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 = (p 1 , ...,
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
:
- seriesis defined asr=p+qfor everyr=np+nqnn=1,...,N
- seriesis defined asr=p-qfor everyr=np-nqnn=1,...,N
- seriesis defines asr= sup{p,q}for everyr= max{np,nq}nn=1,...,N
- seriesis defined asr= inf{p,q}for everyr= min{np,nq}nn=1,...,N
- inequalitymeans thatp≤qpn≤qfor everyn.n=1,...,N
A function
is called a finite
function if there exist series P
u
(p
)min
, Pmax
∈
Z
N
such that:
impliesu(p)≠0
Pmin≤p≤Pmax.
Operations of convolution and correlation are only
defined for finite functions.
Consider functions
u
,
v
and series Pmin
, Pmax
Qmin
, Qmax
∈
Z
N
such that:
u
(p
)
≠
0P
.min
≤
p
≤
Pmax
v
(q
)
≠
0Q
min
≤
q
≤
Qmax
.Definitions of linear correlation and linear
convolution for functions
u
and
v
are given
below.
Linear Convolution
If function
is the convolution
of
w
=
u
*
v
u
and
v
, then:
w
(r
)
≠
0R
min
≤
r
≤
R
max
R
min
=
Pmin
+ Qmin
R
max
=
Pmax
+ Qmax
.If
, then:
R
min
≤
r
≤
R
max
w
(r
) =
∑
u
(t
)·v
(r
−
t
)
t
∈
Z
N
such that
T
min
≤
t
≤
T
max
T
min
=
sup{Pmin
,
r
−
Qmax
}T
max
=
inf{Pmax
,
r
−
Qmin
}.Linear Correlation
If function
w
=
u
•
v
is the
correlation of
u
and
v
, then:
w
(r
)
≠
0R
min
≤
r
≤
R
max
R
min
=
Qmin
- Pmax
R
max
=
Qmax
- Pmin
If
, then:
R
min
≤
r
≤
R
max
w
(r
) =
∑
u
(t
)·v
(r
+t
)
t
∈
Z
N
T
min
≤
t
≤
T
max
T
min
=
sup{Pmin
, Qmin
−
r
}T
max
=
inf{Pmax
, Qmax
−
r
}Representation of the functions
convolution and correlation functions is described in theData Allocation.
u
,
v
,
w
as the input/output data for the Intel® oneAPI Math Kernel Library