Mathematics Colloquia and Seminars

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An important problem in Computer Graphics is shape deformation: given a source shape and some user defined constraints, the goal is to generate a deformed shape which fulfills the constraints and preserves the local details of the original shape. A general way of solving this problem is to use space deformations, which are defined as a map from the source domain to R2 or R3. Especially useful for this task are conformal and harmonic maps, since they generate smooth deformations which do not have extra oscillations beyond those required by the constraints, and preserve local details. From this point of view, deformation reduces to finding conformal and harmonic maps between simply connected domains. In general, this problem does not have a closed form solution, and approximating the solution directly by discretizing the domain is too computationally expensive to be done in realtime. We present a method for approximating conformal and harmonic maps between domains as a linear combination of special basis functions, which are derived from the free space Green's functions. We show how this method can be successfully applied to 2D and 3D shape deformation and deformation transfer.