The rst error analysis of the geometric hashing technique was done by Grimson, Huttenlocher and Jacobs [GHJ91]. They used a uniform model for sensor error, and
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concluded several things: rst, that when sensor error is taken into account and a particular image triplet is chosen in the recognition stage, then the regions in the hash table that are consistent with the sensed position of any fourth image point (step (ii)) are ellipses whose center and axes are dependent on conguration of the image basis.
Thus, the error regions themselves cannot be taken into account at the preprocessing stage, but rather must be computed in step (ii) of the recognition stage, and all model bases in the region incremented.
Second, they derived the probability that a single random image basis would match any model basis as follows: Suppose the probability that a single random image point will be in a region consistent with any model point is on average. Let us x the model basis that we are interested in. The probability that a single image point will fall in any region consistent with this particular model basis is
p = 1;(1;)m
since there are m places in the index table where this basis appears, and the image point must avoid all of them. However, there are n image points, so the probability that this particular model basis gets at leastk votes is
wk = 1;kX;1
i=0
nk
!
pk(1;p)n;k
This is the probability that a single random image basis matches a xed model basis.
There are m(m;1)(m;2) bases in the hash table (we will use mh3i to denote this expression), so probability that this image basis will contribute at least k votes to any model basis is
1;Pfimage basis contributes k to no model basisg
= 1;(1;wk)mh3i
= 1;
"
kX;1 i=0
nk
!
pk(1;p)n;k
#mh3i
This is the probability that in a single pass through the recognition stage of the geometric hashing algorithm, the image basis being tested will nd a match of at least size k at random. They presented an analogous analysis for alignment, which is identical except the roles of n and m are switched. Thus they conclude that the probability of an overall false positive was greater for the geometric hashing case than for alignment, because n > m prevails rather generally.
The dierence in the positions ofn and m in their analysis was based on the assump-tion that alignment counts at most one image point per model disk whereas geometric hashing counts all image points that appear in the model disk. This is equivalent to the distinction between Schemes 1 and 3 that was discussed in Chapter 5. However, the geometric hashing scheme can be easily modied to use either collection method by keeping track of whether a point has already been collected from that particular
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location; in fact, this is the method generally used. The alignment method can also easily use either collection scheme. Therefore, the probability of error is equivalent for either method when using comparable collection schemes.
Interestingly enough, we have shown the opposite of the conclusion of [GHJ91] with regard to the probability of error as a function of collection method. We concluded in Chapter 5 that the performance of Scheme 1 (counting all image points that fall in a weight disk) is better than that of Scheme 3 (at most one image point per weight disk); this is because even when the clutter is very high, the expected number of points falling inside a correctly hypothesized weight disk is always greater than the expected number of points falling inside a random weight disk regardless of the clutter level.
Therefore, the expected value of the sum of points appearing in the disk will always be higher if the disk is correct. The other collection scheme saturates with noise once there is a high probability that at least one image point will appear per weight disk.
This nding does not contradict the analysis in [GHJ91] since the weighting scheme used in that analysis was based on a uniform model for sensor noise.
To apply our error analysis we would have to project entire error ellipses into the hash table as described in [GHJ91], but in the Gaussian error model case, the ellipses would be smaller and we would increment weights for model bases instead of votes.
Now we can appreciate why it was important to limit the Gaussian weight disk to a nite size. If the distribution were unbounded, we would have to go through the entire table and contribute some small weight to every basis, thus changing the run time of the original geometric hashing algorithm. Applying our method to this domain results in being able to derive triples of (, PF, PD) for the termination step of the geometric hashing algorithm (step (b)).
Furthermore, we can easily calculate the probability that a particular image basis will match any model basis, as was done in [GHJ91]. We already mentioned that the geometric hashing technique can be considered a \ltering" step which provides candidate model to image basis correspondences to some more expensive verication step. Then the technique would be considered to break down once the number of matches it oers up is too high.
Suppose we are willing to verify (by alignment or any other verication technique) all bases that pass our threshold, as long as there arek of them. Then, an overall false positive is the combined event that the three image points being tested do not arise from the model, yet more than k model bases \look good". An overall true positive is the combined event that the three image points do arise from the model, that k model bases pass the test, and of these, one of them is the correct one. We will call these combined events F and D, and
PfFg = 1;Pki=0mh3ii PiF(1;PF)mh3i;i PfDg = PDPki=0;1
mh3i i
PiF(1;PF)mh3i;i
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