and LO-RANSAC [@Chum2003Locally]. Further, this algorithm inserts degeneracy and local optimization processes after generating the minimal-sample model.

During the sampling step, USAC uses a weighted sampling algorithm named PROSAC [@Chum2005Matching], where the initial correspondences are reordered at first based on the descending sort order of brute-force matching scores and correspondences with higher scores are preserved. At the checking stage of the model (homography matrix or fundamental matrix), a correspondence is judged as inlier by Eq. \[eq:LRF3\] with the threshold $t_{\mathbf{H}}$ or by the equation $$\label{eq:LRF3'}
{\frac{{{\left( {\mathbf{y}^{'T}}{\mathbf{F}_{i}}\mathbf{y} \right)}^{2}}}{\left( {\mathbf{F}_{i}}\mathbf{y} \right)_{1}^{2}+\left( {\mathbf{F}_{i}}\mathbf{y} \right)_{2}^{2}+\left( \mathbf{F}_{i}^{T}{\mathbf{y}^{'}} \right)_{1}^{2}+\left( \mathbf{F}_{i}^{T}{\mathbf{y}^{'}} \right)_{2}^{2}}}\le {{t}_\mathbf{F}},$$ where $\mathbf{F}_{i}$ is the $i$th hypothetic fundamental matrix, $t_\mathbf{F}$ is the threshold and ${\bf y} = {[\begin{array}{*{20}{c}}
    {\bf x}&1
    \end{array}]^T}$ (similarly for $\mathbf{y}^{'}$). After generating the minimal-sample model, USAC verifies whether the model is interesting by the SPRT test [@matas2005randomized]. The likelihood ratio can be computed after evaluating $n$ correspondences as $$\label{eq:LRF10}
{{{\xi }_{n}}=\prod\limits_{i=1}^{n}{\frac{p({{r}_{i}}|{\mathbf{H}_{b}})}{p({{r}_{i}}|{\mathbf{H}_{g}})}}},$$ where $\mathbf{H}_{g}$ and $\mathbf{H}_{b}$ respectively represent a “good” model and a “bad” model, $r_{i}$ is equal to 1 if $c_i$ is consistent with the generated model and 0 otherwise, $p(1|\mathbf{H}_g)$ is approximated by the inlier ratio and $p(r_i|\mathbf{H}_b)$ follows a Bernoulli distribution. If the $\xi_n$ is higher than an 