Fractional‐order tensor regularisation for image inpainting

Abstract

Compared with classic integer-order calculus, fractional calculus is a more powerful mathematical method that non-linearly preserves and enhances image features in different frequency bands. In order to extend fractional-in-space diffusion scheme with matrix-valued diffusivity to perform superior image inpainting, the authors build the new fractional-order tensor regularisation (FTR) model by utilising the newly defined fractional-order structure tensor (FST) to control the regularisation process. The proposed model is derived as a process that minimises a functional proportional to the FST composed of the inner product of the fractional derivative vector and its transposition; hence, the new model not only inherits genuine anisotropism of tensor regularisation, but is also better equipped to handle subtle details and complex structures because of the characteristics of fractional calculus. To minimise the proposed functional, the corresponding Euler-Lagrange equation is deduced, and the anisotropism of the proposed model is analysed accordingly. Fractional-order derivative masks in positive x and y directions and negative x and y directions are implemented according to the shifted Grumwald-Letnikov definition, and a proper iterative numerical scheme is analysed. According to experimental results on various test images, the proposed FTR inpainting model demonstrates superior inpainting performance both in noiseless and noisy scenarios.