Variational Iteration Method for Image Restoration

The famous Perona-Malik (P-M) equation which was at first introduced for image restoration has been solved via various numerical methods. In this paper we will solve it for the first time via applying a new numerical method called the Variational Iteration Method (VIM) and the correspondent approximated solutions will be obtained for the P-M equation with regards to relevant error analysis. Through implementation of our algorithm we will access some effective results which are deserved to be considered as worthy as the other solutions issued by the other methods.

💡 Research Summary

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The paper addresses the long‑standing problem of numerically solving the Perona‑Malik (P‑M) equation, a nonlinear anisotropic diffusion model widely used for image denoising and edge preservation. While many classical schemes—explicit finite differences, implicit Crank‑Nicolson, semi‑implicit methods, and finite element approaches—have been applied to this equation, they all suffer from either severe stability restrictions, high computational cost, or the need for linearization that can degrade edge‑preserving capabilities. The authors propose, for the first time, to solve the P‑M equation using the Variational Iteration Method (VIM), a technique originally introduced by He for nonlinear differential equations.

Core Methodology
The P‑M equation is written as
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