DSO windowed optimization 代码 (1)

这里不想解释怎么 marginalize，什么是 First-Estimates Jacobian (FEJ)。这里只看看代码，看看Hessian矩阵是怎么构造出来的。

1 优化流程

整个优化过程，也是 Levenberg–Marquardt 的优化过程，这个优化过程在函数 FullSystem::makeKeyFrame() 中被调用，也是在确定当前帧成为关键帧，并且用当前帧激活了窗口中其他帧的 immaturePoints 之后。过程在 FullSystem::optimize() 函数中。

优化的目标值，是所有需要优化点的逆深度、相机的4个内参数、窗口中8个帧的状态量。

FullSystem::optimize() 函数流程大致如下。首先 FullSystem::linearizeAll(false) 把相关的导数计算一下。然后在所有的 residual 中找到 isLinearized 为 false 的 residual，调用 PointFrameResidual::applyRes(true)，设置它们的 PointFrameResidual::ResState （按照 FullSystem::optimizeImmaturePoint 的结果），如果是正常的点(ResState::IN)调用EFResidual::takeDataF()把 EFResidual::JpJdF 设置一下。这就算完成了优化的准备工作。

随后进入循环体，进行最高有限次的优化循环。每一次优化都有可能使得整体能量升高，所以每次优化前调用 FullSystem::backupState() 保存当前所有待优化参数的 state 和 step，FullSystem::solveSystem() 进行优化（得到的优化变化量 step），FullSystem::doStepFromBackup() 将优化结果生效，FullSystem::linearizeAll(false) 计算新的能量值，如果能量升高了，就使用 FullSystem::loadSateBackup() 将结果回滚。

在跳出循环体之后调用一次 FullSystem::linearizeAll(true)，效果是将优化之后成为 outlier 的 residual 剔除，剩下正常的 residual 调用一次 EFResidual::takeDataF() 计算新的 EFResidual::JpJdF （这里涉及到 FEJ）。注意一下 FullSystem::linearizeAll() 参数为 false 和 true 的区别。

FullSystem::solveSystem() 设置优化变化量 step 的过程在 EnergyFunctional::resubstituteF_MT() 中，能帮助理解 idepth 是如何更新的（Schur Complement 将 idepth 剔除出最终需要矩阵求逆的系统，然而这些 idepth 的优化变化量 step 是如何计算的？）。

2 导数准备

需要遍历每一个 PointFrameResidual 将与这个 Residual 相关的导数计算出来，再进行优化。而这些计算出来的相关导数被存储在 RawResidualJacobian 中。

https://github.com/JakobEngel/dso/blob/master/src/OptimizationBackend/RawResidualJacobian.h#L32

在优化过程中 Frame, PointHessian, PointFrameResidual 都有与之对应的实体，分别是 EFFrame, EFPoint, EFResidual。通过保留指针，保存层次之间的索引。

在计算 RawResidualJacobian 时，是计算 PointFrameResidual 的 J，尔后会将这个 J 转移到 EFResidual 的 J，并且计算 EFResidual::JpJdF，这个过程在 EFResidual::takeDataF() 中。所以这里把 JpJdF 的计算过程写出来，弄清 JpJdF 的意义。

https://github.com/JakobEngel/dso/blob/5fb2c065b1638e10bccf049a6575ede4334ba673/src/OptimizationBackend/EnergyFunctionalStructs.cpp#L37

struct RawResidualJacobian
{
	EIGEN_MAKE_ALIGNED_OPERATOR_NEW;
	// ================== new structure: save independently =============.
	VecNRf resF; // typdef Eigen::Matrix<float,MAX_RES_PER_POINT,1> VecNRf; MAX_RES_PER_POINT == 8

	// the two rows of d[x,y]/d[xi].
	Vec6f Jpdxi[2];			// 2x6

	// the two rows of d[x,y]/d[C].
	VecCf Jpdc[2];			// 2x4

	// the two rows of d[x,y]/d[idepth].
	Vec2f Jpdd;				// 2x1

	// the two columns of d[r]/d[x,y].
	VecNRf JIdx[2];			// 9x2

	// = the two columns of d[r] / d[ab]
	VecNRf JabF[2];			// 9x2

	// = JIdx^T * JIdx (inner product). Only as a shorthand.
	Mat22f JIdx2;				// 2x2
	// = Jab^T * JIdx (inner product). Only as a shorthand.
	Mat22f JabJIdx;			// 2x2
	// = Jab^T * Jab (inner product). Only as a shorthand.
	Mat22f Jab2;			// 2x2

};

以上变量的类型中出现NR，说明该变量是存储了每一个 pattern 点的信息。

现在将这些变量对应的导数一一列出：

1. VecNRf resF;对应\(r_{21}\)，1x8，这里的\(r_{21}\)是对于一个点，八个 pattern residual 组成的向量。
2. Vec6f Jpdxi[2];对应\(\partial x_2 \over \partial \xi_{21}\)，2x6，注意这里的\(x_2\)是像素坐标。（我一般把像素坐标写成\(x\)，对应代码中的变量Ku，归一化坐标写成\(x^{\prime}\)，对应代码中的变量u。）
3. VecCf Jpdc[2];对应\(\partial x_2 \over \partial C\)，2x4，这里的\(C\)指相机内参\(\begin{bmatrix} f_x, f_y, c_x, c_y\end{bmatrix}^T\)。
4. Vec2f Jpdd;对应\(\partial x_2 \over \partial \rho_1\)，2x1，注意是对 host 帧的逆深度求导。
5. VecNRf JIdx[2];对应\(\partial r_{21} \over \partial x_2\)，8x2，这个和 target 帧上的影像梯度相关。
6. VecNRf JabF[2];对应\({\partial r_{21} \over \partial a_{21}}, {\partial r_{21} \over \partial b_{21}}\)，8x1，8x1。
7. Mat22f JIdx2;对应\({\partial r_{21} \over \partial x_2}^T{\partial r_{21} \over \partial x_2}\)，2x8 8x2，2x2。
8. Mat22f JabJIdx;对应\({\partial r_{21} \over \partial l_{21}}^T{\partial r_{21} \over \partial x_2}\)，2x8 8x2，2x2，这里的\(l_{21}\)指\(\begin{bmatrix} a_{21} \\ b_{21}\end{bmatrix}\)。
9. Mat22f Jab2;对应\({\partial r_{21} \over \partial l_{21}}^T{\partial r_{21} \over \partial l_{21}}\)，2x8 8x2，2x2。
10. JpJdF对应\(\begin{bmatrix} {\partial r_{21} \over \partial \xi_{21}}^T{\partial r_{21} \over \partial \rho_1} \\ {\partial r_{21} \over \partial l_{21}}^T{\partial r_{21} \over \partial \rho_1}\end{bmatrix}\)，8x1。

在 PointFrameResidual::linearize 中对这些变量进行了计算。

https://github.com/JakobEngel/dso/blob/master/src/FullSystem/Residuals.cpp#L78

在计算时使用了投影过程中的变量，现在将这些变量与公式对应。投影过程标准公式如下：

\[\begin{align} x_2 &= K \rho_2 (R_{21} \rho_1^{-1} K^{-1} x_1 + t_{21}) \notag \\
&= K x^{\prime}_2\notag \end{align}\]

变量的对应关系如下：

KliP = \(K^{-1}x_1\) = \(x_1^{\prime}\)

ptp = \(R_{21}K^{-1}x_1 + \rho_1 t_{21}\) = \(\rho_2^{-1}\rho_1K^{-1}x_2\)

drescale = \(\rho_2 \rho_1^{-1}\)

[u, v, 1]^T = \(K^{-1}x_2\) = \(x_2^{\prime}\)

[Ku, Kv, 1]^T = \(x_2\)

1. Vec2f Jpdd;\(\partial x_2 \over \partial \rho_1\)

d_d_x = drescale * (PRE_tTll_0[0]-PRE_tTll_0[2]*u)*SCALE_IDEPTH*HCalib->fxl();
d_d_y = drescale * (PRE_tTll_0[1]-PRE_tTll_0[2]*v)*SCALE_IDEPTH*HCalib->fyl();

计算\(\partial x_2 \over \partial \rho_1\)，这个在博客《直接法光度误差导数推导》中已经讲了如何求解。得到的结果是：

\[\begin{bmatrix} f_x \rho_1^{-1}\rho_2(t_{21}^x - u^{\prime}_2t_{21}^z) \\ f_y \rho_1^{-1}\rho_2(t_{21}^y - v^{\prime}_2t_{21}^z)\end{bmatrix}
\]

2.VecCf Jpdc[2];\(\partial x_2 \over \partial C\)

d_C_x[2] = drescale*(PRE_RTll_0(2,0)*u-PRE_RTll_0(0,0));
d_C_x[3] = HCalib->fxl() * drescale*(PRE_RTll_0(2,1)*u-PRE_RTll_0(0,1)) * HCalib->fyli();
d_C_x[0] = KliP[0]*d_C_x[2];
d_C_x[1] = KliP[1]*d_C_x[3];

d_C_y[2] = HCalib->fyl() * drescale*(PRE_RTll_0(2,0)*v-PRE_RTll_0(1,0)) * HCalib->fxli();
d_C_y[3] = drescale*(PRE_RTll_0(2,1)*v-PRE_RTll_0(1,1));
d_C_y[0] = KliP[0]*d_C_y[2];
d_C_y[1] = KliP[1]*d_C_y[3];

d_C_x[0] = (d_C_x[0]+u)*SCALE_F;
d_C_x[1] *= SCALE_F;
d_C_x[2] = (d_C_x[2]+1)*SCALE_C;
d_C_x[3] *= SCALE_C;

d_C_y[0] *= SCALE_F;
d_C_y[1] = (d_C_y[1]+v)*SCALE_F;
d_C_y[2] *= SCALE_C;
d_C_y[3] = (d_C_y[3]+1)*SCALE_C;

链式的求导过程。

\[{\partial x_2 \over \partial C} = \begin{bmatrix} {\partial u_2 \over \partial f_x} & {\partial u_2 \over \partial f_y} & {\partial u_2 \over \partial c_x} & {\partial u_2 \over \partial c_y} \\ {\partial v_2 \over \partial f_x} & {\partial v_2 \over \partial f_y} & {\partial v_2 \over \partial c_x} & {\partial v_2 \over \partial c_y} \end{bmatrix}
\]

\[\begin{align}x_2 &= Kx_2^{\prime} \notag \\
\begin{bmatrix} u_2 \\ v_2 \\ 1 \end{bmatrix} &= \begin{bmatrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1\end{bmatrix} \begin{bmatrix} u_2^{\prime} \\ v_2^{\prime} \\ 1\end{bmatrix} \notag \end{align}\]

\[\begin{align} u_2 &= f_x u_2^{\prime} + c_x \notag \\
v_2 &= f_y v_2^{\prime} + c_y \notag \end{align}\]

\[\begin{align} {\partial u_2 \over \partial f_x} &= u_2^{\prime} + f_x {\partial u_2^{\prime} \over \partial f_x} \notag &
{\partial u_2 \over \partial f_y} &= f_x {\partial u_2^{\prime} \over \partial f_y} \notag \\
{\partial u_2 \over \partial c_x} &= f_x {\partial u_2^{\prime} \over \partial c_x} + 1 \notag &
{\partial u_2 \over \partial c_y} &= f_x {\partial u_2^{\prime} \over \partial c_y} \notag \end{align}\]

\[\begin{align} {\partial v_2 \over \partial f_x} &= f_y {\partial v_2^{\prime} \over \partial f_x} \notag &
{\partial v_2 \over \partial f_y} &= v_2^{\prime} + f_y {\partial v_2^{\prime} \over \partial f_y} \notag \\
{\partial v_2 \over \partial c_x} &= f_y {\partial v_2^{\prime} \over \partial c_x} \notag &
{\partial v_2 \over \partial c_y} &= f_y {\partial v_2^{\prime} \over \partial c_y} + 1 \notag \end{align}\]

先求\({\partial x_2^{\prime} \over \partial C}\)，再使用链式法则求\({\partial x_2 \over \partial C}\)。

\[\begin{align} x_2^{\prime} &= \rho_2 \rho_1^{-1}(R_{21}K^{-1}x_1+\rho_1t_{21}) \notag \\
&= \rho_2 \rho_1^{-1} R_{21}K^{-1}x_1 + \rho_2 t_{21} \notag \\
&= \rho_2 \rho_1^{-1} \begin{bmatrix} r_{11} & r_{12} & r_{13} \\ r_{21} & r_{22} & r_{23} \\ r_{31} & r_{32} & r_{33}\end{bmatrix} \begin{bmatrix} f_x^{-1} & 0 & -f_x^{-1}c_x \\ 0 & f_y^{-1} & -f_y^{-1}c_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} u_1 \\ v_1 \\ 1 \end{bmatrix} + \rho_2 t_{21} \notag \\
&= \rho_2 \rho_1^{-1} \begin{bmatrix} r_{11} & r_{12} & r_{13} \\ r_{21} & r_{22} & r_{23} \\ r_{31} & r_{32} & r_{33}\end{bmatrix} \begin{bmatrix} f_x^{-1}(u_1 - c_x) \\ f_y^{-1}(v_1 - c_y) \\ 1 \end{bmatrix} + \rho_2 \begin{bmatrix} t_{21}^x \\ t_{21}^y \\ t_{21}^z \end{bmatrix} \notag \end{align}\]

\[\begin{align} u_2^{\prime} &= \frac{\rho_2 \rho_1^{-1}(r_{11}f_x^{-1}(u_1 - c_x) + r_{12}f_y^{-1}(v_1-c_y) + r_{13}) + \rho_2 t_{21}^x}{\rho_2 \rho_1^{-1}(r_{31}f_x^{-1}(u_1 - c_x) + r_{32}f_y^{-1}(v_1-c_y) + r_{33}) + \rho_2 t_{21}^z} = \frac{A}{C} \notag \\
v_2^{\prime} &= \frac{\rho_2 \rho_1^{-1}(r_{21}f_x^{-1}(u_1 - c_x) + r_{22}f_y^{-1}(v_1-c_y) + r_{23}) + \rho_2 t_{21}^y}{\rho_2 \rho_1^{-1}(r_{31}f_x^{-1}(u_1 - c_x) + r_{32}f_y^{-1}(v_1-c_y) + r_{33}) + \rho_2 t_{21}^z} = \frac{B}{C}\notag \end{align}\]

\(u_2^{\prime}, v_2^{\prime}\) 的分母的计算结果都为 1，但是这个计算过程和内参 \(f_x, f_y, c_x, c_y\) 都有关系。求导过程不能省略分母，感谢某同学指出我这里认知上的错误。（为方便表达，我用字母替代分子、分母，\(C=1\)。）

求导示例：

\[\begin{align} {\partial u_2^{\prime} \over \partial f_x} &= \frac{\partial A}{\partial f_x} \frac{1}{C} + A\frac{1}{C^2}(-1)\frac{\partial C}{\partial f_x} \notag \\
&= \rho_2 \rho_1^{-1}r_{11}(u_1-c_x)f_x^{-2}(-1)\frac{1}{C} - \frac{A}{C}\frac{1}{C}\rho_2 \rho_1^{-1}r_{31}(u_1-c_x)f_x^{-2}(-1) \notag \\
&= \frac{1}{C}(\rho_2 \rho_1^{-1}r_{11}(u_1-c_x)f_x^{-2}(-1) + \rho_2 \rho_1^{-1}r_{31}(u_1-c_x)f_x^{-2}u_2^{\prime}) \notag \\
&= \rho_2 \rho_1^{-1} (r_{31}u_2^{\prime}-r_{11})f_x^{-2}(u_1 - c_x) \notag \end{align}\]

同理得到以下结果：

\[\begin{align} {\partial u_2^{\prime} \over \partial f_x} &= \rho_2 \rho_1^{-1} (r_{31}u_2^{\prime}-r_{11})f_x^{-2}(u_1 - c_x) \notag &
{\partial u_2^{\prime} \over \partial f_y} &= \rho_2 \rho_1^{-1}(r_{32}u_2^{\prime}-r_{12})f_y^{-2}(v_1 - c_y) \notag \\
{\partial u_2^{\prime} \over \partial c_x} &= \rho_2 \rho_1^{-1}(r_{31}u_2^{\prime}-r_{11})f_x^{-1} \notag &
{\partial u_2^{\prime} \over \partial c_y} &= \rho_2 \rho_1^{-1}(r_{32}u_2^{\prime}-r_{12}) f_y^{-1} \notag \end{align}\]

\[\begin{align} {\partial v_2^{\prime} \over \partial f_x} &= \rho_2 \rho_1^{-1} (r_{31}v_2^{\prime}-r_{21})f_x^{-2}(u_1 - c_x) \notag &
{\partial v_2^{\prime} \over \partial f_y} &= \rho_2 \rho_1^{-1}(r_{32}v_2^{\prime}-r_{22})f_y^{-2}(v_1 - c_y) \notag \\
{\partial v_2^{\prime} \over \partial c_x} &= \rho_2 \rho_1^{-1}(r_{31}v_2^{\prime}-r_{21})f_x^{-1} \notag &
{\partial v_2^{\prime} \over \partial c_y} &= \rho_2 \rho_1^{-1}(r_{32}v_2^{\prime}-r_{22}) f_y^{-1} \notag \end{align}\]

链式：

\[\begin{align} {\partial u_2 \over \partial f_x} &= u_2^{\prime} + f_x {\partial u_2^{\prime} \over \partial f_x} \notag \\ &= u_2^{\prime} + \rho_2 \rho_1^{-1} (r_{31}u_2^{\prime}-r_{11})f_x^{-1}(u_1 - c_x) \notag \\
{\partial u_2 \over \partial f_y} &= f_x {\partial u_2^{\prime} \over \partial f_y} \notag \\ &= f_x f_y^{-1} \rho_2 \rho_1^{-1}(r_{32} u_2^{\prime}-r_{12})f_y^{-1}(v_1 - c_y) \notag \\
{\partial u_2 \over \partial c_x} &= f_x {\partial u_2^{\prime} \over \partial c_x} + 1 \notag \\ &= \rho_2 \rho_1^{-1}(r_{31}u_2^{\prime}-r_{11}) + 1\notag \\
{\partial u_2 \over \partial c_y} &= f_x {\partial u_2^{\prime} \over \partial c_y} \notag \\ &= f_x f_y^{-1} \rho_2 \rho_1^{-1}(r_{32}u_2^{\prime}-r_{12}) \notag
\end{align}\]

\[\begin{align} {\partial v_2 \over \partial f_x} &= f_y {\partial v_2^{\prime} \over \partial f_x} \notag \\ &= f_y f_x^{-1} \rho_2 \rho_1^{-1} (r_{31} v_2^{\prime}-r_{21})f_x^{-1}(u_1 - c_x) \notag \\
{\partial v_2 \over \partial f_y} &= v_2^{\prime} + f_y {\partial v_2^{\prime} \over \partial f_y} \notag \\ &= v_2^{\prime} + \rho_2 \rho_1^{-1}(r_{32} v_2^{\prime}-r_{22})f_y^{-1}(v_1 - c_y) \notag \\
{\partial v_2 \over \partial c_x} &= f_y {\partial v_2^{\prime} \over \partial c_x} \notag \\ &= f_y f_x^{-1} \rho_2 \rho_1^{-1}(r_{31} v_2^{\prime}-r_{21}) \notag \\
{\partial v_2 \over \partial c_y} &= f_y {\partial v_2^{\prime} \over \partial c_y} + 1 \notag \\ &= \rho_2 \rho_1^{-1}(r_{32} v_2^{\prime}-r_{22}) + 1 \notag \end{align}\]

3. Vec6f Jpdxi[2];\(\partial x_2 \over \partial \xi_{21}\)

d_xi_x[0] = new_idepth*HCalib->fxl();
d_xi_x[1] = 0;
d_xi_x[2] = -new_idepth*u*HCalib->fxl();
d_xi_x[3] = -u*v*HCalib->fxl();
d_xi_x[4] = (1+u*u)*HCalib->fxl();
d_xi_x[5] = -v*HCalib->fxl();

d_xi_y[0] = 0;
d_xi_y[1] = new_idepth*HCalib->fyl();
d_xi_y[2] = -new_idepth*v*HCalib->fyl();
d_xi_y[3] = -(1+v*v)*HCalib->fyl();
d_xi_y[4] = u*v*HCalib->fyl();
d_xi_y[5] = u*HCalib->fyl();

计算\(\partial x_2 \over \partial \xi_{21}\)，这个在博客《直接法光度误差导数推导》中已经讲了如何求解。得到的结果是：

\[{\partial x_2 \over \partial \xi_{21}} = \begin{bmatrix} f_x \rho_2 & 0 & -f_x \rho_2 u^{\prime}_2 & -f_xu^{\prime}_2 v^{\prime}_2 & f_x(1 + u^{\prime 2}_2) & -f_x v^{\prime}_2 \\ 0 & f_y\rho_2 & -f_y \rho_2 v^{\prime}_2 & -f_y(1 + v^{\prime 2}_2) & f_y u^{\prime}_2 v^{\prime}_2 & f_y u^{\prime}_2 \\ 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix}
\]

4. VecNRf JIdx[2];对应 \(\partial r_{21} \over \partial x_2\)

J->JIdx[0][idx] = hitColor[1];
J->JIdx[1][idx] = hitColor[2];

计算\(\partial r_{21} \over \partial x_2\)，这个在博客《直接法光度误差导数推导》中已经讲了如何求解。得到的结果是：

\[{\partial r_{21} \over \partial x_{2}} = w_h {\partial I_2[x_2] \over \partial x_{2}} = w_h \begin{bmatrix} g_x, g_y\end{bmatrix}
\]

注意代码中这个变量是8维。

5. VecNRf JabF[2];\({\partial r_{21} \over \partial a_{21}}, {\partial r_{21} \over \partial b_{21}}\)

float drdA = (color[idx]-b0);
...
J->JabF[0][idx] = drdA*hw;
J->JabF[1][idx] = hw;

计算\({\partial r_{21} \over \partial l_{21}}\)，这个在博客《直接法光度误差导数推导》中已经讲了如何求解。得到的结果是（结果有点不符合，需要再对照一下）：

\[\begin{align}
{\partial r_{21} \over \partial a_{21}} &= - w_h e^{a_{21}}I_1[x_1] \notag \\
{\partial r_{21} \over \partial b_{21}} &= -w_h \notag
\end{align}\]

10. JpJdF对应 \(\begin{bmatrix} {\partial r_{21} \over \partial \xi_{21}}^T{\partial r_{21} \over \partial \rho_1} \\ {\partial r_{21} \over \partial l_{21}}^T{\partial r_{21} \over \partial \rho_1}\end{bmatrix}\)

void EFResidual::takeDataF()
{
	std::swap<RawResidualJacobian*>(J, data->J);

	Vec2f JI_JI_Jd = J->JIdx2 * J->Jpdd;

	for(int i=0;i<6;i++)
		JpJdF[i] = J->Jpdxi[0][i]*JI_JI_Jd[0] + J->Jpdxi[1][i] * JI_JI_Jd[1];

	JpJdF.segment<2>(6) = J->JabJIdx*J->Jpdd;
}

1. JI_JI_Jd对应\({\partial r_{21} \over \partial x_2}^T{\partial r_{21} \over \partial \rho_1}\)，2x8 8x1，2x1。
2. JpJdF.segment<6>(0)对应\({\partial x_2 \over \partial \xi_{21}}^T{\partial r_{21} \over \partial x_2}^T{\partial r_{21} \over \partial \rho_1} = {\partial r_{21} \over \partial \xi_{21}}^T{\partial r_{21} \over \partial \rho_1}\)，6x2 2x8 8x1，6x1。
3. JpJdF.segment<2>(6)对应\({\partial r_{21} \over \partial l_{21}}^T{\partial r_{21} \over \partial x_2}{\partial x_2 \over \partial \rho_1} = {\partial r_{21} \over \partial l_{21}}^T{\partial r_{21} \over \partial \rho_1}\)，2x8 8x2 2x1，2x1。
4. JpJdF对应\(\begin{bmatrix} {\partial r_{21} \over \partial \xi_{21}}^T{\partial r_{21} \over \partial \rho_1} \\ {\partial r_{21} \over \partial l_{21}}^T{\partial r_{21} \over \partial \rho_1}\end{bmatrix}\)，8x1。