{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-quaternion-based-certifiably-optimal","title":"A Quaternion-based Certifiably Optimal Solution to the Wahba Problem with Outliers","arxiv_id":"1905.12536","date":"2019-05-29","proceeding":"ICCV 2019 10","authors":["Heng Yang","Luca Carlone"],"abstract":"The Wahba problem, also known as rotation search, seeks to find the best rotation to align two sets of vector observations given putative correspondences, and is a fundamental routine in many computer vision and robotics applications. This work proposes the first polynomial-time certifiably optimal approach for solving the Wahba problem when a large number of vector observations are outliers. Our first contribution is to formulate the Wahba problem using a Truncated Least Squares (TLS) cost that is insensitive to a large fraction of spurious correspondences. The second contribution is to rewrite the problem using unit quaternions and show that the TLS cost can be framed as a Quadratically-Constrained Quadratic Program (QCQP). Since the resulting optimization is still highly non-convex and hard to solve globally, our third contribution is to develop a convex Semidefinite Programming (SDP) relaxation. We show that while a naive relaxation performs poorly in general, our relaxation is tight even in the presence of large noise and outliers. We validate the proposed algorithm, named QUASAR (QUAternion-based Semidefinite relAxation for Robust alignment), in both synthetic and real datasets showing that the algorithm outperforms RANSAC, robust local optimization techniques, global outlier-removal procedures, and Branch-and-Bound methods. QUASAR is able to compute certifiably optimal solutions (i.e. the relaxation is exact) even in the case when 95% of the correspondences are outliers.","url_abs":"https://arxiv.org/abs/1905.12536v4","url_pdf":"https://arxiv.org/pdf/1905.12536v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"a-quaternion-based-certifiably-optimal","repo_url":"https://github.com/MIT-SPARK/STRIDE","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}},{"paper_slug":"a-quaternion-based-certifiably-optimal","repo_url":"https://github.com/MIT-SPARK/TEASER-plusplus","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"a-quaternion-based-certifiably-optimal","repo_url":"https://github.com/MIT-SPARK/estimation-contracts","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"BSD-2-Clause"}},{"paper_slug":"a-quaternion-based-certifiably-optimal","repo_url":"https://github.com/StrikerCC/Teaser-plusplus-testing","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1905.12536","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1905.12536"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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