a new bilevel optimization problem for the image restoration

This paper presents a novel bilevel optimization approach for a nonlinear partial differential equation. the approach aims to enhance the quality of image denoising  by estimating certain parameters within this equation. our work deals with both analytical and numerical results. analytically, we establish the existence of a solution to the bilevel optimization problem and apply the alternating direction method of multipliers algorithm to approximate this solution. furthermore, the method fine-tunes the restoration process, effectively reducing noise while preserving crucial image features. finally, numerical results validate the performance of our method, surpassing traditional denoising approaches. this research makes an important contribution to image restoration, paving the way for high-quality practical applications.