A Fractional Graph La+Ψ approach to Image Reconstruction
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We present a variational method for ill-posed problems, which embeds the fractional power of the standard graph Laplacian operator into the regularization term. We explore the dependence of the regularizer on a preliminary approximation of the solution, which is obtained using various existing reconstruction methods Ψ from the literature. As a result, the regularization term becomes both dependent on and adaptive to the observed data, noise, and the choice of the fractional exponent. We present a selected numerical example problem on 2D computerized tomography, for which we consider various reconstruction techniques Ψ, including Filtered Back Projection, Total Variation, and a trained deep neural network. Incorporating the fractional power of the graph Laplacian operator into the regularization term significantly enhances the quality of the approximated solutions for each method Ψ. Additionally, we show that our proposal behaves as a regularization method and is also stable with respect to variations in the noise level.