Multidimensional discrete convolution

In signal processing, multidimensional convolution refers to the mathematical operation between two functions f and g of n-dimensions that produces a third function, also of n-dimensions. Multidimensional convolution is a direct extension of the 1-Dimensional convolution case.

Definition

Similar to the 1-Dimensional case, an asterisk is used to represent the convolution operation. The number of dimensions in the given operation is reflected in the number of asterisks. For example, the following represents a 2-Dimensional convolution:

${\displaystyle y(n_{1},n_{2})=x(n_{1},n_{2})**h(n_{1},n_{2})}$ 

Problem Statement & Basics

Motivation & Applications

Row-Column Decomposition with Separable Signals

Separable Signals

A signal is said to be separable if it can be written as the product of multiple 1-Dimensional signals ^[1]. Mathematically, this is expressed as the following:

${\displaystyle x(n_{1},n_{2},...,n_{M})=x(n_{1})x(n_{2})...x(n_{M})}$ 

Some readily recognizable separable signals include the unit step function, and the delta-dirac impulse function.

${\displaystyle u(n_{1},n_{2},...,n_{M})=u(n_{1})u(n_{2})...u(n_{M})}$  (unit step function)

${\displaystyle \delta (n_{1},n_{2},...,n_{M})=\delta (n_{1})\delta (n_{2})...\delta (n_{M})}$  (delta-dirac impulse function)

Overlap and Add and Overlap and Save

The Helix Transform

Similar to row-column decomposition, the helix transform computes the multidimensional convolution by incorporating one-dimensional convolutional properties and operators. Instead of using the separability of signals,

References

1. Dudgeon, Dan; Mersereau, Russell (1983), Multidimensional Digital Signal Processing, Prentice-Hall, p. 8