Fractal-based methods in analysis

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Fractals are mathematical entities that have useful multiscale properties; no matter the level to which we zoom in on a fractal, we always reveal more detail, structure, and self-similarity. "Fractal-based methods" refers not just to the theory and tools of fractals themselves, but also to the driving philosophy behind this mathematics. Recently, this philosophy has been used to treat problems far removed from classical fractals. In many application areas, phenomena of interest can have important features at multiple scales (spatial or temporal). For physical models, one might consider continuum or macroscale models (which capture no lower level information), mesoscale or nanoscale models (which include information about groups of atoms and molecules), molecular dynamical models (which include information about individual atoms), or quantum mechanical models (which includes information about electrons). In recent years, scientists have become able to gather data at multiple scales, for pathological tissues or tumors, for example. These scientists now seek to formulate and analyze compatible multiscale models. When measurements are available, scientists often seek to solve the "inverse problem" of estimating the values (or functional form) of the parameters in these models. The first goal of fractal imaging was to represent a given target image by a fractal. Subsequent research sought to perform image processing tasks within this framework. The program seeks to develop (1) fractal-based frameworks for treating problems in biomedical science (such as tumor detection, modeling, and analysis), ecological science (such as ecosystem modeling and analysis), environmental science (such as pollution-driven population dynamics), and physics/engineering (such as heat flow and vibrations); and (2) new imaging frameworks based on some very recent generalizations of the original fractal imaging notions. The work is unified by a common philosophical approach and by the fact that measurements (of a tumor, say) may well come in the form of an image, in which case the related analysis can involve elements of (1) and (2).