Convolution and Correlation Mathematical Notation and Definitions

The following notation is necessary to explain the underlying mathematical definitions used in the text:

`R = (-∞, +∞)`	The set of real numbers.
`Z = {0, ±1, ±2, ...}`	The set of integer numbers.
`Z^N = Z× ... ×Z`	The set of N-dimensional series of integer numbers.
`p = (p₁, ..., p_N) ∈Z^N`	N-dimensional series of integers.
`u`:Z^N→R	Function `u` with arguments from Z^N and values from R.
`u`(`p`) = `u`(`p`₁, ..., `p`_N)	The value of the function `u` for the argument (`p`₁, ..., `p`_N).
`w = u*v`	Function `w` is the convolution of the functions `u`, `v`.
`w = u•v`	Function `w` is the correlation of the functions `u`, `v`.

Given series p, q∈Z^N:

series r = p + q is defined as rⁿ = pⁿ + qⁿ for every n=1,...,N
series r = p - q is defined as rⁿ = pⁿ - qⁿ for every n=1,...,N
series r = sup{p, q} is defines as rⁿ = max{pⁿ, qⁿ} for every n=1,...,N
series r = inf{p, q} is defined as rⁿ = min{pⁿ, qⁿ} for every n=1,...,N
inequality p≤q means that pⁿ≤qⁿ for every n=1,...,N.

A function u(p) is called a finite function if there exist series P^min, P^max∈Z^N such that:

u(p) 
        ≠ 0

implies

 P^min≤p≤ P^max.

Operations of convolution and correlation are only defined for finite functions.

Consider functions u, v and series P^min, P^maxQ^min, Q^max∈Z^N such that:

u(p) ≠ 0 implies P^min≤p≤ P^max.
v(q) ≠ 0 implies Q^min≤q≤ Q^max.

Definitions of linear correlation and linear convolution for functions u and v are given below.

Linear Convolution

If function w = u*v is the convolution of u and v, then:

w(r) ≠ 0 implies R^min≤r≤R^max,
where R^min = P^min + Q^min and R^max = P^max + Q^max.

If R^min≤r≤R^max, then:

w(r) = ∑u(t)·v(r−t)is the sum for all t∈Z^N such that T^min≤t≤T^max,
where T^min = sup{P^min, r− Q^max} and T^max = inf{P^max, r− Q^min}.

Linear Correlation

If function w = u•v is the correlation of u and v, then:

w(r) ≠ 0 implies R^min≤r≤R^max,
where R^min = Q^min - P^max and R^max = Q^max - P^min.

If R^min≤r≤R^max, then:

w(r) = ∑u(t)·v(r+t)is the sum for all t∈Z^N such that T^min≤t≤T^max,
where T^min = sup{P^min, Q^min−r} and T^max = inf{P^max, Q^max−r}.

Representation of the functions u, v, was the input/output data for the Intel® oneAPI Math Kernel Library convolution and correlation functions is described in theData Allocation.