直接法光度误差导数推导
DSO 代码中初始化的部分。CoarseInitializer 将第一帧作为 ref frame，第二帧作为 new frame。ref frame 的 idepth (inverse depth) 一开始的时候都设置为1，随后在确定 new frame 相对 ref frame 之间相对位姿、光度变化过程中设定为正确值。
new frame 相对 ref frame 之间存在 8 个参数需要确定，前 6 个参数是 se(3)，后 2 个参数是光度仿射变换的参数。
本文参考高翔博士的《DSO详解》：https://zhuanlan.zhihu.com/p/29177540。
此处以及以后 DSO 相关的文章对导数的定义如下：
\[\begin{align} \frac{\partial \mathbf{f}}{\partial \mathbf{x}} = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \dots & \frac{\partial f_1}{\partial x_m} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \dots & \frac{\partial f_2}{\partial x_m} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_p}{\partial x_1} & \frac{\partial f_p}{\partial x_2} & \dots & \frac{\partial f_p}{\partial x_m} \end{bmatrix} \end{align}
\]
1 光度仿射变换
光度仿射变换是将两帧之间辐射值进行对应。对应的参数有两个 \(\begin{bmatrix} a, b \end{bmatrix}\)。
将影像 1 的辐射值变换到影像 2 中：
\[\begin{align} I_2 = a_{21} I_1 + b_{21} \end{align}
\]
\(I_1\) 表示影像 1 中的辐射值，\(I_2\) 表示影像 2 中的辐射值。
系统中其他地方求光度误差的方法，是按照 AffLight::fromToVecExposure 和计算光度误差的语句计算的。所以实际上相对光度仿射变换参数应该是如下方法计算（\(\Delta t_1, \Delta t_2\) 是曝光时间）：
\[\begin{align} a_{21} &= {e^{a_2} \Delta t_2 \over e^{a_1} \Delta t_1} \\
b_{21} &= b_2 - a_{21} b_1 \end{align}\]
但是需要注意到现在是初始化，设定了影像 1 的光度变换参数为 \(\begin{bmatrix} 0, 0 \end{bmatrix}\)。所以相对光度仿射变换参数可以退化成下面的形式：
\[\begin{align} a_{21} &= {e^{a_2} \Delta t_2 \over \Delta t_1} \\
b_{21} &= b_2 \end{align}\]
2 光度误差
光度误差可以用以下公式计算：
\[\begin{align} r = w_h (I_2[x_2] - (a_{21}I_1[x_1] + b_{21})) \end{align}
\]
其中 \(w_h\) 是 Huber 权重，\(I_1, I_2\) 分别是影像1、2，\(x_1, x_2\) 分别是空间中一点 \(X\) 在影像上的像素坐标。
其中 \(x_2\) 可以写作
\[\begin{align} x_2 = f(x_1, \xi_{21}, \rho_1) \end{align}
\]
即 \(x_2\) 是由 \(x_1\) 投影而来，投影的过程中需要两帧之间的相对位姿 \(\xi_{21}\) 和 \(x_1\) 在 1 中的逆深度 \(\rho_1\)。
3 导数
3.1 光度仿射变换导数 \({\partial r_{21} \over \partial a_{21}}, {\partial r_{21} \over \partial b_{21}}\)
从最简单的光度误差参数开始求导数：
\[\begin{align}
{\partial r_{21} \over \partial a_{21}} &= - w_h I_1[x_1] \\
{\partial r_{21} \over \partial b_{21}} &= -w_h
\end{align}\]
我认为 DSO 代码在这里写错了，按照 DSO 代码，应该如此：
\[\begin{align} {\partial r_{21} \over \partial a_{21}} = - w_h a_{21}I_1[x_1] \end{align}
\]
我怀疑是不是我看错了，其实 DSO 优化 \(a_2\)，这样就有
\[\begin{align} {\partial r_{21} \over \partial a_{2}} = - w_h a_{2}I_1[x_1] \end{align}
\]
但是实际并不是这样的，代码是正确的。DSO 优化的参数就是 \(a_{21}\)，从这里将曝光时间加入到初始值中，就可以看出优化的参数就是相对光度变换参数。
3.2 相对位姿导数 \({\partial r_{21} \over \partial \xi_{21}}\) 和逆深度导数 \({\partial r_{21} \over \partial \rho_{1}}\)
随后求光度误差对相对位姿和逆深度的导数：
\[{\partial r_{21} \over \partial \xi_{21}}, {\partial r_{21} \over \partial \rho_{1}}
\]
通过链式法则：
\[\begin{align}
{\partial r_{21} \over \partial \xi_{21}} &= {\partial r_{21} \over \partial x_{2}} {\partial x_{2} \over \partial \xi_{21}} \\
{\partial r_{21} \over \partial \rho_{1}} &=
{\partial r_{21} \over \partial x_{2}} {\partial x_{2} \over \partial \rho_{1}}
\end{align}\]
\({\partial r_{21} \over \partial x_{2}}\) 可以方便求得：
\[\begin{align} {\partial r_{21} \over \partial x_{2}} = w_h {\partial I_2[x_2] \over \partial x_{2}} = w_h \begin{bmatrix} g_x \\ g_y\end{bmatrix} \end{align}
\]
\(g_x, g_y\) 是影像 \(I_2\) 在 \(x_2\) 处的梯度。
现在，我们需要求的是 \({\partial x_{2} \over \partial \xi_{21}}\) 和 \({\partial x_{2} \over \partial \rho_1}\)。
分别列两张影像的投影方程如下：
\[\begin{align} \begin{cases}
\rho_1^{-1} x_1 &= K X_1 \\
\rho_2^{-1} x_2 &= K (R_{21} X_1 + t_{21})
\end{cases} \end{align}\]
整理一下
\[\begin{align} X_1 &= \rho_1^{-1} K^{-1} x_1 \\
x_2 &= K \rho_2 (R_{21} \rho_1^{-1} K^{-1} x_1 + t_{21}) \notag \\
&= K x^{\prime}_2 \end{align}\]
上式中的 \(x^{\prime}_2\) 就是归一化平面坐标，可以写作
\[x^{\prime}_2 = \begin{bmatrix}u^{\prime}_2, v^{\prime}_2, 1 \end{bmatrix} ^T
\]
通过链式法则，整理一下，我们需要计算的\({\partial x_{2} \over \partial \xi_{21}}\) 和 \({\partial x_{2} \over \partial \rho_1}\)：
\[\begin{align}{\partial x_{2} \over \partial \xi_{21}} &= {\partial x_{2}\over \partial x^{\prime}_{2}} {\partial x^{\prime}_{2} \over \partial \xi_{21}} \\
{\partial x_{2} \over \partial \rho_1} &= {\partial x_{2}\over \partial x^{\prime}_{2}} {\partial x^{\prime}_{2} \over \partial \rho_1} \end{align}\]
先计算两者的共同部分 \({\partial x_{2}\over \partial x^{\prime}_{2}}\)。
\[\begin{align} x_2 &= K x^{\prime}_2 \\
\begin{bmatrix} u_2 \\ v_2 \\ 1 \end{bmatrix}&= \begin{bmatrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1\end{bmatrix} \begin{bmatrix} u^{\prime}_2 \\ v^{\prime}_2 \\ 1 \end{bmatrix} \end{align}\]
于是有
\[\begin{align}{\partial x_{2}\over \partial x^{\prime}_{2}} &= \begin{bmatrix} {\partial u_{2}\over \partial u^{\prime}_{2}} & {\partial u_{2}\over \partial v^{\prime}_{2}} & {\partial u_{2}\over \partial 1} \\ {\partial v_{2}\over \partial u^{\prime}_{2}} & {\partial v_{2}\over \partial v^{\prime}_{2}} & {\partial v_{2}\over \partial 1} \\ {\partial 1\over \partial u^{\prime}_{2}} & {\partial 1\over \partial v^{\prime}_{2}} & {\partial 1\over \partial 1}\end{bmatrix}\notag \\
&= \begin{bmatrix} f_x & 0 & 0 \\ 0 & f_y & 0 \\ 0 & 0 & 0 \end{bmatrix} \end{align}\]
随后分别计算 \({\partial x^{\prime}_{2} \over \partial \xi_{21}}\) 和 \({\partial x^{\prime}_{2} \over \partial \rho_1}\) 。
3.2.1 推导 \({\partial x^{\prime}_{2} \over \partial \rho_1}\)
\[\begin{align} {\partial x^{\prime}_{2} \over \partial \rho_1} = \begin{bmatrix} {\partial u^{\prime}_{2} \over \partial \rho_1} \\ {\partial v^{\prime}_{2} \over \partial \rho_1} \\ 0\end{bmatrix} \end{align}
\]
\[\begin{align} x^{\prime}_2 = \rho_2 (R_{21} \rho_1^{-1} K^{-1} x_1 + t_{21}) \end{align}
\]
将 \(R_{21} K^{-1} x_1\) 看做整体，矩阵 \(A\)，矩阵 \(A\) 有三行。
\[\begin{align} A = \begin{bmatrix} a_1^T \\ a_2^T \\ a_3^T \end{bmatrix} \end{align}
\]
\(x^{\prime}_2\) 就可以写成如下形式
\[\begin{align} \begin{bmatrix} u^{\prime}_2 \\ v^{\prime}_2 \\ 1 \end{bmatrix} = \rho_2 \begin{bmatrix} \rho_1^{-1} a_1^Tx_1 + t_{21}^x \\ \rho_1^{-1} a_2^Tx_1 + t_{21}^y \\ \rho_1^{-1} a_3^Tx_1 + t_{21}^z \end{bmatrix} \end{align}
\]
可得 2 中的逆深度
\[\begin{align} \rho_2 = (\rho_1^{-1} a_3^Tx_1 + t_{21}^z)^{-1} \end{align}
\]
代入
\[\begin{align} u^{\prime}_2 &= \rho_2(\rho_1^{-1}a_1^Tx_1 + t_{21}^x) \notag \\
&= {{\rho_1^{-1}a_1^Tx_1 + t_{21}^x} \over {\rho_1^{-1} a_3^Tx_1 + t_{21}^z}} \notag \\
&= {{a_1^Tx_1 + \rho_1 t_{21}^x} \over {a_3^Tx_1 + \rho_1 t_{21}^z}} \end{align}\]
\[\begin{align} {\partial u^{\prime}_{2} \over \partial \rho_1} &= t_{21}^x {1 \over {a_3^Tx_1 + \rho_1 t_{21}^z}} \notag \\ &\ \ + (a_1^Tx_1 + \rho_1 t_{21}^x) {1 \over {(a_3^Tx_1 + \rho_1 t_{21}^z)^2}} (-1) t_{21}^z \notag \\
&= \rho_1^{-1} {1 \over {\rho_1^{-1}a_3^Tx_1 + t_{21}^z}}(t_{21}^x - {{a_1^Tx_1 + \rho_1 t_{21}^x} \over {a_3^Tx_1 + \rho_1 t_{21}^z}}t_{21}^z)\notag \\
&= \rho_1^{-1}\rho_2(t_{21}^x - u^{\prime}_2t_{21}^z) \end{align}\]
同理
\[\begin{align} {\partial v^{\prime}_{2} \over \partial \rho_1} = \rho_1^{-1}\rho_2(t_{21}^y - v^{\prime}_2t_{21}^z) \end{align}
\]
结论
\[\begin{align} {\partial x^{\prime}_{2} \over \partial \rho_1} = \begin{bmatrix} \rho_1^{-1}\rho_2(t_{21}^x - u^{\prime}_2t_{21}^z) \\ \rho_1^{-1}\rho_2(t_{21}^y - v^{\prime}_2t_{21}^z) \\ 0\end{bmatrix} \end{align}
\]
3.2.2 推导 \({\partial x^{\prime}_{2} \over \partial \xi_{21}}\)
\[\begin{align} x^{\prime}_2 = \rho_2(R_{21}X_1 + t_{21}) = \rho_2 X_2 \end{align}
\]
参考高翔博士的《SLAM十四讲》 4.3.5 的 \(\partial (Tp) \over \partial \xi\)。
可知
\[\begin{align}{\partial X_2 \over \partial \xi_{21}} &= \begin{bmatrix} {\partial X_2 \over \partial \xi_{21}} \\ {\partial Y_2 \over \partial \xi_{21}} \\ {\partial Z_2 \over \partial \xi_{21}}\end{bmatrix} \notag \\
&= \begin{bmatrix} 1 & 0 & 0 & 0 & Z_2 & - Y_2 \\ 0 & 1 & 0 & -Z_2 & 0 & X_2 \\ 0 & 0 & 1 & Y_2 & -X_2 & 0\end{bmatrix} \end{align}\]
\[\begin{align} {\partial x^{\prime}_{2} \over \partial \xi_{21}} = \begin{bmatrix} {\partial \rho_2 \over \partial \xi_{21}} X_2 \\ {\partial \rho_2 \over \partial \xi_{21}} Y_2 \\ {\partial \rho_2 \over \partial \xi_{21}} Z_2 \end{bmatrix} + \rho_2 {\partial X_2 \over \partial \xi_{21}} \end{align}
\]
\[\begin{align} {\partial \rho_2 \over \partial \xi_{21}} = {\partial {1 \over Z_2}\over \partial \xi_{21}} = {-1 \over Z_2^2} \begin{bmatrix} 0 & 0 & 1 & Y_2 & -X_2 & 0\end{bmatrix} \end{align}
\]
\[\begin{align} {\partial x^{\prime}_2 \over \partial \xi_{21}} &= \begin{bmatrix} 0 & 0 & -{X_2 \over Z_2^2} & - {X_2 Y_2\over Z_2^2} & {X_2^2 \over Z_2^2} & 0 \\ 0 & 0 & -{Y_2 \over Z_2^2} & -{Y_2^2 \over Z_2^2} & {X_2 Y_2 \over Z_2^2} & 0 \\ 0 & 0 & -{Z_2 \over Z_2^2} & -{Y_2 Z_2\over Z_2^2} & {X_2 Z_2 \over Z_2^2} & 0\end{bmatrix} \notag \\
& \ \ + \begin{bmatrix} {1 \over Z_2} & 0 & 0 & 0 & 1 & -{Y_2 \over Z_2} \\ 0 & {1 \over Z_2} & 0 & -1 & 0 & {X_2 \over Z_2} \\ 0 & 0 & {1 \over Z_2} & {Y_2 \over Z_2} & -{X_2 \over Z_2} & 0\end{bmatrix} \notag \\
&= \begin{bmatrix} {1 \over Z_2} & 0 & -{X_2 \over Z_2^2} & -{X_2 Y_2 \over Z_2^2} & 1 + {X_2^2 \over Z_2^2} & -{Y_2 \over Z_2} \\ 0 & {1 \over Z_2} & -{Y_2 \over Z_2^2} & -(1 + {Y_2^2 \over Z_2^2}) & {X_2 Y_2 \over Z_2^2} & {X_2 \over Z_2} \\ 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix} \notag \\
&= \begin{bmatrix} \rho_2 & 0 & -\rho_2 u^{\prime}_2 & -u^{\prime}_2 v^{\prime}_2 & 1 + u^{\prime 2}_2 & -v^{\prime}_2 \\ 0 & \rho_2 & -\rho_2 v^{\prime}_2 & -(1 + v^{\prime 2}_2) & u^{\prime}_2 v^{\prime}_2 & u^{\prime}_2 \\ 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix} \end{align}\]
最后一步是将空间坐标系坐标 \(X\) 用归一化坐标系坐标 \(u^{\prime}_2\) 和逆深度 \(\rho_2\) 替换。
3.2.3 替换求得 \({\partial x_{2} \over \partial \xi_{21}}\) 和 \({\partial x_{2} \over \partial \rho_1}\)
\[\begin{align} {\partial x_{2} \over \partial \rho_1} &= {\partial x_{2}\over \partial x^{\prime}_{2}} {\partial x^{\prime}_{2} \over \partial \rho_1} \notag \\
&= \begin{bmatrix} f_x & 0 & 0 \\ 0 & f_y & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \rho_1^{-1}\rho_2(t_{21}^x - u^{\prime}_2t_{21}^z) \\ \rho_1^{-1}\rho_2(t_{21}^y - v^{\prime}_2t_{21}^z) \\ 0\end{bmatrix} \notag \\
&= \begin{bmatrix} f_x \rho_1^{-1}\rho_2(t_{21}^x - u^{\prime}_2t_{21}^z) \\ f_y \rho_1^{-1}\rho_2(t_{21}^y - v^{\prime}_2t_{21}^z) \\ 0\end{bmatrix} \end{align}\]
\[\begin{align} {\partial x_{2} \over \partial \xi_{21}} &= {\partial x_{2}\over \partial x^{\prime}_{2}} {\partial x^{\prime}_{2} \over \partial \xi_{21}} \notag \\
&= \begin{bmatrix} f_x & 0 & 0 \\ 0 & f_y & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \rho_2 & 0 & -\rho_2 u^{\prime}_2 & -u^{\prime}_2 v^{\prime}_2 & 1 + u^{\prime 2}_2 & -v^{\prime}_2 \\ 0 & \rho_2 & -\rho_2 v^{\prime}_2 & -(1 + v^{\prime 2}_2) & u^{\prime}_2 v^{\prime}_2 & u^{\prime}_2 \\ 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix} \notag \\
&= \begin{bmatrix} f_x \rho_2 & 0 & -f_x \rho_2 u^{\prime}_2 & -f_xu^{\prime}_2 v^{\prime}_2 & f_x(1 + u^{\prime 2}_2) & -f_x v^{\prime}_2 \\ 0 & f_y\rho_2 & -f_y \rho_2 v^{\prime}_2 & -f_y(1 + v^{\prime 2}_2) & f_y u^{\prime}_2 v^{\prime}_2 & f_y u^{\prime}_2 \\ 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix} \end{align}\]
3.2.4 最终得到 \({\partial r_{21} \over \partial \xi_{21}}\) 和 \({\partial r_{21} \over \partial \rho_{1}}\)
\[\begin{align}
{\partial r_{21} \over \partial \rho_1} &= {\partial r_{21} \over \partial x_{2}} {\partial x_{2} \over \partial \rho_1} \notag \\
&= w_h {\partial I_2[x_2] \over \partial x_2} {\partial x_2 \over \partial \rho_1} \notag \\
&= w_h \begin{bmatrix} g_x & g_y & 0\end{bmatrix} \begin{bmatrix} f_x \rho_1^{-1}\rho_2(t_{21}^x - u^{\prime}_2t_{21}^z) \\ f_y \rho_1^{-1}\rho_2(t_{21}^y - v^{\prime}_2t_{21}^z) \\ 0\end{bmatrix} \notag \\
&= w_h(g_x f_x \rho_1^{-1}\rho_2(t_{21}^x - u^{\prime}_2t_{21}^z) + g_y f_y \rho_1^{-1}\rho_2(t_{21}^y - v^{\prime}_2t_{21}^z))
\end{align}\]
\[\begin{align}
{\partial r_{21} \over \partial \xi_{21}} &= {\partial r_{21} \over \partial x_{2}} {\partial x_{2} \over \partial \xi_{21}} \notag \\
&= w_h {\partial I_2[x_2] \over \partial x_2} {\partial x_2 \over \partial \xi_{21}} \notag \\
&= w_h \begin{bmatrix} g_x & g_y & 0\end{bmatrix} \notag \\
& \ \ \begin{bmatrix} f_x \rho_2 & 0 & -f_x \rho_2 u^{\prime}_2 & -f_x u^{\prime}_2 v^{\prime}_2 & f_x(1 + u^{\prime 2}_2) & -f_x v^{\prime}_2 \\ 0 & f_y \rho_2 & -f_y \rho_2 v^{\prime}_2 & -f_y(1 + v^{\prime 2}_2) & f_y u^{\prime}_2 v^{\prime}_2 & f_y u^{\prime}_2 \\ 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix}\notag \\
&= w_h \begin{bmatrix} g_x f_x \rho_2 \\ g_y f_y \rho_2 \\ -g_x f_x \rho_2 u^{\prime}_2 - g_y f_y \rho_2 v^{\prime}_2 \\ -g_x f_x u^{\prime}_2 v^{\prime}_2 -g_y f_y(1 + v^{\prime 2}_2) \\ g_x f_x(1 + u^{\prime 2}_2) + g_y f_y u^{\prime}_2 v^{\prime}_2 \\ -g_x f_x v^{\prime}_2 + g_y f_y u^{\prime}_2\end{bmatrix}^T
\end{align}\]
4 代码中变量的含义
函数 Vec3f CoarseInitializer::calcResAndGS(...)。
dp0, dp1, dp2, dp3, dp4, dp5 是光度误差对 se(3) 六个量的导数，即 \({\partial r_{21} \over \partial \xi_{2}}\)。
dp6, dp7 是光度误差对辐射仿射变换的参数的导数，即 \({\partial r_{21} \over \partial a_{2}}\) 和 \({\partial r_{21} \over \partial b_{2}}\)。
dd 是光度误差对逆深度的导数，即 \({\partial r_{21} \over \partial \rho_1}\)。
