Combining Multiple Directions in the Total Variation Function: Application to Image Inpainting Using the L-BFGS Algorithm

In image processing, several tasks require the  replacing of missing or deteriorated pixels in digital images through algorithms capable of estimating plausible values for the damaged regions. Total Variation (TV)-based formulations have been widely used for this purpose, generally relying on horizontal and vertical differences between adjacent pixels. In this work, we propose a new formulation that combines four TV directions into a single objective function, enabling a more expressive representation of local structures to improve the quality of restored images. The resulting unconstrained differentiable nonlinear optimization problem is solved using the limited-memory quasi-Newton (L-BFGS) algorithm. Numerical experiments on diverse image sets and noise types show that our approach outperforms the classical TV formulation in terms of standard image quality metrics such as Peak Signal-to-Noise Ratio (PSNR) and the Structural Similarity Index Metric (SSIM), without increasing computational cost.