LSD-SLAM 权重更新公式推导

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LSD-SLAM权重更新推导

参考帧：逆深度为 $d$，相机内参为 $K$，像素坐标为 $(u,v)$，相机坐标系下坐标为 $(x,y,z)$。参考帧相机坐标系下对应的坐标为：
$$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\frac{1}{d}{K}^{-1}\left[\begin{array}{c}u\\ v\\ 1\end{array}\right]\text{\hspace{1em}(2)}$$
当前帧：像素坐标为 $(u^{\prime },v^{\prime })$，相机坐标系下坐标为 $(x^{\prime },y^{\prime },z^{\prime })$。
$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right]=R\left[\begin{array}{c}x\\ y\\ z\end{array}\right]+t=R\frac{1}{d}{K}^{-1}\left[\begin{array}{c}u\\ v\\ 1\end{array}\right]+t\text{\hspace{1em}(3)}$$

R1dK−1[uv1]=1d[a1a2a3a4a5a6a7a8a9][1fx0−cxfx01fy−cyfy001]=1d[x′′y′′z′′](4) R\frac{1}{d}K^{-1}\left[\begin{matrix}u\\v\\1\end{matrix}\right]= \frac{1}{d} \left[\begin{matrix} a_1&a_2&a_3\\ a_4&a_5&a_6\\ a_7&a_8&a_9 \end{matrix}\right] \left[\begin{matrix} \frac{1}{f_x}&0&-\frac{c_x}{f_x}\\ 0&\frac{1}{f_y}&-\frac{c_y}{f_y}\\ 0&0&1 \end{matrix}\right]= \frac{1}{d}\left[\begin{matrix}x''\\y''\\z''\end{matrix}\right]\tag4 Rd1​K−1⎣⎡​uv1​⎦⎤​=d1​⎣⎡​a1​a4​a7​​a2​a5​a8​​a3​a6​a9​​⎦⎤​⎣⎡​fx​1​00​0fy​1​0​−fx​cx​​−fy​cy​​1​⎦⎤​=d1​⎣⎡​x′′y′′z′′</