PeronaMalik

The Perona–Malik equation is a nonlinear, anisotropic diffusion model introduced by Pietro Perona and Jitendra Malik in 1990 for image denoising and edge-preserving smoothing. It aims to reduce noise while preserving important structures such as edges, making it a popular tool in image processing and computer vision.

Mathematically, the equation describes the evolution of image intensity I(x,y,t) by the partial differential equation ∂I/∂t = ∇·(c(|∇I|) ∇I). The diffusion coefficient c(s) is a decreasing function of the gradient magnitude, so diffusion is strong in homogeneous regions and weak across edges. Common choices include c(s) = exp(-(s/κ)^2) and c(s) = 1/(1+(s/κ)^2), where κ is a parameter that sets edge sensitivity.

In practice, the equation is solved iteratively using finite-difference methods. It can smooth within regions while reducing noise without blurring edges excessively, and it has been used as a preprocessing step in various vision tasks and in medical imaging. The idea of edge-aware diffusion has inspired a broad family of methods and has influenced subsequent research in nonlinear filtering and image restoration.

The Perona–Malik model has also faced mathematical scrutiny, as certain formulations can be ill-posed, leading to instability or non-unique solutions. This spurred developments toward well-posed nonlinear diffusion approaches and related techniques such as total variation flows. Extensions by researchers like Weickert proposed coherence-enhancing diffusion and other edge-preserving methods that build on the same core principle.