Algorithm 1 Circular model-fitting with outlier rejection
Input:
points	▹ Set of points to be fitted
T	▹ Threshold used to compute the cost function
FAILURE_PROBABILITY	▹ Probability of not finding a correct model
INLIER_PROPORTION	▹ Proportion of inliers in data
Output:
$best2\_a,best2\_b,best2\_R$	▹ Best model parameters found
Initialization
1: $best\_C\leftarrow {10}^{12}$	▹ Initialize to a large number
2: $N\leftarrow log(FAILURE\_PROBABILITY)/log(1-INLIER\_PROPORTIO{N}^3)$
Find model
3: for$i=1$ to N do
Find possible model
4: Take 3 points randomly
5: Build matrices $\mathbf{A}$ and $\mathbf{b}$ using equations (4) and (5) and the 3 sampled points
6: Find model parameters  $a,b,R$  using equations (6) and (7)
7: Compute the cost function  C  using equations (8) to ()
If this possible model is better than the previous one, we keep it
8: if ($C<best\_C$) then
9:  $best\_C\leftarrow C$
10:  $best\_a,best\_b,best\_R\leftarrow a,b,R$
11: end if
12: end for
Refine the model using inliers
13: Select the points  $(x_i,y_i)$  such that  $e_i<T$  using (8)
14: Build matrices $\mathbf{A}$ and $\mathbf{b}$ using equations (4) and (5) and the selected inliers
15: Find model parameters  $best2\_a,best2\_b,best2\_R$  using equations (6) to (7)
16: return $best2\_a,best2\_b,best2\_R$