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Compute Graphics Forum (Proceedings of the Symposium on Geometry Processing 2013)

Rigid registration of two geometric data sets is essential in many applications, including robot navigation, surface reconstruction, and shape matching. Most commonly, variants of the Iterative Closest Point (ICP) algorithm are employed for this task. These methods alternate between closest point computations to establish correspondences between two data sets, and solving for the optimal transformation that brings these correspondences into alignment. A major difficulty for this approach is the sensitivity to outliers and missing data often observed in 3D scans. Most practical implementations of the ICP algorithm address this issue with a number of heuristics to prune or re-weight correspondences. However, these heuristics can be unreliable and difficult to tune, which often requires substantial manual assistance. We propose a new formulation of the ICP algorithm that avoids these difficulties by formulating the registration optimization using sparsity inducing norms. Our new algorithm retains the simple structure of the ICP algorithm, while achieving superior registration results when dealing with outliers and incomplete data.

Solving for optimal rigid alignment for incomplete geometry. (a) Traditional least-squares ICP does not distinguish between inliers and outliers, resulting in poor alignment. (b) The $\ell_1$-ICP is more robust, but still cannot cope with the large amount of correspondence outliers. We show that $\ell_p$-ICP, with $p \in [0,1]$ robustly handles large amounts of noise and outliers. Bottom rows: Illustration of sparsity-inducing norms. ($\alpha$) Regularization of the input vector (black-framed) with an $\ell_p$-norm leads to increased sparsity as we decrease the value of $p$. ($\beta$) The vector of complements provides an indication of how much its corresponding entry contributes in the optimization. Large outliers (red) contribute progressively less as we decrease $p$.