Image decomposition and denoising using fractional‐order partial differential equations

In this study, the authors propose a fractional derivative‐based image decomposition and denoising model which decomposes the image into the cartoon component (the component formed by homogeneous regions and with sharp boundaries) and the texture (or noise) component. The cartoon component is modelled by a function of the fractional‐order total bounded variation, while the texture component is modelled by an oscillatory function, bounded in the negative Sobolev space norm. The authors give the corresponding minimisation functional, after some transformations, and then the resulting fractional‐order partial differential equation can be solved using the Fourier transform. By symmetry and asymmetry of the fractional‐order derivative, some generalisations and variants of the proposed model are also introduced. Finally, the authors implement the algorithm by the fractional‐order finite difference in the frequency‐domain. The experimental results demonstrate that the proposed models make objective and visual improvements compared with other standard approaches in the task of decomposition and denoising.