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found that relevant textural indices are given by the information dimension D(1) and the curvature parameter c = 2|D (1)|/(D(0) D(1)), where D(q) is de ned by D(q) = (q)/(q 1), with an obvious modi cation for q = 1 In particular, these indices allow us to separate the three strata present in these logs For instance, D(1) is correlated with the degree of heterogenity: a formation which is more heterogenous translates into smaller D(1) [SAU 97] 118 Edge detection 1181 Introduction In the multifractal approach to edge detection, an image is modeled by a measure , or, more precisely, a Choquet capacity [LEV 98] A Choquet capacity is roughly a measure which does not need to satisfy the additivity requirement This distinction will not be essential for our discussion below, and the reader may safely assume that we are dealing with measures See 1 for more precise information on this The basic assumptions underlying the multifractal approach to image segmentation are as follows: The relevant information for the analysis can be extracted from the H lder regularity of Three levels contribute to the perception of the image: the pointwise H lder regularity of at each point, the variation of the H lder regularity of in local neighborhoods and the global distribution of the regularity in the whole scene The analysis should be translation and scale invariant Let us brie y compare the multifractal approach to traditional methods such as mathematical morphology (MM), and gradient based methods, or more generally image multiscale analysis (IMA): As in MM and IMA, translation and scale invariance principles are ful lled There is no so-called local comparison principle or local knowledge principle , ie the decision of classifying a point as an edge point is not based only on local information On the contrary, it is considered useful to analyze information about whole parts of the image at each point The most important difference between traditional and multifractal methods lies in the way they deal with regularity While the former aims at obtaining smoother versions of the image (possibly at different scales) in order to remove irregularities, the latter tries to extract information directly from the singularities Edges, for instance, are not considered as points where large variations of the signal still exist after smoothing, but as points whose regularity is different from the background regularity in the raw data Such an approach makes sense for complex images, in which the relevant structures are themselves irregular Indeed, an implicit assumption of MM and IMA is that the useful information lies at boundaries between originally smooth regions, so that it is natural to lter the image However, there are cases (eg.

nodes that they call insider and outsider nodes The insider nodes are assumed to be trusted and are the ones which execute the IDS Thus all the explanation above applies to insider nodes The authors had to consider these types of special ad hoc networks since the proposed IDS placement algorithms (such as random placement, GO-DOM, approximation to optimal) are not secure to attacks from untrusted nodes Rather the security of these algorithms is derived from the fact that the nodes executing these algorithms are trusted The same ideas can be applied to general ad hoc networks provided secure versions of the IDS placement algorithms can be designed This was the reason why we have described the algorithms more generally here An important assumption in [80, 83] is that the nodes that execute IDS in the promiscuous mode operate perfectly, but this is not always true A node may fail to capture and analyze the packets perfectly due to many reasons This failure may happen due to faults in the node, power-saving operations that the node might use or if the attack has been designed to evade the IDS, or if the attacker successfully launches a DoS attack on the node, or if the node does not receive the packet due to the vagaries of promiscuous mode monitoring such as collisions (see Section 552), poor transmission quality in wireless links, and so on Such a failure model is considered in [81], where the authors extend the algorithms proposed in [80, 83] The authors, however, assume that a node is not compromised and hence never reports falsely on the occurrence of malicious packets To address such failure, the authors propose to make use of redundancy whereby more than one node in a neighborhood monitors packets promiscuously We show this in Figure 54 In this gure the shaded nodes indicate the nodes that are executing IDS We see that node A is protected by four other nodes that execute IDS As a result, traf c emanating from node A will always be analyzed if at least three of the four nodes (C, D, E, and F) have not failed at any point in time A question is related to the number of nodes in a neighborhood, denoted as k, that have to execute IDS to reach the desirable level of detection The authors address this by relating the number k to the desired detection probability Using this number k, the authors provide extensions to the mathematical models used in the previous techniques to ensure that every node is covered by k other nodes that execute IDS In addition, the approximation algorithms are also extended by considering a node as satis ed if it is covered by more than k nodes In each iteration, the approximation algorithms then select the node with the maximum number of unsatis ed neighbors to execute the IDS.

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