Iterative Finite Difference Method with Adaptive Median Filtering

The finite difference method is a numerical technique for solving partial differential equations by discretizing differential equations. In MATLAB implementation, this typically involves: first discretizing continuous space and time into grid points, then replacing derivatives with difference approximations at each grid point to transform differential equations into difference equations. Key implementation steps include defining grid spacing (dx, dt), initializing boundary conditions, and constructing difference matrices using vectorized operations for computational efficiency.

Iterative methods are commonly employed to solve difference equations, particularly when direct solutions are computationally expensive or infeasible. Widely-used iterative techniques include Jacobi iteration, Gauss-Seidel iteration, and Successive Over-Relaxation (SOR) iteration. These methods share the characteristic of progressively refining solutions from initial guesses through update iterations until convergence criteria are met. MATLAB implementations often utilize while-loops with tolerance checks, where matrix operations update solution vectors based on specific iteration formulas - for example, Gauss-Seidel leverages triangular matrices for sequential updates.

Adaptive median filtering is an enhanced median filtering algorithm particularly effective for removing impulse noise (e.g., salt-and-pepper noise) from images. Unlike standard median filtering, it dynamically adjusts the filtering window size based on local noise characteristics, thereby better preserving image details while removing noise. In MATLAB implementation, this involves scanning each pixel, calculating statistical properties within local windows using functions like medfilt2 or custom windowing functions, and conditionally expanding window sizes until noise-free pixels are found or maximum window size is reached.

The combined application of finite difference methods and iterative techniques enables efficient solutions to complex PDE problems, while adaptive median filtering provides flexible solutions for image denoising. MATLAB's robust numerical computing capabilities, including built-in functions for matrix operations and image processing toolboxes, make the implementation of these methods highly efficient and accessible.