Some trade-offs and a new algorithm for geometric hashing

(a) (b) Affine coordinate plane Image Plane α β β α Image Plane plane coordinate Affine Figure 4. The increases error regions and the effect of the error partitioning: The measured locations of the image basis are inside the error regions around their true locations (three small squares at left and bottom of (a)) but the transformation error implies that verification of some other point must be done with a larger region (large square). Correspondingly , the parallelogram-shaped voting region in the affine plane is large ((b)). With to the partitioned error method, the error is divided into some regions (here, d = 2 and we get 4 regions), and much smaller error regions (c) and voting area (d) are implied. Note that the poor reliability of the bare algorithm was indeed improved with the partitioned voting, but was associated with a very substantial increase in time. One cause for the poor performance is that no further verification was done and the vote obtained by the hashing itself, served as a score. To improve this weak point, we increased the candidate list length to 10, implemented a variation of Jacobs verification algorithm [8], and used it to make the final decision. We found that searching in a 6-dimensional error space is too costly and instead searched in two 3-dimensional spaces which are projections of the full space (a technique similar to that used by Cass [3]). The use of projections reduces the complexity of the verification from O(mnn 6) to O(mnn 3), and practically, increases the running time only by about 5%. We also found that it always identified the correct match, so it seems that the decrease in reliability is not substantial (2 middle columns). Note that verification by itself is not sufficient to get high reliability,. We tested it and found that in almost all cases, the reason was that the candidate list did not contain the correct hypothesis. Reliability was highest when both partitioned voting and verification were used. The next step is to improve the speed by stopping after a relatively small, fixed, number (k) of randomly drawn image bases. Requiring that a correct image basis is selected with probability 1 ? , implies that k = log log(1?(mc n) 3) ? log ? n mc 3 Note that 1 ? is lower than the probability of successful detection of a correct match. Setting = 0:01 …