Coresets for Triangulation

• Published in: IEEE transactions on pattern analysis and machine intelligence Vol. 40; no. 9; pp. 2095 - 2108
• Main Authors: Zhang, Qianggong, Chin, Tat-Jun
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.09.2018; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Approximation; Approximation algorithms; Approximation error; Cameras; Computer vision; Convergence; Coresets; generalised linear programming; Image reconstruction; Indexes; Iterative methods; multiple view geometry; Optimization; Three-dimensional displays; Triangulation
• ISSN: 0162-8828; 1939-3539; 2160-9292
• DOI: 10.1109/TPAMI.2017.2750672

Multiple-view triangulation by ℓ ∞ minimisation has become established in computer vision. State-of-the-art ℓ ∞ triangulation algorithms exploit the quasiconvexity of the cost function to derive iterative update rules that deliver the global minimum. Such algorithms, however, can be computationally costly for large problem instances that contain many image measurements, e.g., from web-based photo sharing sites or long-term video recordings. In this paper, we prove that ℓ ∞ triangulation admits a coreset approximation scheme, which seeks small representative subsets of the input data called coresets . A coreset possesses the special property that the error of the ℓ ∞ solution on the coreset is within known bounds from the global minimum. We establish the necessary mathematical underpinnings of the coreset algorithm, specifically, by enacting the stopping criterion of the algorithm and proving that the resulting coreset gives the desired approximation accuracy. On large-scale triangulation problems, our method provides theoretically sound approximate solutions. Iterated until convergence, our coreset algorithm is also guaranteed to reach the true optimum. On practical datasets, we show that our technique can in fact attain the global minimiser much faster than current methods.