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Abstract
In this work, we use bivariate splines to find the approximations ofthe solutions to two variational models, the ROF model and theTV-$L^p$ model. The reason to use bivariate splines is because ofthe simplicity of their construction, their accuracy of evaluationand their capability to approximate functions defined on domains ofcomplex shape. We start by showing that both the ROF model and theTV-$L^p$ model have solutions in the spline space, and the solutionsare unique and stable. Then we go on to prove that the solutions inthe spline space approximate the solutions in the Sobolev space orthe $BV$ space. Two iterative numerical algorithms are given tocompute the bivariate spline solutions and their convergence areproved. Numerical examples of the applications of the bivariatespline approximations in image inpainting, image resizing, wrinkleremoving and image denoising are given. The convergence of theiterative numerical algorithms is examined. Finally, we propose anedge-adaptive triangulation algorithm which triangulates an imageaccording to its edges. To find the edges, we use the Chan-VeseActive Contour Model, which is also a variational model.