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The sum measure, max and iso capacities are the basic tools used for analyzing images There are cases where speci c capacities, and/or more elaborate schemes than the one just described, must be designed in order to get robust results See [LEV 98] for more details 119 Change detection in image sequences using multifractal analysis In [CAN 96], the authors propose a multifractal approach to the problem of change detection in image sequences Multifractal analysis proves to be useful for detecting changes without any a priori knowledge of the object to be extracted If a change occurs in an incoming image, it is re ected in the global description provided by the multifractal spectrum graph The abscissa of the spectrum part ( , fh ( )) that has changed makes it possible to extract binary images corresponding to the changes As can be seen in Figure 1122c and Figure 1123b, the extracted changes using multifractal analysis are much more relevant than the absolute difference between the two images

Figure 1122 (a) First image, (b) second image and (c) absolute difference (pixel to pixel) between the two images

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Page 59 the project execution plan as a standard company procedure that is incorporated into our QA system However, developed procedures, no matter how carefully devised and controlled, do not ensure that the tool works As the final step of our program, the potential users of our system both in our own organization and in our supply chain have to be trained to use the system In the following chapters, we will continue to explore in depth our understanding of customer-based project management This chapter provided a basic understanding of some of the unique characteristics of the project management process and the roles they will be assuming Now, with the stage set, the balance of this book will continue exploring the processes companies must develop to become attuned to the needs of their customer base Successful project management becomes successful customer management

1110 Image reconstruction In [TUR 02], the authors describe an interesting method for image reconstruction that mixes multifractal analysis and more traditional diffusion-like techniques The image I (or, more precisely, the modulus of its gradient) is rst modeled as a so-called log-Poisson multifractal measure This means that its spectrum f admits the following parametric form: f ( ) = f + 1 log (2 f ) ,

where is the lowest observed exponent, f the associated spectrum value and := log[1 + /(2 f )] The justi cation for considering such a model is that it seems to describe many natural images with reasonable accuracy For most images, it is observed that f 1, and this is the value chosen for the rest of the method The points in the image having exponent are the most singular ones In many cases, the set comprising all these points, which is called the most singular manifold (MSM), is strongly related to the edges of the images This observation and the fact that the whole spectrum may be computed once is known suggest that the MSM contains the most relevant information, and that the whole image may be reconstructed from it In order to implement this idea, a linear operator is applied to the MSM This operator must satisfy the following natural constraints: it should be translation-invariant, isotropic, and allow us to recover the original power (Fourier) spectrum of the image Under these constraints, we may show that there is essentially exactly one possible operator, and that the image I may be reconstructed through the following formula: c(f ) = where c is the Fourier transform of c := I I0 , I0 is the mean of I, f denotes frequency, v0 (x) = c(x) if x belongs to the MSM, and v0 (x) = 0 otherwise Reconstruction of real-world images, such as outdoor scenes or natural textures, is surprisingly good considering the simplicity of the method if v0 (f ) , f2

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