Multiscale Data Representations in Geometric and Nonlinear Settings

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The principle investigator and his collaborators plan to construct, streamline and analyze a suite of multiscale data representation methods in various nonlinear and geometric settings, as well as their applications.Some of these multiscale representations include: free-form jet subdivision surfaces,multiscale representation of vector and tensor fields on free-form subdivision surfaces, wavelet-like transform of nonlinear range data, e.g. time series or spatial arrays of data taking values at a manifold.The PI intends to explore theoretical questions such as:When a traditional spline or subdivision method is modified to apply to data which satisfy specific nonlinear constraints, how would the curviness of the underlying nonlinear manifold affect the stability, approximation and smoothness behavior of the original method? The PI and his collaborators are alsodeveloping software tools for the fast prototyping and computational analysis of these novel multiscale methods, as well as applying them to real datasets. Fueled by the advances in various sensing technologies (in optics, syntheture aperture radar, etc.), new forms of data type begin to arise in many different significant fields in science and engineering, and it is the task of mathematicians to help preparing the world to make good use of such data. Obvious application areas include material sciences, diffusion tensor imaging, hyperspectral imagery, computer aided design/animation, robotics (motion planning), just to name a few.In each of these technologies, efficient multiscale methods is a must for processing tasks such as the compression, registration, fast search, browsing, structure reconstruction, etc.