differences between points ${\bf{p}}_i$ and ${\bf{q}}_i$ in the 3D space. However, Fig.\[fig\_newmu\] demonstrates that by using this method, the updating rate of $\mu$ may be slow in situations when the range of depths is large. To achieve faster convergence rate, we update the scale factor $\mu$ by comparing the projected image coordinates of ${\bf{p}}_i$, which are denoted as ${{\bf{v}}_i}$, and the real image points ${\bf{x}}_i$. Denoting matrices ${\bf{B}} = \left[ {{{\bf{v}}_1} - {{\bf{x}}_{o}},...,{{\bf{v}}_N} - {{\bf{x}}_o}} \right]_{3\times N}$ and ${\bf{C}} = \left[ {{{\bf{x}}_1} - {{\bf{x}}_o},...,{{\bf{x}}_N} - {{\bf{x}}_o}} \right]_{3\times N}$. $\mu$ is updated by

$$%\mu = {{{\left\| {{{\rm{vector(}\bf{A}\rm{)}}}} \right\|}}} / {{{\left\| {{{\rm{vector(}\bf{S}\rm{)}}}} \right\|}}}.
\Delta \mu  = \left\| {{\rm{vector}}({\bf{C}})} \right\|/\left\| {{\rm{vector}}({\bf{B}})} \right\|
\label{eq_mu1}$$

$${\mu _{{\rm{new}}}} = {\mu _{{\rm{old}}}}\Delta \mu
\label{eq_mu2}$$

Finally, points ${\bf{p}}_i$ are updated according to Eq..

Proof of Convergence
--------------------

We first provide the mathematical proof of the convergence of R1PP$n$P when not using $1/\lambda_i$ as weights in the objective function . $k$ denotes the number of iterations, ${\bf{q}}_i^{(k+1)}$ is obtained by vertically projecting ${\bf{p}}_i^{(k)}$ to the line of sight $i$, and ${\bf{q}}_i^{(k+1)}$ and ${\bf{q}}_i^{(k)}$ are on the line of sight $i$. Hence, the three points, ${\bf{p}}_i^{(k)}$,${\bf{q}}_i^{(k)}$, and ${\bf{q}}_i^{(k+1)}$ comprise a right-angled triangle. Therefore, for each index $i,i \neq o$,

$$\left\| 