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2 changes: 1 addition & 1 deletion chapters/backend2.tex
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Expand Up @@ -115,7 +115,7 @@ \subsection{Residuals and Jacobians}

According to the principle of graph optimization, the equation will not hold exactly in practice. Hence, we set up the least-square error and then, as before, discuss the derivative of the error with respect to the optimized variable. Here, we move the $\bm{T}_{ij}$ of the above equation to the right side of the equation to construct the error $\bm{e}_{ij}$:
\begin{equation}
\bm{e}_{ij} = \Delta \bm{\xi}_{ij} \ln \left( \bm{T}_{ij}^{-1} \bm{T}_i^{-1} \bm{T}_j \right)^\vee
\bm{e}_{ij} = \ln \left( \bm{T}_{ij}^{-1} \bm{T}_i^{-1} \bm{T}_j \right)^\vee
\end{equation}

Note that there are two optimization variables: $\bm{\xi}_i$ and $\bm{\xi}_j$, so we find the derivative of $\bm{e}_{ij}$ about these two variables. According to the derivation method of Lie algebra, give $\bm{\xi}_i$ and $\bm{\xi}_j$ a left disturbance: $ \bm{\delta \xi}_i$ and $\bm{\delta \xi}_j$. Then the error becomes:
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2 changes: 1 addition & 1 deletion chapters/mapping.tex
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Expand Up @@ -571,7 +571,7 @@ \subsection{Inverse Depth Filter}
\item There may be points that are very far away in some outdoor applications or even at infinity. It is difficult to cover these points in our initial value, and there will be some numerical difficulties in describing the large depth values with Gaussian distribution.
\end{enumerate}

Thus, the inverse depth came into being. People found in the simulation that the hypothesis that the inverse depth forms a Gaussian distribution is more effective {\cite{Civera2008}}. Later, in practical applications, the inverse depth also has better numerical stability, which gradually becomes a general and standard practice in the existing SLAM systems {\cite{Forster2014, Engel2014, Mur- Artal2015}}.
Thus, the inverse depth came into being. People found in the simulation that the hypothesis that the inverse depth forms a Gaussian distribution is more effective {\cite{Civera2008}}. Later, in practical applications, the inverse depth also has better numerical stability, which gradually becomes a general and standard practice in the existing SLAM systems {\cite{Forster2014, Engel2014, Mur-Artal2015}}.

It is not complicated to change the demonstration program from positive depth to inverse depth. Just change $d$ to the inverse depth $d^{-1}$ in the previous depth's derivation. We also leave this change as an exercise for readers to complete.

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2 changes: 1 addition & 1 deletion chapters/vo1.tex
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Expand Up @@ -872,7 +872,7 @@ \subsection{Discussion}
\section{3D−2D:PnP}
PnP (Perspective-n-Point) is a method to solve 3D to 2D motion estimation. It describes how to estimate the camera's pose when the $n$ 3D space points and their projection positions are known. As mentioned earlier, the 2D-2D epipolar geometry method requires eight or more point pairs (take the eight-point method as an example), and there have problems with initialization, pure rotation, and scale. However, if the 3D position of one of the feature points in the two images is known, then we need at least three pairs (and at least one additional point to verify the result) to estimate the camera motion. The 3D position of the feature point can be determined by triangulation or the depth map of an RGB-D camera. Therefore, in binocular or RGB-D visual odometry, we can directly use PnP to estimate camera motion. While in the monocular case, initialization must be conducted before using PnP. The 3D-2D method does not require epipolar constraints and can obtain better motion estimation in a few matching points. It is the most important pose estimation method.
There are many ways to solve PnP problems, for example, P3P {\cite{GaoHouTangEtAl2003}}, direct linear transformation (DLT), EPnP (Efficient PnP) {\cite{LepetitMoreno-NoguerFua2008 }}, UPnP {\cite{Penate-SanchezAndrade-CettoMoreno-Noguer2013}}, etc. In addition, nonlinear optimization can be used to construct a least-square problem and iteratively solve it, which is commonly called the bundle adjustment. Let's look at DLT first, and then we will explain the bundle adjustment approach.
There are many ways to solve PnP problems, for example, P3P {\cite{GaoHouTangEtAl2003}}, direct linear transformation (DLT), EPnP (Efficient PnP) {\cite{LepetitMoreno-NoguerFua2008}}, UPnP {\cite{Penate-SanchezAndrade-CettoMoreno-Noguer2013}}, etc. In addition, nonlinear optimization can be used to construct a least-square problem and iteratively solve it, which is commonly called the bundle adjustment. Let's look at DLT first, and then we will explain the bundle adjustment approach.
\subsection{Direct Linear Transformation}
Consider such a problem: we know the 3D positions of a point set and their projections in the camera, now we want to find the camera's pose. This problem can be used to solve the camera pose when a given map and image are given. If the 3D point is regarded as a point in another camera coordinate system, it can also solve the two cameras' relative motion problem. We will start with simple questions.
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21 changes: 21 additions & 0 deletions ref.bib
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Expand Up @@ -5806,6 +5806,27 @@ @InProceedings{Handa2014
month = {May},
}


@InProceedings{Marton2009,
author = {Zoltan Csaba Marton and Radu Bogdan Rusu and Michael Beetz},
title = {{On Fast Surface Reconstruction Methods for Large and Noisy Datasets}},
booktitle = {Proceedings of the IEEE International Conference on Robotics and Automation (ICRA)},
year = {2009},
address = {Kobe, Japan},
month = {May 12-17},
}

@Article{Alexa2003,
author = {Alexa, Marc and Behr, Johannes and Cohen-Or, Daniel and Fleishman, Shachar and Levin, David and Silva, Claudio T.},
title = {Computing and rendering point set surfaces},
journal = {IEEE Transactions on visualization and computer graphics},
year = {2003},
volume = {9},
number = {1},
pages = {3--15},
publisher = {IEEE},
}

@Comment{jabref-meta: databaseType:bibtex;}

@Comment{jabref-meta: grouping:
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