Bivariate splines for image enhancements based on variational models
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In this work, we use bivariate splines to find the approximations of the solutions to two variational models, the ROF model and the TV-$L^p$ model. The reason to use bivariate splines is because of the simplicity of their construction, their accuracy of evaluation and their capability to approximate functions defined on domains of complex shape. We start by showing that both the ROF model and the TV-$L^p$ model have solutions in the spline space, and the solutions are unique and stable. Then we go on to prove that the solutions in the spline space approximate the solutions in the Sobolev space or the $BV$ space. Two iterative numerical algorithms are given to compute the bivariate spline solutions and their convergence are proved. Numerical examples of the applications of the bivariate spline approximations in image inpainting, image resizing, wrinkle removing and image denoising are given. The convergence of the iterative numerical algorithms is examined. Finally, we propose an edge-adaptive triangulation algorithm which triangulates an image according to its edges. To find the edges, we use the Chan-Vese Active Contour Model, which is also a variational model.