A vector-valued PDE-constrained image inpainting model

In image inpainting, the identification and inpainting of local detail features and the preservation of global features are crucial. Models based on fractional-order partial differential equations exhibit rich evolutionary behaviors. These behaviors enable them to effectively comprehend image details. Additionally, these models possess a certain sharpening effect in image inpainting. However, they are also prone to issues such as inaccurate identification of large-scale features and over-sharpening. The optimal control model proposed in this paper uses the total variation energy of image global features as the objective function. It also employs the spatial fractional-order vector-valued Cahn–Hilliard equation as the constraint, aiming to achieve a balanced effect between local detail restoration and preservation of global features. The paper aims to optimize the objective function by designing numerical computation schemes for non-convex constraint conditions using \(L_{2}\) $L_2$ gradient flow, \(H^{-1}\) $H^{-1}$ gradient flow, and convex splitting. The Split Bregman method is used for further optimization, and a dynamic grayscale adjustment strategy is introduced to maintain grayscale discrimination capability while enhancing computational efficiency. Numerical experiments demonstrate that the new model exhibits certain advantages over other inpainting methods in terms of PSNR values. It also shows strong competitiveness in terms of SSIM values, especially in severely damaged images where it demonstrates greater stability. Compared to traditional fractional-order equation models, the proposed model captures global features and incorporates the dynamic grayscale adjustment strategy, resulting in significantly reduced computation time.