consider the effective number of correspondences as [@briales2017convex]. Specifically, the effective number of correspondences for ${{n}_{\pi }}$ point-to-plane, ${{n}_{l}}$ point-to-line, and ${{n}_{p}}$ point-to-point correspondences is calculated as $N = {{n}_{\pi }} + 2{{n}_{l}} + 3{{n}_{p}}$. Given an $N$, we randomly generate a combination of ${n}_{\pi }$, ${{n}_{l}}$ and ${{n}_{p}}$ whose effective number of correspondences is $N$.

  Algorithm                      time ($ms$)
  ----------------------------- -------------
  Our least-squares algorithm       2.95
  DLSSol                            20.0
  Our minimal algorithm             0.296
  DMinSol                           0.599

  : Computational time. For the least-squares problem, we consider the time to solve (13), which is independent of the number of correspondences.[]{data-label="table:time_vs_orig"}

**Effect of intermediate unknowns** We introduce intermediate unknowns to simplify the least-squares and the minimal problems. To verify their benefit, we evaluate the performance of the direct least-squares solution (DLSSol) from solving (13), and the direct minimal solution (DMinSol) from solving the equation system from the numerator of (9) for the minimal problem. We employed the algorithm in [@larsson2017efficient] to generate the solvers. For the least-squares problem, we run 2000 trails for each $N \in [7,15]$. Fig. \[fig:vs\_DLSSol\] shows the results. Directly solving (13) can recover the global minimizer of (4) in theory. However, the large mean errors of DLSSol verify that this polynomial solver 