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in medical imaging, satellite or radar imaging) where the meaningful features are essentially singular As in MM, and contrarily to IMA, the multifractal approach does not assume that there is a universal scheme for image analysis Rather, depending on what we are looking for, different measures may be used to describe the image Both MM and IMA consider the relative values of the gray levels as the basic information Here the H lder regularity is considered instead This again is justi ed in situations where the important information lies in the singularity structure of the image Throughout the rest of this section, we make the following assumption: f := fh = fg The simplest approach then consists of de ning a measure (or, often, a sequence of Choquet capacities on the image), calculating its multifractal spectrum, and classifying each point according to the corresponding value of ( , f ( )), both in a geometric and a probabilistic fashion The value of gives a local information about the pointwise regularity: for a xed capacity, an ideal step edge point in an image without noise is characterized by a given value The value of f ( ) yields a global information: a point on a smooth contour belongs to a set T whose dimension is 1, a point contained in a homogenous region has f ( ) = 2, etc The probabilistic interpretation of f ( ) corresponds to the fact that a point in a homogenous region is a frequent event, an edge-point is a rare event, and, for instance, a corner an even rarer event (see Figures 1119 and 1120) Indeed, if too many edge points are detected, it is in general more appropriate to describe these points as belonging to a homogenous (textured) zone

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some peaks above 1 and up to 3 The exponent values in [0, 1], which imply that the signal is continuous but not differentiable, con rm the visual impression of great irregularity We can go beyond this qualitative comment by observing that the periods where something occurs have a de nite signature in terms of the exponents: they are characterized by a dramatic increase of l followed by very small values, below 02 Let us examine some examples The most remarkable point of the local regularity graph is its maximum at abscissa 2,018, with an amplitude of 3 The most singular points, ie those with the smallest exponent, are situated just after this maximum: the exponent is around 02 for the abscissae between 2,020 and 2,050, and of the order of 005 between points 2,075 and 2,100 These two values are distinctly below the exponent average, which is 04 (the variance being 0036) Calculations show that less than 10% of the log points possess an exponent smaller than 02 This remarkable behavior corresponds to the crash of October 19, 1987 which occurs at abscissa 2,036, in the middle of the rst zone of points with low regularity after the maximum: the most irregular days of the entire signal are thus, as expected, situated in the weeks which followed the crash It is worthwhile noting that this fact is much more apparent on the regularity signal than on the original log, where only the crash appears clearly, with the subsequent period not displaying remarkable features Let us now consider another area which contains many points of low regularity along with some isolated regular points (ie having l > 1) It corresponds to the zone between abscissae 4,450 and 4,800: this period approximatively corresponds to the Asian crisis that took place between January 1997 and June 1998 (analysts do not agree upon the exact dates of the beginning and the ending of this crisis: some of them date its beginning in mid-1997 or its end towards the end of 1999, or much later) On the graph of the log, we can observe that this period seems slightly more.

irregular than others In terms of exponents, we notice that it contains two maxima, with values greater than 1, both followed by low regularity points: this area comprises a high proportion of irregular points, since 12% of its points have an exponent lower than 015 This proportion is three times higher than that observed in the whole log The analysis just performed has remained at an elementary level However, it has allowed us to show that major events have repercussions on the evolution of the local H lder exponent and that the graph of l emphasizes features not easily visible on the original log

Other schemes have also been proposed for dealing with the problem of detecting uncooperative nodes Marti et al [84] proposed a simple scheme in which each node utilizes a watchdog for monitoring its neighbors and identifying misbehaving nodes (eg nodes that do not forward packets) Each node utilizes a pathrater that ensures that routes used for routing do not contain the misbehaving nodes The effectiveness of this scheme is limited because each node acts independently and there is no cooperation among nodes in exchanging information about misbehaving nodes This scheme was shown through simulations to increase throughput up to 27 percent in the presence of misbehaving nodes while increasing overhead by a few percentage points So far we have been assuming that each node acts independently in order to capture and analyze traf c, but as discussed earlier, cooperation amongst the nodes executing the IDS functions might result in more ef cient monitoring of the network We discuss this aspect next 542 Cooperative Intrusion Detection

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