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The source could populate the seqnum eld above using a hash function on a node s unique identity pseudonym In order to populate the tr_dest eld, the source needs to share secret trapdoor keys with the destination This secret trapdoor key will ensure that only the destination can obtain the information Any node on route to the destination will not be able to obtain this information since they do not possess the relevant keys This ensures that the identity of the destination is kept secret Finally, to populate the onion, the source has different choices In one option the source randomly picks a symmetric key Ks, which it uses to encrypt its own address (or a special source tag) and populates the resulting ciphertext in this eld Following this the RREQ packet is broadcast by the source

(1013)

Moreover, the support of the invariant measure is the IFS attractor Let us now consider the practical case in which the fractal object is a digital image The normalized value of a pixel B of the image corresponds to the probability of the Borel subset B of R2 According to (1012), the value of a pixel B of the image 1 n is equal to the sum of the values of pixels i (B) in n 1 , multiplied by the probabilities pi The invariant measure associated with the IFS attractor can also be obtained by iterating the three following operations a great number of times, initialized on an arbitrary point x0 of R2 : choose a transformation i with the probability pi ; calculate x1 = i (x0 ); replace x0 by x1

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3 Recall that a measure is, in the physical sense of the term, a measurable quantity (eg the light intensity) that allows us to associate weights with the different points of its support (the support of the measure is the set of the points on which it is de ned)

Page 206 which we initiated a project The sooner a firm is able to nail down a firm and mutually agreeable product concept, the sooner they can begin the project specific engineering needed to realize the project s goals At this point, uncertainty begins to moderate to the point when planning and engineering has advanced far enough to articulate a firm product offering Here the amount of project specific planning and engineering is usually proportional to the reduction of the uncertainty involved in the project Product thinking becomes the project organization s answer to the uncertainty that is mutually held by contractor and client alike Note that as more time elapses, the product firms up to the point where specific schedules and costs can be estimated and a clearer understanding of project risks articulated All these activities are often precursors to an actual development process; that is, while this process seems lengthy, it is a necessary component of understanding the scope of the project and recognizing precisely how to add value to the client organization The bottom line is to be able to present the potential customer with a detailed plan for project execution, based on cumulative learning and full input from all stakeholders involved in the project

When the number of iterations is suf ciently high, the points are distributed on a compact set R2 de ning the attractor of the IFS The density (frequency of the visits) of each pixel of the attractor de nes the invariant measure (see [ELT 87] and the chaos game in [BARN 88]), whose form is controlled by the set of the prede ned probabilities pi 1026 Inverse problem The de nition of the inverse problem can be stated as follows: given an object A belonging to H(R2 ) and a measure on A, how can we nd the IFS and the set of probabilities pi , for which A is the attractor and the invariant measure Various works have attempted to solve this constrained optimization problem, the dif cultly of which lies in its large dimensionality and in the irregularity of the function to be minimized The proposed solutions use different techniques based on genetic algorithms [LUT 93], wavelets [RIN 94] and other approaches [BARN 86, CAB 92, KRO 92, LEV 90, MAN 89, VRS 90] 103 Application to natural image processing: image coding 1031 Introduction The goal of this section is to introduce the basic methods that make it possible to associate a natural image with a nally contracting transformation W whose attractor is an approximation of the image itself If this inverse problem is solved, we can

then talk about image compression, since storing the coef cients of W requires less information than storing the original image It is also referred to as lossy compression, owing to the fact that the attractor constitutes only one approximation of the original image The compression of an image by fractals relies on a transformation called fractal transformation, which consists of transforming the image by a nally contracting operator, so that its visual aspect remains almost unchanged For that, the image transformation is made up of N elementary sub-transformations, each one operating on a block of the image, in the following way (see Figure 104) The image is partitioned in N blocks rn called destination blocks:

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