In this article we discuss the implementation of the combined first and second order total variation inpainting that was introduced by Papafitsoros and Schönlieb. We describe the algorithm we use (split Bregman) in detail, and we give some examples that indicate the difference between pure first and pure second order total variation inpainting.
In this paper we study the one dimensional second order total generalised variation regularisation (TGV) problem with L 2 data fitting term. We examine some properties of this model and we calculate exact solutions using simple piecewise affine functions as data terms. We investigate how these solutions behave with respect to the TGV parameters and we verify our results using numerical experiments.
We study a general class of infimal convolution type regularisation functionals suitable for applications in image processing. These functionals incorporate a combination of the total variation (TV) seminorm and L p norms. A unified well-posedness analysis is presented and a detailed study of the one dimensional model is performed, by computing exact solutions for the corresponding denoising problem and the case p = 2. Furthermore, the dependency of the regularisation properties of this infimal convolution approach to the choice of p is studied. It turns out that in the case p = 2 this regulariser is equivalent to Huber-type variant of total variation regularisation. We provide numerical examples for image decomposition as well as for image denoising. We show that our model is capable of eliminating the staircasing effect, a well-known disadvantage of total variation regularisation. Moreover as p increases we obtain almost piecewise affine reconstructions, leading also to a better preservation of hat-like structures.
We study a general class of infimal convolution type regularisation functionals suitable for applications in image processing. These functionals incorporate a combination of the total variation seminorm and norms. A unified well-posedness analysis is presented and a detailed study of the one-dimensional model is performed, by computing exact solutions for the corresponding denoising problem and the case . Furthermore, the dependency of the regularisation properties of this infimal convolution approach to the choice of p is studied. It turns out that in the case this regulariser is equivalent to the Huber-type variant of total variation regularisation. We provide numerical examples for image decomposition as well as for image denoising. We show that our model is capable of eliminating the staircasing effect, a well-known disadvantage of total variation regularisation. Moreover as p increases we obtain almost piecewise affine reconstructions, leading also to a better preservation of hat-like structures.
Abstract. In this work we introduce a formulation for a non-local Hessian that combines the ideas of higher-order and non-local regularization for image restoration, extending the idea of non-local gradients to higher-order derivatives. By carefully choosing the weights, the model allows to improve on the current state of the art higher-order method, Total Generalized Variation, with respect to overall quality and particularly close to jumps in the data. In the spirit of recent work by Brezis et al., our formulation also has analytic implications: for a suitable choice of weights, it can be shown to converge to classical second-order regularizers, and in fact allows a novel characterization of higher-order Sobolev and BV spaces.
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