f(m,n)f(m,n) is separable if f(m,n)=fv(m)fh(n)f(m,n)=f_v(m)f_h(n)

  • where fv(m)f_v(m): changes vertically, fh(n)f_h(n): changes horizontally

Separable image = Rank 1 matrix

  • Rank 1 matrix = product of 1D column vector and 1D row vector
  • H=[10−120−210−1]=[121][10−1]=hxhyTH= \begin{bmatrix} 1 & 0 & -1 \\ 2 & 0 & -2 \\ 1 & 0 & -1 \end{bmatrix} = \begin{bmatrix} 1\\2\\1\end{bmatrix} \begin{bmatrix} 1&0&-1\end{bmatrix} = h_x h_y^T

2D DSFT of separable signal = product of 1D DSFT of each 1D component

References:

  1. https://en.wikipedia.org/wiki/Multidimensional_discrete_convolution#Separable_signals