Wednesday, October 12, 2005

Convolution of Gaussians

Gaussians have an interesting property with respect to convolution. The convolution of a Gaussian (μ1, σ1) and another Gaussian (μ2, σ2) is a new Gaussian (μ3, σ3) with the means and variances being additive; i.e., μ312 and σ3^2= σ1^2 + σ2^2. (Jaynes, Mathworld).

Consequently, in an image processing application, a Gaussian Blur of radius 3 followed by a Gaussian Blur of radius 4 should produce the same result as a single Gaussian Blur of radius 5 (give or take some rounding error).


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