gaussianfilter

Gaussian filter, named after the Gaussian function, is a widely used linear smoothing filter in signal and image processing. It reduces random noise and minor details while preserving overall structures. In one dimension, the Gaussian kernel is g(x) = (1/√(2π)σ) exp(-x^2/(2σ^2)). In two dimensions, the kernel is G(x,y) = (1/(2πσ^2)) exp(-(x^2+y^2)/(2σ^2)). The parameter σ controls the amount of smoothing: larger values produce stronger blur and shorter high-frequency content.

Discrete implementations form a finite kernel by sampling the Gaussian at integer coordinates and normalizing so the weights sum to one. The radius r is typically chosen as r ≈ floor(3σ) or floor(4σ), yielding a kernel size of 2r+1. A key property is separability: the 2D Gaussian equals the product of two 1D Gaussians, so a 2D blur can be computed by convolving first with a 1D kernel along rows and then along columns, which improves efficiency.

The Gaussian filter acts as a low-pass filter, attenuating high-frequency components without introducing significant artifacts such as ringing. It is widely used for noise reduction, pre-processing before edge detection, and general smoothing in photography, computer vision, and scientific imaging. In physics terms, Gaussian blur corresponds to the solution of the heat equation and represents diffusion over time, with σ related to diffusion duration.

Limitations include blur of edges and loss of fine detail, and the result is not easily reversible. Practical considerations include choosing σ and kernel size, padding or border handling, and implementation efficiency.