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)》）：

[egin{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}} ight)^T {partial r_{th}^{(i)} over partial xi_{th}} {partial xi_{th} over partial xi_{h}} otag \ &= {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}} otag \ {partial xi_{th} over partial xi_{h}} &= - ext{Ad}_{T_{th}} otag \ {partial xi_{th} over partial xi_{t}} &= I otagend{align}]

复习一下 Schur Complement：

[egin{align} egin{bmatrix} H_{ ho ho} & H_{ ho X} \ H_{X ho} & H_{XX} end{bmatrix} egin{bmatrix} delta ho \ delta X end{bmatrix} &= - egin{bmatrix} J_{ ho}^T r \ J_X^T r end{bmatrix} otag \ egin{bmatrix} H_{ ho ho} & H_{ ho X} \ 0 & H_{XX} - H_{X ho} H_{ ho ho}^{-1} H_{ ho X} end{bmatrix} egin{bmatrix} delta ho \ delta X end{bmatrix} &= - egin{bmatrix} J_{ ho}^T r \ J_X^T r - H_{X ho} H_{ ho ho}^{-1} J_{ ho}^T r end{bmatrix} otag 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 ho} H_{ ho ho}^{-1} H_{ ho X}, H_{X ho} H_{ ho ho}^{-1} J_{ ho}^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，( ho)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_{ ho})是NxM，(J_X)是Nx68。

非 Schur Complement 部分

[H_{XX} = egin{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} ]

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

[egin{align} {partial r over partial C}^T {partial r over partial xi} &= egin{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} egin{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} otag \ &= egin{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} otag end{align} ]

[egin{align} {partial r over partial xi}^T {partial r over partial xi} &= egin{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} egin{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} otag \ &= egin{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} otag end{align}]

[egin{align} J_{X}^Tr &= egin{bmatrix} {partial r over partial C}^T \ {partial r over partial xi}^T end{bmatrix} r otag \ &= egin{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} egin{bmatrix} r^{(1)} \ r^{(2)} \ vdots \ r^{(N)}end{bmatrix} otag \ &= egin{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} otag 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

[egin{align} egin{bmatrix} H_{ ho ho} & H_{ ho X} \ H_{X ho} & H_{XX} end{bmatrix} egin{bmatrix} delta ho \ delta X end{bmatrix} &= - egin{bmatrix} J_{ ho}^T r \ J_X^T r end{bmatrix} otag \ egin{bmatrix} H_{ ho ho} & H_{ ho X} \ 0 & H_{XX} - H_{X ho} H_{ ho ho}^{-1} H_{ ho X} end{bmatrix} egin{bmatrix} delta ho \ delta X end{bmatrix} &= - egin{bmatrix} J_{ ho}^T r \ J_X^T r - H_{X ho} H_{ ho ho}^{-1} J_{ ho}^T r end{bmatrix} otag end{align}]

Hsc：

[H_{X ho} H_{ ho ho}^{-1} H_{ ho X} ]

bsc:

[H_{X ho} H_{ ho ho}^{-1} J_{ ho}^T r ]

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

[egin{align} {partial r over partial C}^T {partial r over partial ho} &= egin{bmatrix} {partial r^{(1)} over partial C}^T & {partial r^{(2)} over partial C}^T & dots & {partial r^{(N)} over partial C}^Tend{bmatrix} egin{bmatrix} {partial r^{(1)} over partial ho^{(1)}} & {partial r^{(1)} over partial ho^{(2)}} & dots & {partial r^{(1)} over partial ho^{(M)}} \ {partial r^{(2)} over partial ho^{(1)}} & {partial r^{(2)} over partial ho^{(2)}} & dots & {partial r^{(2)} over partial ho^{(M)}} \ vdots & vdots & ddots & vdots \ {partial r^{(N)} over partial ho^{(1)}} & {partial r^{(N)} over partial ho^{(2)}} & dots & {partial r^{(N)} over partial ho^{(M)}} end{bmatrix} otag \ &= egin{bmatrix} sum_{i=1}^N {partial r^{(i)} over partial C}^T {partial r^{(i)} over partial ho^{(1)}} & sum_{i=1}^N {partial r^{(i)} over partial C}^T {partial r^{(i)} over partial ho^{(2)}} & dots & sum_{i=1}^N {partial r^{(i)} over partial C}^T {partial r^{(i)} over partial ho^{(M)}} end{bmatrix} otag end{align} ]

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

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

同理

[egin{align} H_{X ho} H_{ ho ho}^{-1} J_ ho^T r otag & = otag egin{bmatrix} left( sum_{i=1}^N {partial r^{(i)} over partial C}^T {partial r^{(i)} over partial ho^{(1)}} ight)left( sum_{i=1}^N {partial r^{(i)} over partial ho^{(1)}}^T{partial r^{(i)} over partial ho^{(1)}} ight)^{-1} & left( sum_{i=1}^N {partial r^{(i)} over partial C}^T {partial r^{(i)} over partial ho^{(2)}} ight) left( sum_{i=1}^N {partial r^{(i)} over partial ho^{(2)}}^T{partial r^{(i)} over partial ho^{(2)}} ight)^{-1} & dots & left( sum_{i=1}^N {partial r^{(i)} over partial C}^T {partial r^{(i)} over partial ho^{(M)}} ight) left( sum_{i=1}^N {partial r^{(i)} over partial ho^{(M)}}^T{partial r^{(i)} over partial ho^{(M)}} ight)^{-1} \ left( sum_{i=1}^N {partial r^{(i)} over partial xi_1}^T {partial r^{(i)} over partial ho^{(1)}} ight) left( sum_{i=1}^N {partial r^{(i)} over partial ho^{(1)}}^T{partial r^{(i)} over partial ho^{(1)}} ight)^{-1} & left( sum_{i=1}^N {partial r^{(i)} over partial xi_1}^T {partial r^{(i)} over partial ho^{(2)}} ight) left( sum_{i=1}^N {partial r^{(i)} over partial ho^{(2)}}^T{partial r^{(i)} over partial ho^{(2)}} ight)^{-1} & dots & left( sum_{i=1}^N {partial r^{(i)} over partial xi_1}^T {partial r^{(i)} over partial ho^{(M)}} ight) left( sum_{i=1}^N {partial r^{(i)} over partial ho^{(M)}}^T{partial r^{(i)} over partial ho^{(M)}} ight)^{-1} \ left( sum_{i=1}^N {partial r^{(i)} over partial xi_2}^T {partial r^{(i)} over partial ho^{(1)}} ight) left( sum_{i=1}^N {partial r^{(i)} over partial ho^{(1)}}^T{partial r^{(i)} over partial ho^{(1)}} ight)^{-1} & left( sum_{i=1}^N {partial r^{(i)} over partial xi_2}^T {partial r^{(i)} over partial ho^{(2)}} ight) left( sum_{i=1}^N {partial r^{(i)} over partial ho^{(2)}}^T{partial r^{(i)} over partial ho^{(2)}} ight)^{-1} & dots & left( sum_{i=1}^N {partial r^{(i)} over partial xi_2}^T {partial r^{(i)} over partial ho^{(M)}} ight) left( sum_{i=1}^N {partial r^{(i)} over partial ho^{(M)}}^T{partial r^{(i)} over partial ho^{(M)}} ight)^{-1} \ vdots & vdots & ddots & vdots \ left( sum_{i=1}^N {partial r^{(i)} over partial xi_8}^T {partial r^{(i)} over partial ho^{(1)}} ight) left( sum_{i=1}^N {partial r^{(i)} over partial ho^{(1)}}^T{partial r^{(i)} over partial ho^{(1)}} ight)^{-1} & left( sum_{i=1}^N {partial r^{(i)} over partial xi_8}^T {partial r^{(i)} over partial ho^{(2)}} ight) left( sum_{i=1}^N {partial r^{(i)} over partial ho^{(2)}}^T{partial r^{(i)} over partial ho^{(2)}} ight)^{-1} & dots & left( sum_{i=1}^N {partial r^{(i)} over partial xi_8}^T {partial r^{(i)} over partial ho^{(M)}} ight) left( sum_{i=1}^N {partial r^{(i)} over partial ho^{(M)}}^T{partial r^{(i)} over partial ho^{(M)}} ight)^{-1} end{bmatrix} otag \ & egin{bmatrix} sum_{i=1}^N {partial r^{(i)} over partial ho^{(1)}}^T r^{(i)} \ sum_{i=1}^N {partial r^{(i)} over partial ho^{(2)}}^T r^{(i)} \ vdots \ sum_{i=1}^N {partial r^{(i)} over partial ho^{(M)}}^T r^{(i)} end{bmatrix} end{align}]