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 $p$.