GaussFilter

Gaussfilter, commonly known as the Gaussian filter, is a linear, shift-invariant smoothing filter whose impulse response is a Gaussian function. In one dimension, the 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)). Convolution with this kernel produces a blurred result that attenuates high-frequency components, thereby reducing noise and small details. The filter is isotropic, so the blur is uniform in all directions, and its strength is governed by the standard deviation σ: larger σ yields stronger smoothing.

In practice, the Gaussian kernel is discretized by sampling and then normalized so the sum equals 1. A common rule of thumb is to choose a kernel radius r ≈ 3σ, resulting in a kernel size of (2r+1) × (2r+1). A key property is separability: G(x,y) = g(x)g(y), which allows the 2D filtering to be performed as two 1D convolutions, increasing computational efficiency.

Applications include noise reduction, pre-processing for edge detection, and general image blurring in photography and computer vision. It is frequently used to reduce detail before further processing, such as feature extraction, while preserving overall image structure better than many other smoothing methods.

Border handling must be considered during implementation (padding, mirroring, or cropping). For large σ or high-resolution images, FFT-based convolution or separable 1D passes are common alternatives. The Gaussfilter is named after Carl Friedrich Gauss and is a foundational tool in signal and image processing.