## Robustness Analysis

Since our features are noisy, we had to allow for a certain error when matching them. In the registration algorithm of Sections 2.3 and 2.4, this is computed from the covariance matrices. The existence of such an error zone allows us to match features that by chance fall in this area. When this probability is sufficiently high, individual false matches can combine themselves (conspiracy) to produce an important matching score.

However, we should note that such a false positive is a correct recognition from a feature point of view (a globally consistent matching) but is incorrect from the image or data point of view. This problem is inherent in the ascendent organization of information: Some important information can be dismissed by simplifying the image as a set of features.

In [28], we developed a generic method to evaluate the probability of matching % features just by chance. The basic framework is the following. Let us assume for the moment that the transformation between the two images is fixed. First, we compute the selectivity y, which is the probability that a random feature (uniformly distributed in the image) falls in a given error zone. Then, we compute the probability that at least % of the m scene features fall in one of the n model error zones. In our analysis, computations are simplified by assuming that all features are randomly distributed.

Now, we will accept a match if there exists one transformation such that at least % features are put into correspondence. Thus, to obtain the mean number of false positives, we just have to integrate over all possible transformations. Let d be the "diameter" of the images. We get the following estimation:

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