移、旋转、缩放。我们做的这些推导都是想要用矩阵的乘法来表示变换,只有这样上面的推导才不显得突兀。而齐次坐标系的出现,也是处于计算机使用矩阵乘法来表示仿射变换的需求。所以理解齐次坐标系,很重要的一点就是为什么我们需要齐次坐标系。 

      对于齐次坐标系,我一直想知道它的几何意义,但由于使用欧氏几何的三维空间来理解,总是不得要领。现在觉得,首先我们要明白这些坐标系都是我们理解这个世界的工具,有些工具对于世界的描述很贴合我们对于世界的想象,所以常用,但另外一些工具是从其它的角度来解读这个世界的,在我们使用这些工具之前,我们需要跳出原先的思维(如齐次坐标系再更高维度的仿射变换映射为低维度的平移),这样才能更好的理解这个世界的本质。

     

    转载于:https://www.cnblogs.com/xin-lover/p/9486341.html

    更多相关内容
  • 平行坐标系的具有良好的数学基础, 其射影几何解释和对偶特性使它很适合用于可视化数据分析。例如series-parallel.data中有如下数据:[[1, 55, 9, 56, 0.46, 18, 6, '良'],[2, 25, 11, 21, 0.65, 34, ...

    parallel

    在 ECharts 中平行坐标系(parallel)是一种常用的可视化高维数据的图表。平行坐标系的具有良好的数学基础, 其射影几何解释和对偶特性使它很适合用于可视化数据分析。

    例如 series-parallel.data 中有如下数据:[

    [1, 55, 9, 56, 0.46, 18, 6, '良'],

    [2, 25, 11, 21, 0.65, 34, 9, '优'],

    [3, 56, 7, 63, 0.3, 14, 5, '良'],

    [4, 33, 7, 29, 0.33, 16, 6, '优'],

    { // 数据项也可以是 Object,从而里面能含有对线条的特殊设置。

    value: [5, 42, 24, 44, 0.76, 40, 16, '优']

    lineStyle: {...},}

    ...

    ]

    数据中,每一行是一个『数据项』,每一列属于一个『维度』。(例如上面数据每一列的含义分别是:『日期』,『AQI指数』, 『PM2.5』, 『PM10』, 『一氧化碳值』, 『二氧化氮值』, 『二氧化硫值』)。

    平行坐标系适用于对这种多维数据进行可视化分析。每一个维度(每一列)对应一个坐标轴,每一个『数据项』是一条线,贯穿多个坐标轴。在坐标轴上,可以进行数据选取等操作。如下:

    5d47cf1ac9451147e97c843f02d52e95.png

    平行坐标系配置方式

    『平行坐标系』的 option 基本配置如下例:option = {

    parallelAxis: [ // 这是一个个『坐标轴』的定义

    {dim: 0, name: schema[0].text}, // 每个『坐标轴』有个 'dim' 属性,表示坐标轴的维度号。

    {dim: 1, name: schema[1].text},

    {dim: 2, name: schema[2].text},

    {dim: 3, name: schema[3].text},

    {dim: 4, name: schema[4].text},

    {dim: 5, name: schema[5].text},

    {dim: 6, name: schema[6].text},

    {dim: 7, name: schema[7].text,

    type: 'category', // 坐标轴也可以支持类别型数据

    data: ['优', '良', '轻度污染', '中度污染', '重度污染', '严重污染']

    }

    ],

    parallel: { // 这是『坐标系』的定义

    left: '5%', // 平行坐标系的位置设置

    right: '13%',

    bottom: '10%',

    top: '20%',

    parallelAxisDefault: { // 『坐标轴』的公有属性可以配置在这里避免重复书写

    type: 'value',

    nameLocation: 'end',

    nameGap: 20

    }

    },

    series: [ // 这里三个系列共用一个平行坐标系

    {

    name: '北京',

    type: 'parallel', // 这个系列类型是 'parallel'

    data: [

    [1, 55, 9, 56, 0.46, 18, 6, '良'],

    [2, 25, 11, 21, 0.65, 34, 9, '优'],

    ...

    ]

    },

    {

    name: '上海',

    type: 'parallel',

    data: [

    [3, 56, 7, 63, 0.3, 14, 5, '良'],

    [4, 33, 7, 29, 0.33, 16, 6, '优'],

    ...

    ]

    },

    {

    name: '广州',

    type: 'parallel',

    data: [

    [4, 33, 7, 29, 0.33, 16, 6, '优'],

    [5, 42, 24, 44, 0.76, 40, 16, '优'],

    ...

    ]

    }

    ]

    };

    需要涉及到三个组件:parallel、parallelAxis、series-parallelparallel:这个配置项是平行坐标系的『坐标系』本身。一个系列(series)或多个系列(如上图中的『北京』、『上海』、『广州』分别各是一个系列)可以共用这个『坐标系』。和其他坐标系一样,坐标系也可以创建多个。位置设置,也是放在这里进行。

    parallelAxis:这个是『坐标系』中的坐标轴的配置。自然,需要有多个坐标轴。其中有 parallelAxis.parallelIndex 属性,指定这个『坐标轴』在哪个『坐标系』中。默认使用第一个『坐标系』。

    series-parallel:这个是『系列』的定义。系列被画到『坐标系』上。其中有 series-parallel.parallelIndex 属性,指定使用哪个『坐标系』。默认使用第一个『坐标系』。

    平行坐标系配置注意项及实例

    配置多个 parallelAxis 时,有些值一样的属性,如果书写多遍则比较繁琐,那么可以放置在 parallel.parallelAxisDefault 里。在坐标轴初始化前,parallel.parallelAxisDefault 里的配置项,会分别融合进 parallelAxis,形成最终的坐标轴的配置。

    如果数据量很大并且发生卡顿的话,建议把 series-parallel.lineStyle.normal.width 设为 0.5(或更小), 可能显著改善性能。

    遇到维度较多的数据的显示时,比如有 50+ 的维度,那么就会有 50+ 个轴。那么可能会页面显示不下。

    那么,可以通过 parallel.axisExpandable 来改善显示效果,如下例子:

    5b2cf554c63df1234291104c748431bd.png点击编辑实例 》》

    展开全文
  • 平行坐标是可视化高维几何和分析多元数据的常用方法。 为了在n维空间中显示一组点,绘制由n条平行线组成的背景,通常是垂直且等距的。所述的点N 维空间被表示为折线与顶点在平行的轴线; 第i 轴上顶点的位置对应于该...

    数据可视化的作业
    用平行坐标绘图可视化分析数据(大概是这个意思吧)

    (说实在我刚看到这个的时候有点不敢相信,因为我甚至不知道这是什么图

    百度得到答案:(真的好官方没有太大用处系列)

    平行坐标是可视化高维几何和分析多元数据的常用方法。

    为了在n维空间中显示一组点,绘制由n条平行线组成的背景,通常是垂直且等距的。所述的点N 维空间被表示为折线与顶点在平行的轴线; 第i 轴上顶点的位置对应于该点的第i个坐标。

    此可视化与时间序列可视化密切相关,除了它应用于轴与时间点不对应的数据,因此没有自然顺序。因此,不同的轴布置可能是有意义的。

    为了表示在高维空间的一个点集, 在N条平行的线的背景下,(一般这N条线都竖直且等距),一个在高维空间的点被表示为一条拐点在N条平行坐标轴的折线,在第K个坐标轴上的位置就表示这个点在第K个维的值。

    平行坐标是信息可视化的一种重要技术。为了克服传统的笛卡尔直角坐标系容易耗尽空间、 难以表达三维以上数据的问题, 平行坐标将高维数据的各个变量用一系列相互平行的坐标轴表示, 变量值对应轴上位置。 为了反映变化趋势和各个变量间相互关系,往往将描述不同变量的各点连接成折线。所以平行坐标图的实质是将 维欧式空间的一个点Xi(xi1,xi2,…,xim) 映射到维平面上的一条曲线。

    平行坐标图可以表示超高维数据。 平行坐标的一个显著优点是其具有良好的数学基础, 其射影几何解释和对偶特性使它很适合用于可视化数据分析。

    浏览一遍之后我大致知道了画图需要的是多维数据了(可是我去哪里找呢555= =

    至于图的话,我只在学长的系统里看见过但是没有了解和研究过,我都不知道要用什么数据去画

    于是,我很懵,我去问了同学得到散点图的回答(显然不是

    于是先去的官网api寻找(我觉得这里可以夸夸自己

    于是找到了一个很像的例子(我有理由确定就是它

    liz
    于是我点了进去大概是这样↓
    平行坐标

    这该死的华丽(但是我觉得哪里怪怪的,也许是坐标轴……然后我挑了下参数,就变成了这个样子↓

    变化

    其实我觉得也没什么变化,放个网址好了:https://observablehq.com/@d3/parallel-coordinates(里面有实例代码

    但是看完代码之后我发现,我没有合适的数据阿(这就很尴尬了

    于是乎就去百度里找数据(要合适我直接做实验报告的确实也是有点难度

    找了很久找到了一个csv数据↓

    City,Consumer Price,Rent,Consumer Price+Rent,Groceries,Restaurant Price,Local Purch. Power
    "Trondheim, Norway",188.91,59.16,142.21,193.94,160.23,67.01
    "Stavanger, Norway",171.32,78.08,137.76,147.74,201.16,78.50
    "Zurich, Switzerland",152.84,74.58,124.67,143.70,138.79,142.77
    "Oslo, Norway",152.03,57.40,117.97,139.22,155.75,99.31
    "Geneva, Switzerland",146.24,79.05,122.06,138.95,135.25,115.54
    "Bern, Switzerland",142.44,57.44,111.85,126.78,103.59,149.28
    "Lucerne, Switzerland",139.94,72.45,115.65,146.52,95.72,105.65
    "Perth, Australia",139.63,43.76,105.13,113.13,131.61,125.08
    "Bergen, Norway",138.79,55.98,108.98,135.46,130.23,97.02
    "Tokyo, Japan",135.23,95.43,120.91,120.42,91.68,90.14
    "Sydney, Australia",132.39,86.25,115.79,121.11,110.98,97.23
    "Adelaide, Australia",129.60,47.30,99.98,120.68,121.44,112.19
    "Monaco, Monaco",128.15,173.08,144.32,89.80,138.14,62.30
    "Copenhagen, Denmark",123.82,48.65,96.77,104.25,139.11,97.43
    "Edinburgh, United Kingdom",122.52,38.21,92.18,92.92,143.02,95.08
    "Melbourne, Australia",121.53,64.37,100.96,112.50,102.37,87.81
    "Dublin, Ireland",119.56,48.30,93.91,109.96,113.19,93.54
    "London, United Kingdom",118.52,87.34,107.30,92.59,117.54,88.58
    "Arhus, Denmark",115.96,65.90,97.94,94.68,132.05,73.97
    "Canberra, Australia",115.89,53.55,93.45,106.77,94.30,115.91
    "Brisbane, Australia",114.70,57.19,94.00,113.64,101.33,100.49
    "Paris, France",113.88,63.47,95.73,95.93,112.89,82.49
    "Malmo, Sweden",112.89,38.45,86.10,92.55,129.79,96.70
    "Toulouse, France",112.28,29.90,82.63,89.10,105.33,82.70
    "Riyadh, Saudi Arabia",112.03,19.91,78.87,138.45,41.65,70.51
    "Amsterdam, Netherlands",110.78,64.61,94.16,69.93,114.88,88.83
    "Darwin, Australia",110.41,56.44,90.98,92.67,111.09,121.91
    "Auckland, New Zealand",110.36,36.80,83.89,98.74,88.77,97.68
    "Gent, Belgium",109.27,36.38,83.04,81.58,109.49,83.02
    "Stockholm, Sweden",109.14,37.75,83.45,87.42,117.66,85.65
    "Brussels, Belgium",109.10,40.63,84.46,91.44,109.75,85.76
    "Aberdeen, United Kingdom",109.08,42.04,84.95,103.80,110.60,119.20
    "Boston, MA, United States",106.80,69.66,93.43,113.24,86.06,109.87
    "Dusseldorf, Germany",106.51,25.77,77.45,77.03,119.73,157.89
    "Wellington, New Zealand",106.02,37.02,81.19,104.20,93.89,90.61
    "Turin, Italy",105.68,40.22,82.12,86.86,107.47,46.42
    "Rome, Italy",105.60,60.03,89.20,87.02,113.68,57.49
    "Vancouver, Canada",105.29,60.49,89.17,103.55,92.54,98.12
    "Edmonton, Canada",105.28,38.95,81.40,104.56,86.97,86.09
    "Brighton, United Kingdom",104.73,54.42,86.62,80.84,87.34,128.05
    "Birmingham, United Kingdom",104.57,42.93,82.39,94.22,90.62,71.95
    "Toronto, Canada",104.53,52.42,85.77,96.13,86.13,104.11
    "Helsinki, Finland",104.08,56.58,86.98,81.00,101.69,96.15
    "Venice, Italy",103.56,31.33,77.56,70.18,89.87,76.23
    "Honolulu, HI, United States",103.46,59.01,87.47,103.67,78.28,90.60
    "Calgary, Canada",103.38,40.93,80.91,99.95,81.08,124.17
    "Sao Paolo, Brazil",102.96,35.60,78.72,62.93,73.58,46.26
    "Tampere, Finland",102.68,30.37,76.65,100.74,98.33,93.84
    "San Francisco, CA, United States",102.52,90.04,98.03,97.84,83.21,89.35
    "Cagliari, Italy",101.88,29.29,75.76,72.93,114.90,66.47
    "Hanover, Germany",101.82,28.11,75.29,82.62,79.95,90.76
    "Belfast, United Kingdom",101.74,22.58,73.25,88.98,113.28,69.18
    "Vienna, Austria",101.47,46.87,81.82,82.49,80.97,99.19
    "Seattle, WA, United States",101.04,48.15,82.01,100.90,82.89,107.23
    "Strasbourg, France",100.86,32.03,76.09,76.03,126.38,79.49
    "Florence, Italy",100.43,41.39,79.18,72.16,94.28,66.66
    "Thessaloniki, Greece",100.39,18.96,71.08,69.76,95.91,42.30
    "Erlangen, Germany",100.39,33.07,76.16,92.92,80.27,28.69
    "Hamburg, Germany",100.18,37.35,77.57,92.62,76.50,90.42
    "Haifa, Israel",100.12,26.52,73.63,78.81,94.18,74.63
    "Reykjavik, Iceland",100.11,31.82,75.53,95.54,90.64,81.22
    "New York, NY, United States",100.00,100.00,100.00,100.00,100.00,100.00
    "Leeds, United Kingdom",98.85,34.95,75.85,81.88,107.45,85.65
    "Phoenix, AZ, United States",98.57,32.76,74.88,85.05,72.83,97.47
    "Washington, DC, United States",98.19,69.37,87.81,89.39,81.26,97.37
    "Linkoping, Sweden",98.06,26.92,72.46,90.51,102.37,114.78
    "Montreal, Canada",97.86,37.57,76.16,94.88,92.23,109.41
    "Bologna, Italy",97.82,38.03,76.31,92.02,89.53,84.84
    "Nice, France",96.89,34.95,74.60,81.58,85.89,85.58
    "Milan, Italy",96.88,49.22,79.73,74.25,106.38,74.06
    "Winnipeg, Canada",96.64,32.72,73.63,96.90,74.52,97.96
    "Athens, Greece",96.49,20.82,69.25,71.85,97.63,48.42
    "Singapore, Singapore",96.08,90.69,94.14,82.76,65.10,93.29
    "Munich, Germany",95.95,46.08,78.00,82.22,87.02,112.58
    "Bordeaux, France",95.92,27.63,71.34,78.45,93.93,67.43
    "Genoa, Italy",95.06,35.04,73.46,82.91,98.27,115.78
    "Manchester, United Kingdom",94.96,36.44,73.90,79.54,90.48,95.97
    "Christchurch, New Zealand",94.35,28.00,70.47,88.50,77.65,82.57
    "Halifax, Canada",94.30,34.13,72.64,95.04,79.02,119.94
    "San Jose, CA, United States",94.11,73.36,86.64,82.33,82.04,113.13
    "Columbus, OH, United States",94.03,20.43,67.54,86.75,75.11,112.75
    "Stuttgart, Germany",94.01,33.18,72.12,68.14,81.10,119.70
    "Marbella, Spain",93.95,30.68,71.18,69.65,99.73,107.63
    "Chania, Greece",93.77,18.01,66.50,69.52,88.97,46.87
    "Ottawa, Canada",93.76,36.87,73.29,96.84,83.34,116.01
    "Brasilia, Brazil",93.08,33.99,71.81,65.89,69.44,32.70
    "Hartford, CT, United States",92.90,47.57,76.59,108.41,72.13,109.10
    "Indianapolis, IN, United States",92.50,31.99,70.72,72.01,69.91,281.77
    "Marseille, France",92.13,31.44,70.28,88.65,82.98,102.35
    "Frankfurt, Germany",92.02,34.75,71.41,65.96,80.00,115.22
    "Rio De Janeiro, Brazil",91.85,39.61,73.05,59.80,69.92,42.76
    "Nicosia, Cyprus",91.80,25.66,68.00,76.36,100.08,65.24
    "London, Canada",91.51,34.41,70.96,78.95,80.36,94.75
    "Lyon, France",91.14,34.41,70.73,82.18,92.00,83.91
    "San Diego, CA, United States",91.05,57.88,79.11,81.09,84.34,109.84
    "Los Angeles, CA, United States",90.87,63.97,81.19,73.58,70.46,122.29
    "San Juan, Puerto Rico",90.78,31.15,69.32,69.25,73.68,30.49
    "Cologne, Germany",90.73,29.58,68.72,74.64,82.78,114.48
    "Minneapolis, MN, United States",90.26,34.78,70.29,72.84,77.16,108.19
    "Tampa, FL, United States",89.97,32.39,69.25,64.01,62.16,90.85
    "Sliema, Malta",89.88,26.35,67.02,70.55,89.05,80.50
    "Berlin, Germany",89.09,35.80,69.91,75.36,71.71,133.19
    "Waterloo, Canada",89.02,29.71,67.67,87.06,67.85,101.67
    "Baltimore, MD, United States",88.99,54.12,76.44,101.95,78.13,118.83
    "Sevilla, Spain",88.96,18.79,63.70,64.69,81.85,73.40
    "Beirut, Lebanon",88.92,37.05,70.25,44.54,76.19,25.73
    "Barcelona, Spain",88.82,33.92,69.06,65.26,88.14,86.59
    "Philadelphia, PA, United States",88.36,58.85,77.74,83.39,78.50,98.96
    "Hong Kong, Hong Kong",88.13,106.22,94.64,85.25,79.06,91.77
    "Madrid, Spain",87.93,39.57,70.52,60.85,88.31,93.96
    "Graz, Austria",87.76,35.14,68.82,77.79,72.87,101.91
    "Dallas, TX, United States",87.62,38.51,69.95,67.93,72.85,113.86
    "Nuremberg, Germany",87.54,30.04,66.84,69.56,78.79,107.62
    "Jerusalem, Israel",87.54,35.97,68.98,70.06,77.82,68.10
    "Houston, TX, United States",86.97,39.70,69.96,74.96,68.89,137.32
    "Bilbao, Spain",86.62,37.70,69.01,60.47,93.40,94.48
    "Leicester, United Kingdom",86.15,27.80,65.15,82.09,67.83,103.13
    "Campinas, Brazil",85.98,23.70,63.56,54.81,66.22,75.34
    "Abu Dhabi, United Arab Emirates",85.84,80.82,84.04,72.59,92.39,134.56
    "Chicago, IL, United States",85.41,49.96,72.65,72.28,74.80,119.92
    "Saint Louis, MO, United States",84.91,23.22,62.71,84.75,64.89,140.12
    "Doha, Qatar",84.73,63.50,77.09,81.63,70.84,153.01
    "Florianopolis, Brazil",83.88,27.46,63.58,69.26,48.84,37.89
    "Pittsburgh, PA, United States",83.72,22.56,61.71,69.83,56.65,135.19
    "Montevideo, Uruguay",83.56,26.22,62.93,66.73,68.83,30.20
    "Dubai, United Arab Emirates",83.39,67.87,77.80,66.31,78.57,149.89
    "Bremen, Germany",82.85,22.77,61.23,70.65,65.58,88.11
    "Porto Alegre, Brazil",82.33,23.75,61.25,50.02,51.01,50.42
    "Cincinnati, OH, United States",82.18,29.11,63.08,72.09,61.97,103.60
    "Valencia, Spain",81.87,19.60,59.46,57.18,73.68,84.53
    "Patras, Greece",81.84,17.58,58.71,63.73,76.59,45.64
    "Curitiba, Brazil",81.10,22.60,60.05,57.17,45.90,48.94
    "Detroit, MI, United States",79.72,30.54,62.02,75.82,67.24,92.61
    "Denver, CO, United States",79.39,48.43,68.25,69.23,64.57,126.10
    "Austin, TX, United States",78.92,45.55,66.91,80.38,64.96,131.95
    "Portland, OR, United States",77.22,33.80,61.59,67.34,68.36,123.02
    "Londrina, Brazil",76.68,23.42,57.51,49.26,54.52,30.91
    "Lisbon, Portugal",76.33,29.49,59.47,58.65,62.75,62.83
    "Coimbra, Portugal",75.74,15.84,54.18,68.50,48.93,53.72
    "Split, Croatia",75.60,15.73,54.05,59.97,59.67,51.15
    "Salvador, Brazil",75.53,19.72,55.44,49.95,45.98,33.97
    "Casablanca, Morocco",75.49,36.51,61.46,74.49,54.77,15.00
    "Des Moines, IA, United States",75.45,29.18,58.80,68.64,54.51,104.04
    "Ljubljana, Slovenia",75.42,28.59,58.56,62.73,61.98,56.95
    "Maribor, Slovenia",75.17,28.11,58.23,55.86,56.85,52.19
    "Istanbul, Turkey",74.89,23.21,56.29,55.95,55.29,57.60
    "Astana, Kazakhstan",74.58,36.29,60.80,50.62,38.83,32.01
    "Johannesburg, South Africa",74.23,38.34,61.31,59.72,55.42,89.43
    "Balneario Camboriu, Brazil",74.02,18.98,54.21,48.29,52.10,25.21
    "Las Vegas, NV, United States",73.98,36.00,60.31,66.83,70.80,138.32
    "Manama, Bahrain",71.86,47.24,63.00,63.60,80.96,64.67
    "Zagreb, Croatia",71.49,21.12,53.36,57.85,60.09,54.58
    "Baku, Azerbaijan",71.24,41.03,60.37,53.20,68.17,26.38
    "San Jose, Costa Rica",71.17,25.95,54.90,78.60,52.85,38.83
    "Riga, Latvia",70.75,15.16,50.74,52.59,66.61,34.70
    "Seoul, South Korea",70.53,37.10,58.50,71.05,50.57,121.54
    "Fortaleza, Brazil",70.30,20.94,52.54,51.92,47.81,59.92
    "Bratislava, Slovakia",69.94,29.17,55.27,55.22,50.61,48.64
    "Santiago, Chile",69.89,20.62,52.16,53.54,58.73,44.98
    "Bogota, Colombia",69.50,21.86,52.36,64.95,41.91,26.24
    "Belo Horizonte, Brazil",69.08,21.66,52.01,46.67,60.15,55.99
    "Porto, Portugal",69.05,20.32,51.51,52.65,47.06,52.55
    "Santa Cruz De Tenerife, Spain",68.98,22.67,52.31,54.01,56.80,71.41
    "Novosibirsk, Russia",67.96,26.92,53.19,46.04,77.27,38.33
    "Saint Petersburg, Russia",67.58,31.20,54.49,49.09,73.42,40.98
    "Kaunas, Lithuania",67.44,16.93,49.26,52.66,65.17,31.72
    "Tallinn, Estonia",66.95,15.82,48.55,52.25,60.70,49.68
    "Perm, Russia",66.64,18.04,49.15,59.01,56.54,30.50
    "Budva, Montenegro",66.16,19.29,49.29,53.35,57.43,33.57
    "Amman, Jordan",65.95,14.51,47.44,60.72,51.73,37.85
    "Buenos Aires, Argentina",65.93,21.58,49.97,55.69,57.34,40.33
    "Taipei, Taiwan",65.66,23.16,50.36,82.38,33.92,75.97
    "Jakarta, Indonesia",64.11,28.25,51.21,69.89,39.67,25.60
    "Vilnius, Lithuania",63.84,15.58,46.47,52.08,40.18,34.61
    "Budapest, Hungary",63.41,14.03,45.64,46.92,46.75,37.44
    "Yerevan, Armenia",63.39,25.89,49.89,37.69,57.59,19.61
    "Yekaterinburg, Russia",63.22,21.27,48.12,51.07,72.49,45.59
    "Prague, Czech Republic",62.91,27.59,50.20,49.50,43.30,58.44
    "Debrecen, Hungary",62.47,12.52,44.49,51.07,48.93,43.11
    "Tehran, Iran",62.43,30.99,51.11,56.64,49.01,45.70
    "Brno, Czech Republic",62.35,20.49,47.28,48.46,43.52,68.36
    "Izmir, Turkey",62.12,14.54,45.00,51.08,42.39,46.25
    "Podgorica, Montenegro",61.72,17.49,45.80,44.70,62.72,42.01
    "Medellin, Colombia",61.43,15.28,44.82,45.49,42.01,18.61
    "Antalya, Turkey",61.00,10.27,42.74,44.91,42.74,39.81
    "Kosice, Slovakia",60.86,21.19,46.58,47.90,42.65,54.82
    "Ankara, Turkey",60.73,14.95,44.25,43.66,46.31,58.15
    "Kiev, Ukraine",59.87,31.70,49.73,41.78,61.94,27.80
    "Mexico City, Mexico",59.68,23.22,46.56,53.86,47.77,42.69
    "Lima, Peru",59.52,19.90,45.26,51.37,43.29,34.80
    "Osijek, Croatia",59.50,13.07,42.79,52.02,39.77,45.81
    "Sarajevo, Bosnia And Herzegovina",58.59,13.01,42.19,43.89,44.13,35.81
    "Cordoba, Argentina",57.81,19.29,43.94,52.43,51.76,42.17
    "Tula, Russia",57.72,15.46,42.51,48.96,44.94,51.89
    "Guadalajara, Mexico",57.67,10.28,40.61,47.54,44.78,69.13
    "Tbilisi, Georgia",57.32,15.77,42.37,40.89,52.50,28.71
    "Monterrey, Mexico",56.60,23.53,44.70,49.88,53.05,65.26
    "Belgrade, Serbia",56.21,17.13,42.15,39.73,45.39,30.66
    "Sofia, Bulgaria",55.62,16.24,41.44,45.83,37.78,36.55
    "Kuala Lumpur, Malaysia",55.55,17.71,41.93,55.09,32.09,77.70
    "Damascus, Syria",55.50,14.22,40.65,41.91,41.08,20.52
    "Johor Baharu, Malaysia",55.32,7.56,38.13,47.53,33.04,38.85
    "Wroclaw, Poland",54.85,19.34,42.07,42.05,42.99,58.27
    "Guayaquil, Ecuador",54.76,10.14,38.70,44.89,40.14,25.86
    "Beijing, China",54.38,28.71,45.14,61.66,37.49,36.42
    "Shanghai, China",54.20,29.86,45.44,51.64,42.60,47.04
    "Ulaanbaatar, Mongolia",53.86,24.01,43.12,57.94,37.81,22.57
    "Quito, Ecuador",53.69,11.66,38.56,47.42,39.26,23.85
    "Warsaw, Poland",53.64,22.81,42.55,42.07,45.74,64.83
    "Cairo, Egypt",53.40,17.34,40.42,45.39,48.99,20.64
    "Poznan, Poland",53.20,13.93,39.07,42.48,41.12,58.48
    "Novi Sad, Serbia",52.70,10.53,37.52,36.62,50.99,35.95
    "Iasi, Romania",52.66,13.28,38.48,43.50,34.27,31.37
    "Bangkok, Thailand",52.61,26.42,43.18,63.49,29.65,29.50
    "Guangzhou, China",52.44,17.25,39.77,70.42,30.76,25.60
    "Varna, Bulgaria",52.13,11.51,37.51,45.17,38.79,34.13
    "Katowice, Poland",51.87,19.49,40.21,43.11,41.41,57.95
    "Gdansk, Poland",51.64,20.34,40.37,40.86,43.63,64.99
    "Cluj-napoca, Romania",51.35,10.94,36.81,45.02,36.15,38.79
    "Bucharest, Romania",50.97,15.28,38.12,39.77,41.79,36.15
    "Banja Luka, Bosnia And Herzegovina",50.12,11.38,36.18,40.89,45.75,43.60
    "Brasov, Romania",50.05,9.04,35.29,39.77,35.41,33.48
    "Constanta, Romania",49.97,14.32,37.14,36.51,36.55,37.55
    "Timisoara, Romania",49.90,10.66,35.78,44.77,32.78,38.57
    "Plovdiv, Bulgaria",49.05,10.38,35.13,39.39,32.68,31.18
    "Quezon City, Philippines",48.83,13.86,36.24,50.04,28.11,28.25
    "Krakow, Poland",48.63,19.48,38.14,38.02,34.91,61.36
    "Szczecin, Poland",48.59,12.85,35.73,38.88,41.24,53.60
    "Hanoi, Vietnam",47.83,40.93,45.34,50.12,31.48,20.82
    "Chisinau, Moldova",47.81,12.35,35.05,38.59,40.27,27.56
    "Cebu, Philippines",47.61,8.16,33.41,47.26,21.60,35.28
    "Minsk, Belarus",47.40,15.41,35.89,35.32,53.72,22.16
    "Manila, Philippines",47.34,13.47,35.15,48.24,23.02,27.02
    "Makati, Philippines",47.17,28.45,40.43,40.15,34.74,20.06
    "Managua, Nicaragua",46.57,11.08,33.80,45.10,31.34,41.13
    "Skopje, Macedonia",46.51,11.74,33.99,35.19,35.78,34.39
    "Esfahan, Iran",45.56,19.60,36.22,47.53,34.47,49.90
    "Nis, Serbia",44.85,11.02,32.68,31.88,37.15,28.62
    "Lublin, Poland",42.40,13.98,32.17,34.50,32.67,60.48
    "Gurgaon, India",41.64,12.26,31.07,36.11,26.13,63.86
    "Ho Chi Minh City, Vietnam",41.09,29.65,36.97,48.66,24.37,45.15
    "Algiers, Algeria",40.60,13.23,30.75,42.50,31.32,23.70
    "Lahore, Pakistan",40.09,7.75,28.45,37.26,31.69,23.46
    "Delhi, India",37.93,12.08,28.63,35.23,26.57,62.71
    "Davao, Philippines",37.59,11.36,28.15,39.78,21.57,44.20
    "Mumbai, India",36.62,21.51,31.18,35.97,26.86,57.91
    "Bangalore, India",34.92,9.80,25.87,35.32,19.80,65.65
    "Indore, India",34.85,5.75,24.38,32.90,16.09,89.92
    "Chandigarh, India",33.95,6.98,24.24,31.63,21.68,52.41
    "Ahmedabad, India",33.12,6.47,23.53,29.26,24.03,50.67
    "Hyderabad, India",32.67,7.26,23.52,32.57,20.06,72.32
    "Madurai, India",32.05,5.65,22.55,29.99,13.11,25.02
    "Kolkata, India",31.97,8.81,23.63,35.06,21.02,49.26
    "Pune, India",31.90,9.34,23.78,32.74,20.62,55.50
    "Chennai, India",31.09,9.19,23.21,32.57,17.17,59.10
    "Kochi, India",31.07,5.28,21.79,34.43,16.52,44.85
    "Thiruvananthapuram, India",30.30,5.09,21.23,35.64,17.20,56.05
    "Ludhiana, India",30.20,7.89,22.17,31.27,21.88,55.74
    

    接下来就直接画图就行了

    冲冲冲,画出来的结果↓

    平行坐标图

    展开全文
  • 多维张量的几何理解

    2021-02-13 20:24:32
    运行结果: Tensor("Const_3:0", shape=(3, 4, 2), dtype=float16) 几何表示: 4、四维张量(图像仅用于理解,坐标系axis已经不再适用) #四维张量 const4 = tf.constant([ #第一个3行4列深度为2的三维张量 [[[1, 2],...

    张量理解

    Tensor是Tensorflow中最基础的数据结构,常常翻译为张量,可以理解为n维数组或矩阵,相关函数:

    constant(value, dtype=None, shape=None, name='Const', verify_shape=False)
    

    在这里插入图片描述

    0、零维张量

    import tensorflow as tf
    
    #零维张量
    const0 = tf.constant(1, tf.float16)
    print(const0)
    

    运行结果:

    Tensor("Const:0", shape=(), dtype=float16)
    

    1、一维张量

    #一维张量,长度为4
    const1 = tf.constant([1, 2, 3, 4], tf.float16)
    print(const1)
    

    运行结果:

    Tensor("Const_1:0", shape=(4,), dtype=float16)
    

    几何表示:
    在这里插入图片描述

    一维张量没有行和列的概念,只有长度的概念。上述的const1就是长度为4的一维张量,或者称为向量。
    上面的图仅为示意,代表一维张量只有axis=0这个方向,并不是指这是一个4行的向量。事实上,tensorflow在做一些运算时,反而经常把1行N列的二维张量简化成一个长度为N的一维向量。

    2、二维张量

    #二维张量,3行4列
    const2 = tf.constant([
                         [1, 2, 3, 4],
                         [5, 6, 7, 8],
                         [9, 10, 11, 12]
                         ], tf.float16)
    print(const2)
    

    运行结果:

    Tensor("Const_2:0", shape=(3, 4), dtype=float16)
    

    几何表示:
    在这里插入图片描述
    3、三维张量

    #三维张量,3行4列深度为2的张量
    const3 = tf.constant([
                         [[ 1,  2], [ 3,  4], [ 5,  6], [ 7,  8]],
                         [[11, 12], [13, 14], [15, 16], [17, 18]],
                         [[21, 22], [23, 24], [25, 26], [27, 28]]
                         ], tf.float16)
    print(const3)
    

    运行结果:

    Tensor("Const_3:0", shape=(3, 4, 2), dtype=float16)
    

    几何表示:
    在这里插入图片描述

    4、四维张量(图像仅用于理解,坐标系axis已经不再适用)

    #四维张量
    const4 = tf.constant([
                         #第一个3行4列深度为2的三维张量
                         [[[1,  2], [ 3,  4], [ 5,  6], [ 7,  8]],
                         [[11, 12], [13, 14], [15, 16], [17, 18]],
                         [[21, 22], [23, 24], [25, 26], [27, 28]]
                         ],
                         #第二个3行4列深度为2的三维张量
                         [[[1,  2], [ 3,  4], [ 5,  6], [ 7,  8]],
                         [[11, 12], [13, 14], [15, 16], [17, 18]],
                         [[21, 22], [23, 24], [25, 26], [27, 28]]]
                         ], tf.float16)
    print(const4)
    

    运行结果:

    Tensor("Const_4:0", shape=(2, 3, 4, 2), dtype=float16)
    

    几何表示:
    在这里插入图片描述

    如何判断一个张量的batch数、行、列、深度:

    const4 = tf.constant([
                         #第一个3行4列深度为2的三维张量
                         [[[1,  2], [ 3,  4], [ 5,  6], [ 7,  8]],
                         [[11, 12], [13, 14], [15, 16], [17, 18]],
                         [[21, 22], [23, 24], [25, 26], [27, 28]]
                         ],
                         #第二个3行4列深度为2的三维张量
                         [[[1,  2], [ 3,  4], [ 5,  6], [ 7,  8]],
                         [[11, 12], [13, 14], [15, 16], [17, 18]],
                         [[21, 22], [23, 24], [25, 26], [27, 28]]]
                         ], tf.float16)
    

    从左边开始数连续[的数量,最多有X个[说明是X维张量。上面的例子就是4维张量。
    以三维以上的张量为例:
    从左边开始数连续的[,最后一个[对应的]中一共两个元素,分别为1, 2,说明深度为2。
    接下来向左边数上一级[对应的]中一共有4个元素[1, 2], [ 3, 4], [ 5, 6], [ 7, 8],说明列为4。
    同理继续数上一级,得到3行,2个batch。

    小结:

    shape属性中的元素大于等于3时,可以用3维空间来理解。
    shape=(3, 4, 2`)时,表示3行4列深度为2的张量
    shape=(2, 3, 4, 2)时,表示有2个 3行4列深度为2的张量
    shape=(6, 2, 3, 4, 2)时,表示有6个四维张量,这个四维张量又可以表示为2个 3行4列深度为2的张量。

    shape中的属性分别与axis=0axis=1axis=2axis=3……对应,以此类推。当维度超过3时,上图几何中的坐标系表示就已经错误了。但是对于理解多维是有帮助的。


    作者:X_xxieRiemann 链接:https://www.jianshu.com/p/f34457c222c5 来源:简书 著作权归作者所有。商业转载请联系作者获得授权,非商业转载请注明出处。
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