A novel parabolic model driven by double phase flux operator with variable exponents: Application to image decomposition and denoising

This work tackles a novel parabolic equation driven by a nonlinear operator with double variable exponents, aiming to decompose and denoise images. Our primary approach involves enhancing classical models based on variable exponent operators by considering a novel nonlinear operator having a double-phase flux with unbalanced growth. We begin initially by analyzing the theoretical solvability of our model. Employing the Musielak-Orlicz space, we establish a suitable functional framework for investigating the proposed model. Subsequently, we use the Faedo-Galerkin approach to establish the existence and uniqueness of a weak solution for our problem. Furthermore, we provide a demonstration showcasing how our model preserves solution positivity, emphasizing this as a consistent feature of the proposed model. To evaluate our theoretical results, we present various numerical simulations on some grayscale and medical images (Magnetic Resonance Images (MRI)). These simulations covered aspects like decomposition, robustness and behavior sensitivity of model parameters with visual and quantitative comparisons. The obtained numerical results strongly support the efficiency of the proposed model in preserving features, reducing artifacts and deleting noise in comparison to some well-known existing state-of-the-art models. Furthermore, the proposed model quantitatively surpasses the competitive models in terms of several well-known criteria, namely the Peak Signal-to-Noise Ratio (PSNR), the Structural Similarity Index (SSIM) and the Mean-Square Error (MSE).