with depths of points and propose the objective function of our R1PP$n$P algorithm, that is

$$f({\bf{R}},\mu ,{\lambda _i}) = {\sum\limits_{i = 1,i \ne o}^N {\left( {\frac{1}{{{\lambda _i}}}\left\| {{\lambda _i}{{\bf{x}}_i} - {{\bf{x}}_o} - \mu {\bf{R}}{{\bf{S}}_i}} \right\|} \right)} ^2},
\label{eq_objfun1_lambda}$$

where $1/\lambda_i$ is introduced to adjust the weight of point $i$ to eliminate the inequity among points in Eq..

We estimate $\bf{R}$, $\mu$ and $\lambda_i$ $(i=1,...,N,i\neq o)$ by minimizing the objective function , the variables of which consist of two parts: the camera pose {$\bf{R}$, $\mu$} and the relative depths with respect to the control point $\{ \lambda_i \}$. To describe the following algorithm intuitively, we introduce two sets of points: ${\bf{p}}_i$ and ${\bf{q}}_i$. With a randomly selected control $o$, points ${\bf{p}}_i$ are determined by the camera pose {$\bf{R}$, $\mu$}, and points ${\bf{q}}_i$ are determined by depths $\lambda_i$. We have

$${\bf{p}}_i = {\bf{x}}_o + \mu {\bf{RS}}_i,
\label{eq_p}$$

$${\bf{q}}_i = \lambda_i {\bf{x}}_i.
\label{eq_q}$$

\[ht\]

As shown in Fig. \[fig\_bunnyimg\], points ${\bf{p}}_i$ are attached with the virtual object obtained by rotating and scaling the real object around the control point ${\bf{p}}_o = {\bf{q}}_o = {\bf{x}}_o$. ${\bf{q}}_i$ is the projection of ${\bf{p}}_i$ on the corresponding line of sight. The objective function is equivalent to

$$f({{\bf{p}}_i},{{\bf{q}}_i}) 