Visible to Intel only — GUID: GUID-9E63BAB2-A39A-4107-8547-C66EC533E86D
Visible to Intel only — GUID: GUID-9E63BAB2-A39A-4107-8547-C66EC533E86D
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. |
ZN = Z× ... ×Z | The set of N-dimensional series of integer numbers. |
p = (p1, ..., pN) ∈ZN | N-dimensional series of integers. |
u:ZN→R | Function u with arguments from ZN and values from R. |
u(p) = u(p1, ..., pN) | The value of the function u for the argument (p1, ..., pN). |
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∈ZN:
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 Pmin, Pmax∈ZN such that:
u(p)
≠ 0
implies
Pmin≤p≤ Pmax.
Operations of convolution and correlation are only defined for finite functions.
Consider functions u, v and series Pmin, PmaxQmin, Qmax∈ZN such that:
u(p) ≠ 0 implies Pmin≤p≤ Pmax.
v(q) ≠ 0 implies Qmin≤q≤ Qmax.
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 Rmin≤r≤Rmax,
where Rmin = Pmin + Qmin and Rmax = Pmax + Qmax.
If Rmin≤r≤Rmax, then:
w(r) = ∑u(t)·v(r−t) is the sum for all t∈ZN such that Tmin≤t≤Tmax,
where Tmin = sup{Pmin, r− Qmax} and Tmax = inf{Pmax, r− Qmin}.
Linear Correlation
If function w = u•v is the correlation of u and v, then:
w(r) ≠ 0 implies Rmin≤r≤Rmax,
where Rmin = Qmin - Pmax and Rmax = Qmax - Pmin.
If Rmin≤r≤Rmax, then:
w(r) = ∑u(t)·v(r+t) is the sum for all t∈ZN such that Tmin≤t≤Tmax,
where Tmin = sup{Pmin, Qmin−r} and Tmax = inf{Pmax, Qmax−r}.
Representation of the functions u, v, was the input/output data for the Intel® oneAPI Math Kernel Library (oneMKL) convolution and correlation functions is described in theData Allocation.