at most 40 independent solutions. We adopt the algorithm introduced in [@larsson2017efficient] to generate the polynomial solver which can utilize this property to yield an efficient solution.

After solving (\[equ:23\]), we are then able to compute $\mathbf{t}$ from (\[equ:21\]). The CGR parameters ${{s}_{1}},{{s}_{2}}$ and ${{s}_{3}}$ can be recovered from the definitions in (\[equ:16\]) and (\[equ:17\]). The above formulation treats $\rho ,\alpha ,\beta ,\gamma $ as independent unknowns. However, they are related as they are functions of ${{s}_{1}},{{s}_{2}}$ and ${{s}_{3}}$. To recover the minimizer of (\[equ:12\]), we refine the solution using the Newton-Raphson iteration method in (\[equ:15\]). We summarize our least-squares solution in Algorithm \[alg:LS\].

**Input:** ${{n}_{\pi }}$ point-to-plane, ${{n}_{l}}$ point-to-line, and ${{n}_{p}}$ point-to-point correspondences\
**Output:** $\bf{R}$ and $\bf{t}$\

  ---- ---------------------------------------------------------------------------
  1.   Compute the coefficient matrices $\bf{A}$ and $\bf{B}$ in (\[equ:19\]).
  2.   Compute $\bf{Q}$ in (\[equ:22\]).
  3.   Compute the first order optimal conditions (\[equ:23\])
  4.   Solve the equation system (\[equ:23\]) for $\rho ,\alpha ,\beta ,\gamma$.
  5.   Recover $s_1, s_2, s_3$ using (\[equ:17\]).
  6.   Compute $\bf{t}$ from (\[equ:21\]).
  7.   Refine the solution by Newton-Raphson iteration (\[equ:15\]).
  ---- ---------------------------------------------------------------------------

Minimal Solution {#sec:min}
================

Unlike most pose estimation problems having an unique minimal configuration, our problem has multiple minimal configurations. Specifically, any combination of 