Image smoothing

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.

A median filter replaces an image signal value by the median of nearby values. Unlike a linear average, it can reject isolated extreme observations without averaging both sides of an image edge. The small-radius response depends on neighbourhood geometry and measure; a uniform disk-area median and a circle-boundary median have different time scales.

At a smooth point with nonzero gradient, rotate coordinates so $\nabla u=G{e}_2$. Write $u=u_0+Gy+\frac{1}{2}(A{x}^2+2Bxy+C{y}^2)+O(|(x,y){|}^3)$. A candidate median $u_0+d{h}^2$ has level boundary $y=(d{h}^2-A{x}^2/2)/G+O(h^3)$ in the disk. The area imbalance is $(2d-A/3)h^3/G+o(h^3)$; equal half-areas force $d=A/6$. Here $A=\Delta u-\nabla u^T(D^{2}u)\nabla u/|\nabla u{|}^2$ is the tangential second derivative, giving the formula. The disk uses uniform area measure, not its boundary measure.

Gaussian blur applies convolution with $G_{\sigma }(x)=(2\pi {\sigma }^2{)}^{-n/2}{e}^{-|x{|}^2/(2{\sigma }^2)}$ to an image signal. It averages over a spatial scale $\sigma$, suppressing rapid variation while also blurring image edges. The heat-kernel convolution realizes this filter at time $t={\sigma }^2/2$.

The Fourier transform of a Gaussian and the convolution theorem give this multiplier with the angular-frequency convention $e^{-ix\cdot \xi }$. In cycles-per-length frequencies $k$, it is $e^{-2{\pi }^2{\sigma }^2|k{|}^2}$. Each coordinate has variance ${\sigma }^2$, and higher frequencies receive stronger attenuation.