In theory, the Gaussian function at every point on the image will be non-zero, meaning that the entire image would need to be included in the calculations for each pixel. In practice, when computing a discrete approximation of the Gaussian function, pixels at a distance of more than 3*σ* are small enough to be considered effectively zero. Thus contributions from pixels outside that range can be ignored. Typically, an image processing program need only calculate a matrix with dimensions × (where is the ceiling function) to ensure a result sufficiently close to that obtained by the entire gaussian distribution.

Given two p-dimensional normal probability density functions **G1 ≡ gp (x; a, A)** and **G2 ≡ gp (x; b, B) **

The convolution of these two functions is a normal probability density distribution function with **mean a + b** and **variance A + B**, i.e. **gp (x; a + b, A + B)**: G1 ∗ G2 (z) = gp (z; a + b, A + B)

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