Finite Element Numerical Approximation for Two Image Denoising Models

Finite difference method (FDM) is a well-established variational computational technique to solve problems in image analysis. Compared to the extensively discussed finite difference schemes, very few work has been devoted to finite element method (FEM), which motivates the proposed work. On one hand, FEM has strong physical backgrounds, which allows clear and physically meaningful derivation of difference equations that are easy to implement. On the other hand, combined with the variational methods, the semidiscrete FEM scheme in timescale can give favorable stability and efficiency properties of computations. In this paper, we firstly introduce two classical image denoising models, the Perona–Malik (P–M) model and You–Kaveh (Y–K) model. Then, the finite element numerical algorithm is given for the two models, and the numerical analysis of the algorithm is also presented. These two models correspond to two kinds of nonlinear partial differential equations, the former of which is of fourth order and the latter is of second order. Compared results demonstrate the superiority of the proposed FEM over FDM, in terms of suppressing blocky effects while maintaining the visual quality.