Digital Video Processing, Second Edition

Appendix A. Vector-Matrix Operations in Image and Video Processing

A.1 Two-Dimensional Convolution

In 1D signal processing, FIR filtering, i.e., convolution, of a finite-extent input s(n) and a length 2L + 1 impulse response h(n), that is given by

can be expressed as a vector-matrix product g = Hs, where the matrix H has Toeplitz structure,

The output samples g (–L) to g (–1) and g (N) to g (N + L – 1) have been truncated, so that the input and output vectors are the same length. This vector-matrix form can be easily extended to represent 2D-FIR ...