光流简介

╰+攻爆jí腚メ 2022-07-15 13:12 221阅读 0赞

光流简介

参考维基百科：[光流简介][Link 1]

**光流**(Optical flow or optic flow)是关于视域中的物体[运动检测][Link 2]中的概念。用来描述相对于观察者的运动所造成的观测目标、表面或边缘的运动。光流法在[样型识别][Link 3]、[电脑视觉][Link 4]以及其他[影像处理][Link 5]领域中非常有用，可用于运动检测、物件切割、碰撞时间与物体膨胀的计算、运动补偿编码，或者通过物体表面与边缘进行立体的测量等等。

光流的测算光流法实际是通过检测图像像素点的强度随时间的变化进而推断出物体移动速度及方向的方法。

## 在 2D+*t* 维的情况下（3D 和更高维度亦然），假设位于 ![\{\\displaystyle (x,y,t)\}][displaystyle _x_y_t] 的[体素][Link 6]的亮度是 ![\{\\displaystyle I(x,y,t)\}][displaystyle I_x_y_t]。该体素在两个图像帧之间移动了 ![\\Delta x][Delta x]、![\\Delta y][Delta y]、![\\Delta t][Delta t]。于是可以得出一个亮度相同的结论： ##

\{\\displaystyle I(x,y,t)=I(x+\\Delta x,y+\\Delta y,t+\\Delta t)\} ![\{\\displaystyle I(x,y,t)=I(x+\\Delta x,y+\\Delta y,t+\\Delta t)\}][displaystyle I_x_y_t_I_x_Delta x_y_Delta y_t_Delta t]

假设该移动很小，那么可以根据[泰勒级数][Link 7]得出：

\{\\displaystyle I(x+\\Delta x,y+\\Delta y,t+\\Delta t)=I(x,y,t)+\{\\frac \{\\partial I\}\{\\partial x\}\}\\Delta x+\{\\frac \{\\partial I\}\{\\partial y\}\}\\Delta y+\{\\frac \{\\partial I\}\{\\partial t\}\}\\Delta t+\} ![\{\\displaystyle I(x+\\Delta x,y+\\Delta y,t+\\Delta t)=I(x,y,t)+\{\\frac \{\\partial I\}\{\\partial x\}\}\\Delta x+\{\\frac \{\\partial I\}\{\\partial y\}\}\\Delta y+\{\\frac \{\\partial I\}\{\\partial t\}\}\\Delta t+\}][displaystyle I_x_Delta x_y_Delta y_t_Delta t_I_x_y_t_frac _partial I_partial x_Delta x_frac _partial I_partial y_Delta y_frac _partial I_partial t_Delta t][H.O.T.][]

因此可以推出：

\{\\displaystyle \{\\frac \{\\partial I\}\{\\partial x\}\}\\Delta x+\{\\frac \{\\partial I\}\{\\partial y\}\}\\Delta y+\{\\frac \{\\partial I\}\{\\partial t\}\}\\Delta t=0\} ![\{\\displaystyle \{\\frac \{\\partial I\}\{\\partial x\}\}\\Delta x+\{\\frac \{\\partial I\}\{\\partial y\}\}\\Delta y+\{\\frac \{\\partial I\}\{\\partial t\}\}\\Delta t=0\}][displaystyle _frac _partial I_partial x_Delta x_frac _partial I_partial y_Delta y_frac _partial I_partial t_Delta t_0]

或

\{\\displaystyle \{\\frac \{\\partial I\}\{\\partial x\}\}\{\\frac \{\\Delta x\}\{\\Delta t\}\}+\{\\frac \{\\partial I\}\{\\partial y\}\}\{\\frac \{\\Delta y\}\{\\Delta t\}\}+\{\\frac \{\\partial I\}\{\\partial t\}\}\{\\frac \{\\Delta t\}\{\\Delta t\}\}=0\} ![\{\\displaystyle \{\\frac \{\\partial I\}\{\\partial x\}\}\{\\frac \{\\Delta x\}\{\\Delta t\}\}+\{\\frac \{\\partial I\}\{\\partial y\}\}\{\\frac \{\\Delta y\}\{\\Delta t\}\}+\{\\frac \{\\partial I\}\{\\partial t\}\}\{\\frac \{\\Delta t\}\{\\Delta t\}\}=0\}][displaystyle _frac _partial I_partial x_frac _Delta x_Delta t_frac _partial I_partial y_frac _Delta y_Delta t_frac _partial I_partial t_frac _Delta t_Delta t_0]

最终可得出结论：

\{\\displaystyle \{\\frac \{\\partial I\}\{\\partial x\}\}V\_\{x\}+\{\\frac \{\\partial I\}\{\\partial y\}\}V\_\{y\}+\{\\frac \{\\partial I\}\{\\partial t\}\}=0\} ![\{\\displaystyle \{\\frac \{\\partial I\}\{\\partial x\}\}V\_\{x\}+\{\\frac \{\\partial I\}\{\\partial y\}\}V\_\{y\}+\{\\frac \{\\partial I\}\{\\partial t\}\}=0\}][displaystyle _frac _partial I_partial x_V_x_frac _partial I_partial y_V_y_frac _partial I_partial t_0]

这里的 \{\\displaystyle V\_\{x\},V\_\{y\}\}![\{\\displaystyle V\_\{x\},V\_\{y\}\}][displaystyle V_x_V_y] 是 \{\\displaystyle x\}![x][] 和 \{\\displaystyle y\}![y][] 方向上的速率，或称为 \{\\displaystyle I(x,y,t)\}![\{\\displaystyle I(x,y,t)\}][displaystyle I_x_y_t] 的光流。而 \{\\displaystyle \{\\tfrac \{\\partial I\}\{\\partial x\}\}\}![\{\\displaystyle \{\\tfrac \{\\partial I\}\{\\partial x\}\}\}][displaystyle _tfrac _partial I_partial x], \{\\displaystyle \{\\tfrac \{\\partial I\}\{\\partial y\}\}\}![\{\\displaystyle \{\\tfrac \{\\partial I\}\{\\partial y\}\}\}][displaystyle _tfrac _partial I_partial y] 和 \{\\displaystyle \{\\tfrac \{\\partial I\}\{\\partial t\}\}\}![\{\\displaystyle \{\\tfrac \{\\partial I\}\{\\partial t\}\}\}][displaystyle _tfrac _partial I_partial t] 则是图像 \{\\displaystyle (x,y,t)\}![\{\\displaystyle (x,y,t)\}][displaystyle _x_y_t] 在对应方向上的[偏导数][Link 8]。\{\\displaystyle I\_\{x\}\}![\{\\displaystyle I\_\{x\}\}][displaystyle I_x]、\{\\displaystyle I\_\{y\}\}![\{\\displaystyle I\_\{y\}\}][displaystyle I_y] 和 \{\\displaystyle I\_\{t\}\}![\{\\displaystyle I\_\{t\}\}][displaystyle I_t] 的关系可用下式表述：

\{\\displaystyle I\_\{x\}V\_\{x\}+I\_\{y\}V\_\{y\}=-I\_\{t\}\} ![\{\\displaystyle I\_\{x\}V\_\{x\}+I\_\{y\}V\_\{y\}=-I\_\{t\}\}][displaystyle I_x_V_x_I_y_V_y_-I_t]

或

\{\\displaystyle \\nabla I^\{T\}\\cdot \{\\vec \{V\}\}=-I\_\{t\}\} ![\{\\displaystyle \\nabla I^\{T\}\\cdot \{\\vec \{V\}\}=-I\_\{t\}\}][displaystyle _nabla I_T_cdot _vec _V_-I_t]

求解光流的算法有很多，其中最经典的是Lucas-Kanade Method，opencv中实现的也是该算法。

算法简介：

**卢卡斯-金出方法**是一种广泛使用的[光流][Link 9]估计的差分方法，这个方法是由[Bruce D. Lucas][]和[Takeo Kanade][]发明的。它假设光流在像素点的邻域是一个常数，然后使用[最小二乘法][Link 10]对邻域中的所有像素点求解基本的光流方程。

通过结合几个邻近像素点的信息，卢卡斯-金出方法(简称为*L-K方法*)通常能够消除光流方程里的多义性。而且，与逐点计算的方法相比，L-K方法对图像噪声不敏感。不过，由于这是一种局部方法，所以在图像的均匀区域内部，L-K方法无法提供光流信息。

L-K方法假设两个相邻帧的图像内容位移很小，且位移在所研究点*p*的邻域内为大致为常数。所以，可以假设[光流方程][Link 11] 在以*p*点为中心的窗口内对所有的像素都成立。也就是说，局部图像流（速度）向量\{\\displaystyle (V\_\{x\},V\_\{y\})\}![\{\\displaystyle (V\_\{x\},V\_\{y\})\}][displaystyle _V_x_V_y]须满足：

\{\\displaystyle I\_\{x\}(q\_\{1\})V\_\{x\}+I\_\{y\}(q\_\{1\})V\_\{y\}=-I\_\{t\}(q\_\{1\})\} ![\{\\displaystyle I\_\{x\}(q\_\{1\})V\_\{x\}+I\_\{y\}(q\_\{1\})V\_\{y\}=-I\_\{t\}(q\_\{1\})\}][displaystyle I_x_q_1_V_x_I_y_q_1_V_y_-I_t_q_1]

\{\\displaystyle I\_\{x\}(q\_\{2\})V\_\{x\}+I\_\{y\}(q\_\{2\})V\_\{y\}=-I\_\{t\}(q\_\{2\})\} ![\{\\displaystyle I\_\{x\}(q\_\{2\})V\_\{x\}+I\_\{y\}(q\_\{2\})V\_\{y\}=-I\_\{t\}(q\_\{2\})\}][displaystyle I_x_q_2_V_x_I_y_q_2_V_y_-I_t_q_2]

\{\\displaystyle \\vdots \} ![\\vdots][vdots]

\{\\displaystyle I\_\{x\}(q\_\{n\})V\_\{x\}+I\_\{y\}(q\_\{n\})V\_\{y\}=-I\_\{t\}(q\_\{n\})\} ![\{\\displaystyle I\_\{x\}(q\_\{n\})V\_\{x\}+I\_\{y\}(q\_\{n\})V\_\{y\}=-I\_\{t\}(q\_\{n\})\}][displaystyle I_x_q_n_V_x_I_y_q_n_V_y_-I_t_q_n]

其中，\{\\displaystyle q\_\{1\},q\_\{2\},\\dots ,q\_\{n\}\}![\{\\displaystyle q\_\{1\},q\_\{2\},\\dots ,q\_\{n\}\}][displaystyle q_1_q_2_dots _q_n] 是窗口中的像素，\{\\displaystyle I\_\{x\}(q\_\{i\}),I\_\{y\}(q\_\{i\}),I\_\{t\}(q\_\{i\})\}![\{\\displaystyle I\_\{x\}(q\_\{i\}),I\_\{y\}(q\_\{i\}),I\_\{t\}(q\_\{i\})\}][displaystyle I_x_q_i_I_y_q_i_I_t_q_i]是图像在点\{\\displaystyle q\_\{i\}\}![q\_i][q_i]和当前时间对位置*x*，*y*和时间*t*的偏导。

这些等式可以写成[矩阵][Link 12]的形式\{\\displaystyle Av=b\}![\{\\displaystyle Av=b\}][displaystyle Av_b]，此处

\{\\displaystyle A=\{\\begin\{bmatrix\}I\_\{x\}(q\_\{1\})&I\_\{y\}(q\_\{1\})\\\\\[10pt\]I\_\{x\}(q\_\{2\})&I\_\{y\}(q\_\{2\})\\\\\[10pt\]\\vdots &\\vdots \\\\\[10pt\]I\_\{x\}(q\_\{n\})&I\_\{y\}(q\_\{n\})\\end\{bmatrix\}\},\\quad \\quad v=\{\\begin\{bmatrix\}V\_\{x\}\\\\\[10pt\]V\_\{y\}\\end\{bmatrix\}\},\\quad \{\\mbox\{and\}\}\\quad b=\{\\begin\{bmatrix\}-I\_\{t\}(q\_\{1\})\\\\\[10pt\]-I\_\{t\}(q\_\{2\})\\\\\[10pt\]\\vdots \\\\\[10pt\]-I\_\{t\}(q\_\{n\})\\end\{bmatrix\}\}\} ![\{\\displaystyle A=\{\\begin\{bmatrix\}I\_\{x\}(q\_\{1\})&I\_\{y\}(q\_\{1\})\\\\\[10pt\]I\_\{x\}(q\_\{2\})&I\_\{y\}(q\_\{2\})\\\\\[10pt\]\\vdots &\\vdots \\\\\[10pt\]I\_\{x\}(q\_\{n\})&I\_\{y\}(q\_\{n\})\\end\{bmatrix\}\},\\quad \\quad v=\{\\begin\{bmatrix\}V\_\{x\}\\\\\[10pt\]V\_\{y\}\\end\{bmatrix\}\},\\quad \{\\mbox\{and\}\}\\quad b=\{\\begin\{bmatrix\}-I\_\{t\}(q\_\{1\})\\\\\[10pt\]-I\_\{t\}(q\_\{2\})\\\\\[10pt\]\\vdots \\\\\[10pt\]-I\_\{t\}(q\_\{n\})\\end\{bmatrix\}\}\}][displaystyle A_begin_bmatrix_I_x_q_1_I_y_q_1_10pt_I_x_q_2_I_y_q_2_10pt_vdots _vdots _10pt_I_x_q_n_I_y_q_n_end_bmatrix_quad _quad v_begin_bmatrix_V_x_10pt_V_y_end_bmatrix_quad _mbox_and_quad b_begin_bmatrix_-I_t_q_1_10pt_-I_t_q_2_10pt_vdots _10pt_-I_t_q_n_end_bmatrix]

此方程组的等式个数多于未知数个数，所以它通常是[超定][Link 13]的。L-K方法使用[最小二乘法][Link 10]获得一个近似解，即计算一个2x2的方程组：

\{\\displaystyle A^\{T\}Av=A^\{T\}b\} ![\{\\displaystyle A^\{T\}Av=A^\{T\}b\}][displaystyle A_T_Av_A_T_b] 或

\{\\displaystyle \\mathrm \{v\} =(A^\{T\}A)^\{-1\}A^\{T\}b\} ![\{\\displaystyle \\mathrm \{v\} =(A^\{T\}A)^\{-1\}A^\{T\}b\}][displaystyle _mathrm _v_ _A_T_A_-1_A_T_b]

其中，\{\\displaystyle A^\{T\}\}![A^T][A_T]是矩阵\{\\displaystyle A\}![A][]的[转置][Link 14]。即计算：

\{\\displaystyle \{\\begin\{bmatrix\}V\_\{x\}\\\\\[10pt\]V\_\{y\}\\end\{bmatrix\}\}=\{\\begin\{bmatrix\}\\sum \_\{i\}I\_\{x\}(q\_\{i\})^\{2\}&\\sum \_\{i\}I\_\{x\}(q\_\{i\})I\_\{y\}(q\_\{i\})\\\\\[10pt\]\\sum \_\{i\}I\_\{y\}(q\_\{i\})I\_\{x\}(q\_\{i\})&\\sum \_\{i\}I\_\{y\}(q\_\{i\})^\{2\}\\end\{bmatrix\}\}^\{-1\}\{\\begin\{bmatrix\}-\\sum \_\{i\}I\_\{x\}(q\_\{i\})I\_\{t\}(q\_\{i\})\\\\\[10pt\]-\\sum \_\{i\}I\_\{y\}(q\_\{i\})I\_\{t\}(q\_\{i\})\\end\{bmatrix\}\}\} ![\{\\displaystyle \{\\begin\{bmatrix\}V\_\{x\}\\\\\[10pt\]V\_\{y\}\\end\{bmatrix\}\}=\{\\begin\{bmatrix\}\\sum \_\{i\}I\_\{x\}(q\_\{i\})^\{2\}&\\sum \_\{i\}I\_\{x\}(q\_\{i\})I\_\{y\}(q\_\{i\})\\\\\[10pt\]\\sum \_\{i\}I\_\{y\}(q\_\{i\})I\_\{x\}(q\_\{i\})&\\sum \_\{i\}I\_\{y\}(q\_\{i\})^\{2\}\\end\{bmatrix\}\}^\{-1\}\{\\begin\{bmatrix\}-\\sum \_\{i\}I\_\{x\}(q\_\{i\})I\_\{t\}(q\_\{i\})\\\\\[10pt\]-\\sum \_\{i\}I\_\{y\}(q\_\{i\})I\_\{t\}(q\_\{i\})\\end\{bmatrix\}\}\}][displaystyle _begin_bmatrix_V_x_10pt_V_y_end_bmatrix_begin_bmatrix_sum _i_I_x_q_i_2_sum _i_I_x_q_i_I_y_q_i_10pt_sum _i_I_y_q_i_I_x_q_i_sum _i_I_y_q_i_2_end_bmatrix_-1_begin_bmatrix_-_sum _i_I_x_q_i_I_t_q_i_10pt_-_sum _i_I_y_q_i_I_t_q_i_end_bmatrix]

对*i*=1 到 *n*求和。

矩阵\{\\displaystyle A^\{T\}A\}![\{\\displaystyle A^\{T\}A\}][displaystyle A_T_A]通常被称作图像在点*p*的 [结构张量][Link 15]

[Link 1]: https://zh.wikipedia.org/wiki/%E5%8D%A2%E5%8D%A1%E6%96%AF-%E5%8D%A1%E7%BA%B3%E5%BE%B7%E6%96%B9%E6%B3%95
[Link 2]: https://zh.wikipedia.org/w/index.php?title=%E8%BF%90%E5%8A%A8%E6%A3%80%E6%B5%8B&action=edit&redlink=1
[Link 3]: https://zh.wikipedia.org/w/index.php?title=%E6%A8%A3%E5%9E%8B%E8%AD%98%E5%88%AB&action=edit&redlink=1
[Link 4]: https://zh.wikipedia.org/wiki/%E9%9B%BB%E8%85%A6%E8%A6%96%E8%A6%BA
[Link 5]: https://zh.wikipedia.org/wiki/%E5%BD%B1%E5%83%8F%E8%99%95%E7%90%86
[displaystyle _x_y_t]: https://wikimedia.org/api/rest_v1/media/math/render/svg/762804ec2b0ecb79e3259d7ce5e8a79fa07ecbf0
[Link 6]: https://zh.wikipedia.org/wiki/%E4%BD%93%E7%B4%A0
[displaystyle I_x_y_t]: https://wikimedia.org/api/rest_v1/media/math/render/svg/943e368e17aa82dd32682aa0a2f44fc544c5268d
[Delta x]: https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70
[Delta y]: https://wikimedia.org/api/rest_v1/media/math/render/svg/7caae142d915be8ef4d8c423bf91d1f6ea10e8e0
[Delta t]: https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542
[displaystyle I_x_y_t_I_x_Delta x_y_Delta y_t_Delta t]: https://wikimedia.org/api/rest_v1/media/math/render/svg/ddbed7a05ee486795f8df25b6b63814670baed54
[Link 7]: https://zh.wikipedia.org/wiki/%E6%B3%B0%E5%8B%92%E7%BA%A7%E6%95%B0
[displaystyle I_x_Delta x_y_Delta y_t_Delta t_I_x_y_t_frac _partial I_partial x_Delta x_frac _partial I_partial y_Delta y_frac _partial I_partial t_Delta t]: https://wikimedia.org/api/rest_v1/media/math/render/svg/749d10ea42b4fb5f9955342d8e294bed10b7c8fd
[H.O.T.]: https://zh.wikipedia.org/wiki/%E6%91%84%E5%8A%A8%E7%90%86%E8%AE%BA
[displaystyle _frac _partial I_partial x_Delta x_frac _partial I_partial y_Delta y_frac _partial I_partial t_Delta t_0]: https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc4d94f9188324981bba2067476e684467e782d
[displaystyle _frac _partial I_partial x_frac _Delta x_Delta t_frac _partial I_partial y_frac _Delta y_Delta t_frac _partial I_partial t_frac _Delta t_Delta t_0]: https://wikimedia.org/api/rest_v1/media/math/render/svg/bdc292eb494006ef921415bfd9659debcf3b9a05
[displaystyle _frac _partial I_partial x_V_x_frac _partial I_partial y_V_y_frac _partial I_partial t_0]: https://wikimedia.org/api/rest_v1/media/math/render/svg/dcea639f5b077f0b62fb6b3e80cac328371a197b
[displaystyle V_x_V_y]: https://wikimedia.org/api/rest_v1/media/math/render/svg/21998e0b2d26cb691f3f2f19394bc5f5787f3a06
[x]: https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4
[y]: https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d
[displaystyle _tfrac _partial I_partial x]: https://wikimedia.org/api/rest_v1/media/math/render/svg/cc9a3135c6f690d3bac59164e7ffdd94a8e134ea
[displaystyle _tfrac _partial I_partial y]: https://wikimedia.org/api/rest_v1/media/math/render/svg/824a2c1da4733074d8ce7b1c70882799080d9463
[displaystyle _tfrac _partial I_partial t]: https://wikimedia.org/api/rest_v1/media/math/render/svg/64b817827ab67c30ca91a3d7c0049bde9ae00ad6
[Link 8]: https://zh.wikipedia.org/wiki/%E5%81%8F%E5%AF%BC%E6%95%B0
[displaystyle I_x]: https://wikimedia.org/api/rest_v1/media/math/render/svg/eed0dfcd8b1ce6eb2e6b7fbc426b895507d53004
[displaystyle I_y]: https://wikimedia.org/api/rest_v1/media/math/render/svg/6f3eedd0871fd35e1fd8992c53812cd60d1b07c2
[displaystyle I_t]: https://wikimedia.org/api/rest_v1/media/math/render/svg/fc386d951e8ffae76357542f14e160621e6668b1
[displaystyle I_x_V_x_I_y_V_y_-I_t]: https://wikimedia.org/api/rest_v1/media/math/render/svg/262f48c446773db9697b0c4e4b025c6688b755e0
[displaystyle _nabla I_T_cdot _vec _V_-I_t]: https://wikimedia.org/api/rest_v1/media/math/render/svg/f462cbf95a9714d36e0801425adea4ea840b3da0
[Link 9]: https://zh.wikipedia.org/w/index.php?title=%E5%85%89%E6%B5%81&action=edit&redlink=1
[Bruce D. Lucas]: https://zh.wikipedia.org/w/index.php?title=Bruce_D._Lucas&action=edit&redlink=1
[Takeo Kanade]: https://zh.wikipedia.org/w/index.php?title=Takeo_Kanade&action=edit&redlink=1
[Link 10]: https://zh.wikipedia.org/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98%E6%B3%95
[Link 11]: https://zh.wikipedia.org/wiki/%E5%85%89%E6%B5%81%E6%B3%95
[displaystyle _V_x_V_y]: https://wikimedia.org/api/rest_v1/media/math/render/svg/39247d2bdf97bed1059dc3a4253051d430527761
[displaystyle I_x_q_1_V_x_I_y_q_1_V_y_-I_t_q_1]: https://wikimedia.org/api/rest_v1/media/math/render/svg/37bfc1c55493b81f33c3757adb53b93c57e2cd36
[displaystyle I_x_q_2_V_x_I_y_q_2_V_y_-I_t_q_2]: https://wikimedia.org/api/rest_v1/media/math/render/svg/338caf9ca4d7fe5596712d3f526bfa6e16d8a115
[vdots]: https://wikimedia.org/api/rest_v1/media/math/render/svg/f8039d9feb6596ae092e5305108722975060c083
[displaystyle I_x_q_n_V_x_I_y_q_n_V_y_-I_t_q_n]: https://wikimedia.org/api/rest_v1/media/math/render/svg/8df7001467f0acae54e3525c9c6785d9f73ed363
[displaystyle q_1_q_2_dots _q_n]: https://wikimedia.org/api/rest_v1/media/math/render/svg/02258d131c5b0a836eb89b5aba62ecc3268da38a
[displaystyle I_x_q_i_I_y_q_i_I_t_q_i]: https://wikimedia.org/api/rest_v1/media/math/render/svg/733ae911cc0d6cef5e6389a5c8a181b61462777e
[q_i]: https://wikimedia.org/api/rest_v1/media/math/render/svg/2752dcbff884354069fe332b8e51eb0a70a531b6
[Link 12]: https://zh.wikipedia.org/wiki/%E7%9F%A9%E9%98%B5
[displaystyle Av_b]: https://wikimedia.org/api/rest_v1/media/math/render/svg/e1616137729359b315e16bb3a52f56c4101b6285
[displaystyle A_begin_bmatrix_I_x_q_1_I_y_q_1_10pt_I_x_q_2_I_y_q_2_10pt_vdots _vdots _10pt_I_x_q_n_I_y_q_n_end_bmatrix_quad _quad v_begin_bmatrix_V_x_10pt_V_y_end_bmatrix_quad _mbox_and_quad b_begin_bmatrix_-I_t_q_1_10pt_-I_t_q_2_10pt_vdots _10pt_-I_t_q_n_end_bmatrix]: https://wikimedia.org/api/rest_v1/media/math/render/svg/f289ba8a702ce3fa3e32457b02f6a69a28b87a79
[Link 13]: https://zh.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84
[displaystyle A_T_Av_A_T_b]: https://wikimedia.org/api/rest_v1/media/math/render/svg/95fb888e66d0ad314a9e1cb54c20fe443dc26181
[displaystyle _mathrm _v_ _A_T_A_-1_A_T_b]: https://wikimedia.org/api/rest_v1/media/math/render/svg/c0243b8dc5b55074c9a3a6c623970a60fbc13f86
[A_T]: https://wikimedia.org/api/rest_v1/media/math/render/svg/d978c0c4f22282a313e4107f97b9eda56315b258
[A]: https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3
[Link 14]: https://zh.wikipedia.org/wiki/%E8%BD%AC%E7%BD%AE%E7%9F%A9%E9%98%B5
[displaystyle _begin_bmatrix_V_x_10pt_V_y_end_bmatrix_begin_bmatrix_sum _i_I_x_q_i_2_sum _i_I_x_q_i_I_y_q_i_10pt_sum _i_I_y_q_i_I_x_q_i_sum _i_I_y_q_i_2_end_bmatrix_-1_begin_bmatrix_-_sum _i_I_x_q_i_I_t_q_i_10pt_-_sum _i_I_y_q_i_I_t_q_i_end_bmatrix]: https://wikimedia.org/api/rest_v1/media/math/render/svg/7d94a7e72c2c1f8bda30c925e57d543cb4d48145
[displaystyle A_T_A]: https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc54896247ebb4d60e721ac74dcf41614e18133
[Link 15]: https://zh.wikipedia.org/w/index.php?title=%E7%BB%93%E6%9E%84%E5%BC%A0%E9%87%8F&action=edit&redlink=1