A Novel Second-Order Total Variation Model with Biharmonic for High-Quality Image Inpainting

Image reconstruction aims to restore missing parts of the image while keeping geometry and quality intact. The traditional second-order partial differential equation (PDE) and Total Variation (TV) methods can be used to perform such an action; nonetheless, their disadvantages, such as over-smoothing, staircase effect, and loss of cornerstones in image details, have been noted. The major disadvantages of those approaches led to the idea of developing a second-order Total Variation–biharmonic (TV2–BH) model in which the TV method is used in combination with a fourth-order Laplacian diffusion term. The Euler–Lagrange equation is derived as a nonlinear fourth-order PDE and solved using appropriate numerical methods. The experience gained from using the TV2–BH method shows that the proposed approach yields much better results in comparison with the standard inpainting methods due to the preservation of texture patterns and edge information, as well as homogeneous areas and staircase effect minimization. The soundness of the proposed model is duly proved through quantitative assessments based on Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index Measure (SSIM), and Mean Squared Error (MSE). The obtained results confirm that improved reconstruction performance can be achieved by the proposed model compared with classical TV, biharmonic, and TV2 inpainting methods.