Convolution

A function h is the convolution of two functions f and g (denoted h = f * g) if:

• continuous, 1D:    h(x) =

  ∞ ∫ -∞

  f(u) g(x-u) du

• discrete, 1D: h(x) =

  ∞  ∑ u = -∞

  f(u) g(x-u)

  f and g are usually zero in many places of an image

• continuous, 2D: h(x) =

  ∞ ∫ -∞

  ∞ ∫ -∞

  f(u,v) g(x-u, y-v) du dv




• discrete, 2D: h(x) =

  ∞  ∑ u = -∞

  ∞  ∑ u = -∞

  f(u,v) g(x-u, y-v)

• convolve with G_sigma(x), where

  G_sigma(x) = [1/(√(2π)*σ] * e^(-x2/2*σ2)

  smoothing is increased with sigma

• note: f * g' = (f * g)'

• convolve I with derivative of Gaussian function G_sigma(x)

  G_sigma(x)' = (- x)/(√(2π)*σ³) * e^(-x2/2*σ2)