Robust Geometric Model Estimation Based on Scaled Welsch q-Norm

Robust estimation, which aims to recover the geometric transformation from outlier contaminated observations, is essential for many remote sensing and photogrammetry applications. This article presents a novel robust geometric model estimation method based on scaled Welsch <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-norm (<inline-formula> <tex-math notation="LaTeX">$l_{q}$ </tex-math></inline-formula>-norm, <inline-formula> <tex-math notation="LaTeX">$0 < q < 1$ </tex-math></inline-formula>). The proposed algorithm integrates a scaled Welsch weight function into the <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-norm framework. It, thus, inherits all the advantages of the standard <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-norm, i.e., fast and robust. The parameter sensitivity of the standard <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-norm is also largely alleviated by integrating such a weight function. These make the proposed algorithm much superior to RANSAC-type methods in real-life applications. We formulate the new cost function as an augmented Lagrangian function (ALF) and divide the ALF into two subproblems [a <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-norm penalized least-squares (<inline-formula> <tex-math notation="LaTeX">$l_{q}$ </tex-math></inline-formula>LS) problem and a weighted least-squares (WLS) problem] by using alternating direction method of multipliers (ADMM) method. For the WLS problem, we introduce a coarse-to-fine strategy into the iterative reweighted least-squares (IRLS) method. We change the weight function by decreasing its scale parameter. This strategy can largely avoid that the solver gets stuck in local minimums. We adapt the proposed cost into classical remote sensing tasks and develop new robust feature matching (RFM), robust exterior orientation (REO), and robust absolute orientation (RAO) algorithms. Both synthetic and real experiments demonstrate that the proposed method significantly outperforms the other compared state-of-the-art methods. Our method is still robust even if the outlier rate is up to 90%.

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