Merge pull request #49 from hnqiu/master

Fix: duplicate text & a typo in lieGroup.tex

chapters/lieGroup.tex

@@ -387,8 +387,7 @@ \subsection{Derivative on $\mathrm{SO}(3)$}
 \mathop {\min }\limits_{\mathbf{T}} J(\mathbf{T} ) = \sum_{i=1}^{N} \left\| {\mathbf{z}_i - \mathbf{Tp}_i} \right\|^2_2.
 \end{equation}
 
-Because of the noise, the real observed data is not absolutely the same as the one we computed from the observation model, so we can calculate the error of predicted observation with the real one: 
-To solve such an optimized problem (which is a least square problem), we need to calculate the derivative of $J$ by $\mathbf{T}$. We leave the least square problem to the next section. Here we just want to clarify that we normally have some functions that have rotations or transforms as their variables. We have to adjust those rotations or transforms to find a better/best estimation. But, as we mentioned before, since $\mathrm{SO}(3)$ and $\mathrm{SE}(3)$ do not have a well-defined addition (they are just groups), so the derivatives cannot be defined in their common form. If we treat the $\mathbf{R}$ or $\mathbf{T}$ as common matrices, we have to introduce the constraints into our optimization. 
+To solve such an optimization problem (which is a least square problem), we need to calculate the derivative of $J$ by $\mathbf{T}$. We leave the least square problem to the next section. Here we just want to clarify that we normally have some functions that have rotations or transforms as their variables. We have to adjust those rotations or transforms to find a better/best estimation. But, as we mentioned before, since $\mathrm{SO}(3)$ and $\mathrm{SE}(3)$ do not have a well-defined addition (they are just groups), so the derivatives cannot be defined in their common form. If we treat the $\mathbf{R}$ or $\mathbf{T}$ as common matrices, we have to introduce the constraints into our optimization. 
 
 However, from the perspective of Lie algebra, since it consists of vectors, it has a good addition operation. Therefore, there are two ways to solve the problem of derivation using Lie algebra: