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We prove that functions of intrinsic-mode type (a classical models for signals) behave essentially like holomorphic functions: adding a pure carrier frequency eint ensures that the anti-holomorphic part is much smaller than the holomorphic part || P-(f)||L² « P+(f)||L²>. This enables us to use techniques from complex analysis, in particular the unwinding series. We study its stability and convergence properties and show that the unwinding series can provide a high-resolution, noise- robust time-frequency representation. [ABSTRACT FROM AUTHOR]

Multi-time-scale stochastic dynamical systems are ubiquitous in science and engineering, and the reduction of such systems and their models to only their slow components is often essential for scientific computation and further analysis. Rather than being available in the form of an explicit analytical model, often such systems can only be observed as a data set which embodies dynamics on several time scales. We focus on applying and adapting data-mining and manifold learning techniques to detect the slow components in a class of such multiscale data. Traditional data-mining methods are based on metrics (and thus, geometries) which are not informed of the multiscale nature of the underlying system dynamics; such methods cannot successfully recover the slow variables. Here, we present an approach which utilizes both the local geometry and the local noise dynamics within the data set through a metric which is both insensitive to the fast variables and more general than simple statistical averaging. Our analysis of the approach provides conditions for successfully recovering the underlying slow variables, as well as an empirical protocol guiding the selection of the method parameters. Interestingly, the recovered underlying variables are gauge invariant--they are insensitive to the measuring instrument/observation function. [ABSTRACT FROM AUTHOR]

We consider the functional of total variation for functions of either one or two discrete variables. We introduce a new principle for construction of evolution processes that gradually reduce the total variation of a function. The method is derived from analysis of relaxation of the functional along the directions given by the Haar wavelets - an approach that does not require regularization of either the functional or its singular gradient. We further present a rigorous analysis of the analytical properties of the resulting evolution schemes. The purely discrete theory grows out of universal observations about the relation between the total variation and the Haar functions. An additional advantage of the method is that it can be directly implemented as an effective numerical procedure. Moreover, it can be intertwined in a natural way with a nonlinear phase-space filtering which further increases control of the deformation of a signal. Finally, we report that the algorithms can be used as remarkable tools for signal processing and image enhancement. [ABSTRACT FROM AUTHOR]