On a nonlinear laplacian based filter for noise removal

TL;DR

The paper addresses denoising of 1D and 2D data while preserving edges by introducing a nonlinear Laplacian-based filter formulated as a fourth-order PDE. The core approach uses evolution equations, such as $\partial_t u = -\Delta\left(\frac{\Delta u}{(|\Delta u|^2 + \epsilon)^p}\right) - \lambda(u - u_0)$ in 2D (and its 1D analogue) to compute restored data, aiming to avoid the staircase effect inherent to TV denoising. Experiments on smooth and piecewise-smooth 1D signals, 2D functions, and images show the method yields piecewise linear reconstructions that better preserve discontinuities than TV. The work also discusses adaptive parameter selection inspired by ROF and demonstrates the method’s practical potential as a simpler, edge-preserving alternative to TV denoising for both 1D and 2D data.

Abstract

We propose a nonlinear filter for noise removal based on the Laplacian for 1D and 2D data. The method utilizes the solution to a fourth-order nonlinear PDE involving the Laplacian for data reconstruction. Evolution equations are introduced to solve this fourth-order nonlinear equation. Numerical experiments show that the new filter preserves discontinuities while filtering out noise. The restored data are piecewise linear and avoid the staircase effect commonly observed with total variation denoising methods.

Figures (7)

• Figure 1: Original and noisy functions.
• Figure 2: Noisy and restored functions by the new method (left) and the TV method (right).
• Figure 3: Noisy and restored functions by the new method (left) and the TV method (right).
• Figure 4: Original and noisy functions.
• Figure 5: Restored functions by the new method (left) and the TV method (right).