Diffusion image processing

Diffusion filtering evolves an observed image signal using a diffusion equation. The heat equation gives Gaussian image smoothing, while gradient-dependent or tensor-dependent diffusion can reduce transport across image edges. A image smoothing scale, stopping rule and appropriate boundary conditions are part of the filter.

A positive tensor $D=a_n{e}_n{e}_n^T+a_t{e}_t{e}_t^T$ selects different diffusion strengths normal and tangent to an estimated image edge. Taking $0<a_n\ll a_t$ smooths along the image edge while reducing mixing across it. A structure tensor provides robust orientation estimates. Positive eigenvalues ensure forward local parabolicity.

Image smoothing suppresses rapid fluctuations attributed to noise. Linear heat equation image smoothing attenuates Fourier frequencies by $e^{-t|\xi {|}^2}$ but also blurs image edges. Nonlinear filtering uses image signal geometry to distinguish fluctuations from meaningful transitions.