Two-dimensional convolution

All the theory that was developed above for one-dimensional signals can be generalized to two dimensions and applied to images. In most cases, we will treat each dimension separately: if we have a two-dimensional signal f(x,y) we can first apply FT for the variable x keeping y fixed, then for y and obtain a function of two frequency variables. Unfortunately, not all intersting things that can be done in two dimensions can be decomposed into a sequence of two one-dimensional operations.

Two-dimensional discrete convolution can be defined as

$$z[n1,n2] = \sum_{k_1 = - \infty}^{+\infty}\sum_{k_2 = - \infty}^{+\infty} x(k_1, k_2)y(n_1 - k_1, n_2 - k_2)$$

Two-dimensional convolution can be represented as a sequence of two one-dimensional convolutions only if one of the signals is separable, that is if x[n₁,n₂] = x₁[n₁]x₂[n₂] for some one-dimensional signals x₁[n₁] x₂[n₂].