DSO windowed optimization 公式

这里有一个细节，我想了很久才想明白，DSO 中的 residual 联系了两个关键帧之间的相对位姿，但是最终需要优化帧的绝对位姿，中间的导数怎么转换？这里使用的是李群、李代数中的Adjoint。
参考 http://ethaneade.com/lie.pdf 。

需要变通一下，字母太多，表达不方便。此处 \(\xi\) 表示 se(3) 和 affLight 参数。

Adjoint 在其中的使用如下（根据代码推断，具体数学推导看我的博客《Adjoint of SE(3)》）：

\[\begin{align} {\partial r_{th}^{(i)} \over \partial \xi_{h}}^T {\partial r_{th}^{(i)} \over \partial \xi_{h}} &= \left( {\partial r_{th}^{(i)} \over \partial \xi_{th}} {\partial \xi_{th} \over \partial \xi_{h}} \right)^T {\partial r_{th}^{(i)} \over \partial \xi_{th}} {\partial \xi_{th} \over \partial \xi_{h}} \notag \\ &= {\partial \xi_{th} \over \partial \xi_{h}}^T {\partial r_{th}^{(i)} \over \partial \xi_{th}}^T {\partial r_{th}^{(i)} \over \partial \xi_{th}} {\partial \xi_{th} \over \partial \xi_{h}} \notag \\
{\partial \xi_{th} \over \partial \xi_{h}} &= -\text{Ad}_{T_{th}}\notag \\
{\partial \xi_{th} \over \partial \xi_{t}} &= I \notag\end{align}\]

复习一下 Schur Complement：

\[\begin{align} \begin{bmatrix} H_{\rho\rho} & H_{\rho X} \\ H_{X\rho} & H_{XX} \end{bmatrix} \begin{bmatrix} \delta \rho \\ \delta X \end{bmatrix} &= - \begin{bmatrix} J_{\rho}^T r \\ J_X^T r \end{bmatrix} \notag \\
\begin{bmatrix} H_{\rho\rho} & H_{\rho X} \\ 0 & H_{XX} - H_{X\rho} H_{\rho\rho}^{-1} H_{\rho X} \end{bmatrix} \begin{bmatrix} \delta \rho \\ \delta X \end{bmatrix} &= - \begin{bmatrix} J_{\rho}^T r \\ J_X^T r - H_{X\rho} H_{\rho\rho}^{-1} J_{\rho}^T r \end{bmatrix} \notag \end{align}\]

EnergyFuntional::accumulateAF_MT 和 EnergyFunctional::accumulateLF_MT 的目标是计算\(H_{XX}, J_X^T r\)，EnergyFunctional::accumulateSCF_MT 的目标是计算\(H_{X\rho} H_{\rho\rho}^{-1} H_{\rho X}, H_{X\rho} H_{\rho\rho}^{-1} J_{\rho}^T r\)。

\(X\)是68维的，4个相机参数加上8*8个帧状态量。就不写出来了。

这里需要注意一下，\(r\)是Nx1，\(\rho\)是Mx1，\(M \le N\)，即 residual 的数目与需要优化的逆深度的数目不一定相等。\(J_{\rho}\)是NxM，\(J_X\)是Nx68。

非 Schur Complement 部分

\[H_{XX} = \begin{bmatrix} {\partial r \over \partial C}^T {\partial r \over \partial C} & {\partial r \over \partial C}^T {\partial r \over \partial \xi} \\ {\partial r \over \partial \xi}^T {\partial r \over \partial C} & {\partial r \over \partial \xi}^T {\partial r \over \partial \xi} \end{bmatrix}\]

\[\begin{align} {\partial r \over \partial C}^T {\partial r \over \partial C} &= \begin{bmatrix} {\partial r^{(1)} \over \partial C}^T & {\partial r^{(2)} \over \partial C}^T & \dots & {\partial r^{(N)} \over \partial C}^T \end{bmatrix} \begin{bmatrix} {\partial r^{(1)} \over \partial C} \\ {\partial r^{(2)} \over \partial C} \\ \dots \\ {\partial r^{(N)} \over \partial C} \end{bmatrix} \notag \\ &= \sum_{i=1}^{N} {\partial r^{(i)} \over \partial C}^T{\partial r^{(i)} \over \partial C} \notag \end{align}\]

\[\begin{align} {\partial r \over \partial C}^T {\partial r \over \partial \xi} &= \begin{bmatrix} {\partial r^{(1)} \over \partial C}^T & {\partial r^{(2)} \over \partial C}^T & \dots & {\partial r^{(N)} \over \partial C}^T \end{bmatrix} \begin{bmatrix} {\partial r^{(1)} \over \partial \xi_1} & {\partial r^{(1)} \over \partial \xi_2} & \dots & {\partial r^{(1)} \over \partial \xi_8} \\ {\partial r^{(2)} \over \partial \xi_1} & {\partial r^{(2)} \over \partial \xi_2} & \dots & {\partial r^{(2)} \over \partial \xi_8} \\ \vdots & \vdots & \ddots & \vdots \\ {\partial r^{(N)} \over \partial \xi_1} & {\partial r^{(N)} \over \partial \xi_2} & \dots & {\partial r^{(N)} \over \partial \xi_8} \end{bmatrix} \notag \\ &= \begin{bmatrix} \sum_{i=1}^{N} {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \xi_1} & \sum_{i=1}^{N} {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \xi_2} & \dots & \sum_{i=1}^{N} {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \xi_8} \end{bmatrix} \notag \end{align}\]

\[\begin{align} {\partial r \over \partial \xi}^T {\partial r \over \partial \xi} &= \begin{bmatrix} {\partial r^{(1)} \over \partial \xi_1}^T & {\partial r^{(2)} \over \partial \xi_1}^T & \dots & {\partial r^{(N)} \over \partial \xi_1}^T \\ {\partial r^{(1)} \over \partial \xi_2}^T & {\partial r^{(2)} \over \partial \xi_2}^T & \dots & {\partial r^{(N)} \over \partial \xi_2}^T \\ \vdots & \vdots & \ddots & \vdots \\ {\partial r^{(1)} \over \partial \xi_8}^T & {\partial r^{(2)} \over \partial \xi_8}^T & \dots & {\partial r^{(N)} \over \partial \xi_8}^T \end{bmatrix} \begin{bmatrix} {\partial r^{(1)} \over \partial \xi_1} & {\partial r^{(1)} \over \partial \xi_2} & \dots & {\partial r^{(1)} \over \partial \xi_8} \\ {\partial r^{(2)} \over \partial \xi_1} & {\partial r^{(2)} \over \partial \xi_2} & \dots & {\partial r^{(2)} \over \partial \xi_8} \\ \vdots & \vdots & \ddots & \vdots \\ {\partial r^{(N)} \over \partial \xi_1} & {\partial r^{(N)} \over \partial \xi_2} & \dots & {\partial r^{(N)} \over \partial \xi_8} \end{bmatrix} \notag \\
&= \begin{bmatrix} \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T{\partial r^{(i)} \over \partial \xi_1} & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T{\partial r^{(i)} \over \partial \xi_2} & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T{\partial r^{(i)} \over \partial \xi_8} \\ \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T{\partial r^{(i)} \over \partial \xi_1} & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T{\partial r^{(i)} \over \partial \xi_2} & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T{\partial r^{(i)} \over \partial \xi_8} \\ \vdots & \vdots & \ddots & \vdots \\ \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T{\partial r^{(i)} \over \partial \xi_1} & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T{\partial r^{(i)} \over \partial \xi_2} & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T{\partial r^{(i)} \over \partial \xi_8} \end{bmatrix} \notag \end{align}\]

\[\begin{align} J_{X}^Tr &= \begin{bmatrix} {\partial r \over \partial C}^T \\ {\partial r \over \partial \xi}^T \end{bmatrix} r \notag \\ &= \begin{bmatrix} {\partial r^{(1)} \over \partial C}^T & {\partial r^{(2)} \over \partial C}^T & \dots & {\partial r^{(N)} \over \partial C}^T \\ {\partial r^{(1)} \over \partial \xi_1}^T & {\partial r^{(2)} \over \partial \xi_1}^T & \dots & {\partial r^{(N)} \over \partial \xi_1}^T \\ {\partial r^{(1)} \over \partial \xi_2}^T & {\partial r^{(2)} \over \partial \xi_2}^T & \dots & {\partial r^{(N)} \over \partial \xi_2}^T \\ \vdots & \vdots & \ddots & \vdots \\ {\partial r^{(1)} \over \partial \xi_8}^T & {\partial r^{(2)} \over \partial \xi_8}^T & \dots & {\partial r^{(N)} \over \partial \xi_8}^T \end{bmatrix} \begin{bmatrix} r^{(1)} \\ r^{(2)} \\ \vdots \\ r^{(N)}\end{bmatrix} \notag \\
&= \begin{bmatrix} \sum_{i = 1}^{N} {\partial r^{(i)} \over \partial C}^T r^{(i)} \\ \sum_{i = 1}^{N} {\partial r^{(i)} \over \partial \xi_1}^T r^{(i)} \\ \sum_{i = 1}^{N} {\partial r^{(i)} \over \partial \xi_2}^T r^{(i)} \\ \vdots \\ \sum_{i = 1}^{N} {\partial r^{(i)} \over \partial \xi_8}^T r^{(i)} \end{bmatrix} \notag \end{align}\]

所以算这些矩阵就是遍历每一个 residual，累加求和。

Schur Complement 部分

\[\begin{align} \begin{bmatrix} H_{\rho\rho} & H_{\rho X} \\ H_{X\rho} & H_{XX} \end{bmatrix} \begin{bmatrix} \delta \rho \\ \delta X \end{bmatrix} &= - \begin{bmatrix} J_{\rho}^T r \\ J_X^T r \end{bmatrix} \notag \\
\begin{bmatrix} H_{\rho\rho} & H_{\rho X} \\ 0 & H_{XX} - H_{X\rho} H_{\rho\rho}^{-1} H_{\rho X} \end{bmatrix} \begin{bmatrix} \delta \rho \\ \delta X \end{bmatrix} &= - \begin{bmatrix} J_{\rho}^T r \\ J_X^T r - H_{X\rho} H_{\rho\rho}^{-1} J_{\rho}^T r \end{bmatrix} \notag \end{align}\]

Hsc：
\[H_{X\rho} H_{\rho\rho}^{-1} H_{\rho X}\]
bsc:
\[H_{X\rho} H_{\rho\rho}^{-1} J_{\rho}^T r\]

\[\begin{align} J_\rho^TJ_\rho &= {\partial r \over \partial \rho}^T {\partial r \over \partial \rho} \notag \\
&= \begin{bmatrix} {\partial r^{(1)} \over \partial \rho^{(1)}}^T & {\partial r^{(2)} \over \partial \rho^{(1)}}^T & \dots & {\partial r^{(N)} \over \partial \rho^{(1)}}^T \\ {\partial r^{(1)} \over \partial \rho^{(2)}}^T & {\partial r^{(2)} \over \partial \rho^{(2)}}^T & \dots & {\partial r^{(N)} \over \partial \rho^{(2)}}^T \\ \vdots & \vdots & \ddots & \vdots \\ {\partial r^{(1)} \over \partial \rho^{(M)}}^T & {\partial r^{(2)} \over \partial \rho^{(M)}}^T & \dots & {\partial r^{(N)} \over \partial \rho^{(M)}}^T \end{bmatrix} \begin{bmatrix} {\partial r^{(1)} \over \partial \rho^{(1)}} & {\partial r^{(1)} \over \partial \rho^{(2)}} & \dots & {\partial r^{(1)} \over \partial \rho^{(M)}} \\ {\partial r^{(2)} \over \partial \rho^{(1)}} & {\partial r^{(2)} \over \partial \rho^{(2)}} & \dots & {\partial r^{(2)} \over \partial \rho^{(M)}} \\ \vdots & \vdots & \ddots & \vdots \\ {\partial r^{(N)} \over \partial \rho^{(1)}} & {\partial r^{(N)} \over \partial \rho^{(2)}} & \dots & {\partial r^{(N)} \over \partial \rho^{(M)}} \end{bmatrix} \notag \\
&= \begin{bmatrix} \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T{\partial r^{(i)} \over \partial \rho^{(1)}} & \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T{\partial r^{(i)} \over \partial \rho^{(2)}} & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T{\partial r^{(i)} \over \partial \rho^{(M)}} \\ \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T{\partial r^{(i)} \over \partial \rho^{(1)}} & \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T{\partial r^{(i)} \over \partial \rho^{(2)}} & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T{\partial r^{(i)} \over \partial \rho^{(M)}} \\ \vdots & \vdots & \ddots & \vdots \\ \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T{\partial r^{(i)} \over \partial \rho^{(1)}} & \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T{\partial r^{(i)} \over \partial \rho^{(2)}} & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T{\partial r^{(i)} \over \partial \rho^{(M)}} \end{bmatrix} \notag \\
&= \begin{bmatrix} \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T{\partial r^{(i)} \over \partial \rho^{(1)}} & 0 & \dots & 0 \\ 0 & \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T{\partial r^{(i)} \over \partial \rho^{(2)}} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T{\partial r^{(i)} \over \partial \rho^{(M)}} \end{bmatrix} \notag \end{align}\]

\[ \begin{align} {\partial r \over \partial C}^T {\partial r \over \partial \rho} &= \begin{bmatrix} {\partial r^{(1)} \over \partial C}^T & {\partial r^{(2)} \over \partial C}^T & \dots & {\partial r^{(N)} \over \partial C}^T\end{bmatrix} \begin{bmatrix} {\partial r^{(1)} \over \partial \rho^{(1)}} & {\partial r^{(1)} \over \partial \rho^{(2)}} & \dots & {\partial r^{(1)} \over \partial \rho^{(M)}} \\ {\partial r^{(2)} \over \partial \rho^{(1)}} & {\partial r^{(2)} \over \partial \rho^{(2)}} & \dots & {\partial r^{(2)} \over \partial \rho^{(M)}} \\ \vdots & \vdots & \ddots & \vdots \\ {\partial r^{(N)} \over \partial \rho^{(1)}} & {\partial r^{(N)} \over \partial \rho^{(2)}} & \dots & {\partial r^{(N)} \over \partial \rho^{(M)}} \end{bmatrix} \notag \\ &= \begin{bmatrix} \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(1)}} & \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(2)}} & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \end{bmatrix} \notag \end{align} \]

\[\begin{align} {\partial r \over \partial \xi}^T {\partial r \over \partial \rho} &= \begin{bmatrix} {\partial r^{(1)} \over \partial \xi_1}^T & {\partial r^{(2)} \over \partial \xi_1}^T & \dots & {\partial r^{(N)} \over \partial \xi_1}^T \\ {\partial r^{(1)} \over \partial \xi_2}^T & {\partial r^{(2)} \over \partial \xi_2}^T & \dots & {\partial r^{(N)} \over \partial \xi_2}^T \\ \vdots & \vdots & \ddots & \vdots \\ {\partial r^{(1)} \over \partial \xi_8}^T & {\partial r^{(2)} \over \partial \xi_8}^T & \dots & {\partial r^{(N)} \over \partial \xi_8}^T \end{bmatrix} \begin{bmatrix} {\partial r^{(1)} \over \partial \rho^{(1)}} & {\partial r^{(1)} \over \partial \rho^{(2)}} & \dots & {\partial r^{(1)} \over \partial \rho^{(M)}} \\ {\partial r^{(2)} \over \partial \rho^{(1)}} & {\partial r^{(2)} \over \partial \rho^{(2)}} & \dots & {\partial r^{(2)} \over \partial \rho^{(M)}} \\ \vdots & \vdots & \ddots & \vdots \\ {\partial r^{(N)} \over \partial \rho^{(1)}} & {\partial r^{(N)} \over \partial \rho^{(2)}} & \dots & {\partial r^{(N)} \over \partial \rho^{(M)}} \end{bmatrix} \notag \\
&= \begin{bmatrix} \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(1)}} & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(2)}} & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \\ \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T {\partial r^{(i)} \over \partial \rho^{(1)}} & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T {\partial r^{(i)} \over \partial \rho^{(2)}} & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \\ \vdots & \vdots & \ddots & \vdots \\ \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(1)}} & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(2)}} & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \end{bmatrix} \notag \end{align}\]

\[\begin{align} H_{X\rho} H_{\rho\rho}^{-1} H_{\rho X} &= J_X^T J_\rho (J_\rho^TJ_\rho)^{-1} J_\rho^T J_X \notag \\
&= \begin{bmatrix} {\partial r \over \partial C}^T \\ {\partial r \over \partial \xi}^T \end{bmatrix} {\partial r \over \partial \rho} \left( {\partial r \over \partial \rho}^T {\partial r \over \partial \rho} \right)^{-1} {\partial r \over \partial \rho}^T \begin{bmatrix} {\partial r \over \partial C} & {\partial r \over \partial \xi} \end{bmatrix} \notag \\
&= \begin{bmatrix} {\partial r \over \partial C}^T {\partial r \over \partial \rho} \\ {\partial r \over \partial \xi}^T {\partial r \over \partial \rho} \end{bmatrix} \begin{bmatrix} \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T{\partial r^{(i)} \over \partial \rho^{(1)}} \right)^{-1} & 0 & \dots & 0 \\ 0 & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T{\partial r^{(i)} \over \partial \rho^{(2)}} \right)^{-1} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T{\partial r^{(i)} \over \partial \rho^{(M)}} \right)^{-1} \end{bmatrix} \notag \\ & \begin{bmatrix} {\partial r \over \partial \rho}^T {\partial r \over \partial C} & {\partial r \over \partial \rho}^T {\partial r \over \partial \xi} \end{bmatrix} \notag \\
&= \begin{bmatrix} \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(1)}} & \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(2)}} & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \\ \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(1)}} & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(2)}} & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \\ \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T {\partial r^{(i)} \over \partial \rho^{(1)}} & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T {\partial r^{(i)} \over \partial \rho^{(2)}} & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \\ \vdots & \vdots & \ddots & \vdots \\ \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(1)}} & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(2)}} & \dots & \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \end{bmatrix} \notag \\
&\begin{bmatrix} \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T{\partial r^{(i)} \over \partial \rho^{(1)}} \right)^{-1} & 0 & \dots & 0 \\ 0 & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T{\partial r^{(i)} \over \partial \rho^{(2)}} \right)^{-1} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T{\partial r^{(i)} \over \partial \rho^{(M)}} \right)^{-1} \end{bmatrix} \notag \\
&\begin{bmatrix} \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(1)}} \right)^T & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(1)}} \right)^T & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(1)}} \right)^T \\ \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(2)}} \right)^T & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(2)}} \right)^T & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(2)}} \right)^T \\ \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(3)}} \right)^T & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(3)}} \right)^T & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(3)}} \right)^T \\
\vdots & \vdots & \ddots & \vdots \\ \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \right)^T & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \right)^T & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \right)^T \end{bmatrix} \notag \\
&= \begin{bmatrix} \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(1)}} \right)\left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T{\partial r^{(i)} \over \partial \rho^{(1)}} \right)^{-1} & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(2)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T{\partial r^{(i)} \over \partial \rho^{(2)}} \right)^{-1} & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T{\partial r^{(i)} \over \partial \rho^{(M)}} \right)^{-1} \\ \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(1)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T{\partial r^{(i)} \over \partial \rho^{(1)}} \right)^{-1} & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(2)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T{\partial r^{(i)} \over \partial \rho^{(2)}} \right)^{-1} & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T{\partial r^{(i)} \over \partial \rho^{(M)}} \right)^{-1} \\ \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T {\partial r^{(i)} \over \partial \rho^{(1)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T{\partial r^{(i)} \over \partial \rho^{(1)}} \right)^{-1} & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T {\partial r^{(i)} \over \partial \rho^{(2)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T{\partial r^{(i)} \over \partial \rho^{(2)}} \right)^{-1} & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T{\partial r^{(i)} \over \partial \rho^{(M)}} \right)^{-1} \\ \vdots & \vdots & \ddots & \vdots \\ \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(1)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T{\partial r^{(i)} \over \partial \rho^{(1)}} \right)^{-1} & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(2)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T{\partial r^{(i)} \over \partial \rho^{(2)}} \right)^{-1} & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T{\partial r^{(i)} \over \partial \rho^{(M)}} \right)^{-1} \end{bmatrix} \notag \\
&\begin{bmatrix} \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(1)}} \right)^T & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(1)}} \right)^T & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(1)}} \right)^T \\ \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(2)}} \right)^T & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(2)}} \right)^T & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(2)}} \right)^T \\ \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(3)}} \right)^T & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(3)}} \right)^T & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(3)}} \right)^T \\
\vdots & \vdots & \ddots & \vdots \\ \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \right)^T & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \right)^T & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \right)^T \end{bmatrix}\notag \end{align}\]

同理
\[\begin{align} H_{X\rho} H_{\rho\rho}^{-1} J_\rho^T r \notag & = \notag \begin{bmatrix} \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(1)}} \right)\left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T{\partial r^{(i)} \over \partial \rho^{(1)}} \right)^{-1} & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(2)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T{\partial r^{(i)} \over \partial \rho^{(2)}} \right)^{-1} & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial C}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T{\partial r^{(i)} \over \partial \rho^{(M)}} \right)^{-1} \\ \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(1)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T{\partial r^{(i)} \over \partial \rho^{(1)}} \right)^{-1} & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(2)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T{\partial r^{(i)} \over \partial \rho^{(2)}} \right)^{-1} & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_1}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T{\partial r^{(i)} \over \partial \rho^{(M)}} \right)^{-1} \\ \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T {\partial r^{(i)} \over \partial \rho^{(1)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T{\partial r^{(i)} \over \partial \rho^{(1)}} \right)^{-1} & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T {\partial r^{(i)} \over \partial \rho^{(2)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T{\partial r^{(i)} \over \partial \rho^{(2)}} \right)^{-1} & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_2}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T{\partial r^{(i)} \over \partial \rho^{(M)}} \right)^{-1} \\ \vdots & \vdots & \ddots & \vdots \\ \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(1)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T{\partial r^{(i)} \over \partial \rho^{(1)}} \right)^{-1} & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(2)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T{\partial r^{(i)} \over \partial \rho^{(2)}} \right)^{-1} & \dots & \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \xi_8}^T {\partial r^{(i)} \over \partial \rho^{(M)}} \right) \left( \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T{\partial r^{(i)} \over \partial \rho^{(M)}} \right)^{-1} \end{bmatrix} \notag \\
& \begin{bmatrix} \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(1)}}^T r^{(i)} \\ \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(2)}}^T r^{(i)} \\
\vdots \\ \sum_{i=1}^N {\partial r^{(i)} \over \partial \rho^{(M)}}^T r^{(i)} \end{bmatrix} \end{align}\]