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Extra info for Advanced Concepts for Intelligent Vision Systems: 12th International Conference, ACIVS 2010, Sydney, Australia, December 13-16, 2010, Proceedings, Part II
Of course, we have to apply the algorithm to all matched keypoint pairs so that the overall complexity of the topological consistency veriﬁcation is equal to O(n2 + nm3 ). The last step is grouping keypoint pairs (sub-graph detection in the topological graph). We employ the union-ﬁnd algorithm, which has the computational complexity of O(log ∗ n) for a set of size n. There are total nm neighbors for n keypoint pairs so that the overall complexity of the union-ﬁnd algorithm is O(mnlog ∗ n). Altogether, assuming that log ∗ n < f and log ∗ n < m (which is generally true) the computational complexity of the topology-based near-duplicate fragment detection in a pair of images is not larger than O(f n2 ) + O(n2 + nm3 ) + O(f n2 + nm2 ) = O(f n2 + nm3 ) (1) The ﬁrst component (O(f n2 )) of the complexity estimate is related to keypoint matching, while the second one O(nm3 )) corresponds to the analysis of topological consistencies.
The geometric approach captures only parts of objects; the topological approach captures objects with some background. Fig. 5. Near-duplicates matched using the topological approach (Harris-Aﬃne detector and SIFT descriptor applied) randomly selected 1000 distracting image pairs (without similar objects). The database and the method are made publicly available2 . e. the entropy-based approach . pl/~ visible 26 ´ A. Sluzek and M. Paradowski The approach has been tested on our database using publicly available executables3 .
E. pairs of matched keypoints are analyzed topologically within neighborhoods of a limited size only so that the image geometry still plays a certain role. If matched keypoint pairs belong to similar fragments in the matched images, the keypoint neighbors should be similarly distributed in both images (even though the exact geometry can be diﬀerent). Therefore, we propose a heuristic topological constraint representing such similar distributions. The constraint is based on the orientation of vectors connecting a given keypoint to its matched neighbors.