卢卡斯-卡纳德方法

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 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${\displaystyle (V_{x},V_{y})}$须满足：

${\displaystyle I_{x}(q_{1})V_{x}+I_{y}(q_{1})V_{y}=-I_{t}(q_{1})}$

${\displaystyle I_{x}(q_{2})V_{x}+I_{y}(q_{2})V_{y}=-I_{t}(q_{2})}$

${\displaystyle \vdots }$

${\displaystyle I_{x}(q_{n})V_{x}+I_{y}(q_{n})V_{y}=-I_{t}(q_{n})}$

其中，${\displaystyle q_{1},q_{2},\dots ,q_{n}}$ 是窗口中的像素，${\displaystyle I_{x}(q_{i}),I_{y}(q_{i}),I_{t}(q_{i})}$是图像在点${\displaystyle q_{i}}$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的形式${\displaystyle Av=b}$，此处

${\displaystyle A={\begin{bmatrix}I_{x}(q_{1})&I_{y}(q_{1})\\[10pt]I_{x}(q_{2})&I_{y}(q_{2})\\[10pt]\vdots &\vdots \\[10pt]I_{x}(q_{n})&I_{y}(q_{n})\end{bmatrix}},\quad \quad v={\begin{bmatrix}V_{x}\\[10pt]V_{y}\end{bmatrix}},\quad {\mbox{and}}\quad b={\begin{bmatrix}-I_{t}(q_{1})\\[10pt]-I_{t}(q_{2})\\[10pt]\vdots \\[10pt]-I_{t}(q_{n})\end{bmatrix}}}$

此方程组的等式个数多于未知数个数，所以它通常是over-determined的。L-K方法使用最小平方法获得一个近似解，即计算一个2x2的方程组：

${\displaystyle A^{T}Av=A^{T}b}$ 或

${\displaystyle \mathrm {v} =(A^{T}A)^{-1}A^{T}b}$

其中，${\displaystyle A^{T}}$是矩阵${\displaystyle A}$的转置。即计算：

${\displaystyle {\begin{bmatrix}V_{x}\\[10pt]V_{y}\end{bmatrix}}={\begin{bmatrix}\sum _{i}I_{x}(q_{i})^{2}&\sum _{i}I_{x}(q_{i})I_{y}(q_{i})\\[10pt]\sum _{i}I_{y}(q_{i})I_{x}(q_{i})&\sum _{i}I_{y}(q_{i})^{2}\end{bmatrix}}^{-1}{\begin{bmatrix}-\sum _{i}I_{x}(q_{i})I_{t}(q_{i})\\[10pt]-\sum _{i}I_{y}(q_{i})I_{t}(q_{i})\end{bmatrix}}}$

对i=1 到 n求和。

矩阵${\displaystyle A^{T}A}$通常被称作图像在点p的 结构张量。

1. B. D. Lucas and T. Kanade (1981), An iterative image registration technique with an application to stereo vision. 页面存档备份，存于 Proceedings of Imaging Understanding Workshop, pages 121--130
2. Bruce D. Lucas (1984) Generalized Image Matching by the Method of Differences 页面存档备份，存于 (doctoral dissertation)

• The image stabilizer plugin for ImageJ based on the Lucas–Kanade method
• Mathworks Lucas-Kanade Matlab implementation of inverse and normal affine Lucas-Kanade
• FolkiGPU : GPU implementation of an iterative Lucas-Kanade based optical flow
• Lucas-Kanade for the iPhone页面存档备份，存于 by Success Labs. A modified and enhanced port of the OpenCV lkdemo sample application to the iPhone.
• KLT: An Implementation of the Kanade–Lucas–Tomasi Feature Tracker
• Takeo Kanade