Re-weighting and 1-Point RANSAC-Based P<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq1-2871832.gif"/></alternatives></inline-formula>P Solution to Handle Outliers

The ability to handle outliers is essential for performing the perspective-<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq2-2871832.gif"/></alternatives></inline-formula>-point (P<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq3-2871832.gif"/></alternatives></inline-formula>P) approach in practical applications, but conventional RANSAC+P3P or P4P methods have high time complexities. We propose a fast P<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq4-2871832.gif"/></alternatives></inline-formula>P solution named R1PP<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq5-2871832.gif"/></alternatives></inline-formula>P to handle outliers by utilizing a soft re-weighting mechanism and the 1-point RANSAC scheme. We first present a P<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq6-2871832.gif"/></alternatives></inline-formula>P algorithm, which serves as the core of R1PP<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq7-2871832.gif"/></alternatives></inline-formula>P, for solving the P<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq8-2871832.gif"/></alternatives></inline-formula>P problem in outlier-free situations. The core algorithm is an optimal process minimizing an objective function conducted with a random control point. Then, to reduce the impact of outliers, we propose a reprojection error-based re-weighting method and integrate it into the core algorithm. Finally, we employ the 1-point RANSAC scheme to try different control points. Experiments with synthetic and real-world data demonstrate that R1PP<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq9-2871832.gif"/></alternatives></inline-formula>P is faster than RANSAC+P3P or P4P methods especially when the percentage of outliers is large, and is accurate. Besides, comparisons with outlier-free synthetic data show that R1PP<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq10-2871832.gif"/></alternatives></inline-formula>P is among the most accurate and fast P<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq11-2871832.gif"/></alternatives></inline-formula>P solutions, which usually serve as the final refinement step of RANSAC+P3P or P4P. Compared with REPP<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq12-2871832.gif"/></alternatives></inline-formula>P, which is the state-of-the-art P<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq13-2871832.gif"/></alternatives></inline-formula>P algorithm with an explicit outliers-handling mechanism, R1PP<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq14-2871832.gif"/></alternatives></inline-formula>P is slower but does not suffer from the percentage of outliers limitation as REPP<inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq15-2871832.gif"/></alternatives></inline-formula>P.