This dissertation studies the approximation of the continuous total variation based model for image denoising by piecewise polynomial functions on polygonal domains. Our main contributions are the explicit construction of a continuous piecewise linear approximation on rectangular domains, and the construction of a minimizing sequence of bivariate splines of arbitrary degree for a general polygonal domain. For rectangular domains, we propose an alternate discretization of the ROF model and construct the continuous piecewise linear function as the piecewise linear interpolation of the minimizer of the new discrete model. Whereas on general polygonal domains, we use the Galerkin method to define the spline approximation as the minimizer of the ROF functional over a spline space. We then show that when given a suitable family of triangulations, our approach generates a minimizing sequence for the total variation model. In each case, we use an extension argument to show that the approximation converges to the ROF minimizer in the strict topology of the space of bounded variation functions
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