Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

Contents

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
:
  • series
    r
     =
    p
     +
    q
     is defined as
    r
    n
     =
    p
    n
     +
    q
    n
     for every
    n
    =1,...,N
  • series
    r
     =
    p
     -
    q
     is defined as
    r
    n
     =
    p
    n
     -
    q
    n
     for every
    n
    =1,...,N
  • series
    r
     = sup{
    p
    ,
    q
    }
     is defines as
    r
    n
     = max{
    p
    n
    ,
    q
    n
    }
     for every
    n
    =1,...,N
  • series
    r
     = inf{
    p
    ,
    q
    }
     is defined as
    r
    n
     = min{
    p
    n
    ,
    q
    n
    }
     for every
    n
    =1,...,N
  • inequality
    p
    ≤
    q
     means that
    p
    n
    ≤
    q
    n
     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
max
Q
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
,
w
as the input/output data for the
Intel® oneAPI Math Kernel Library
 convolution and correlation functions is described in theData Allocation.

Product and Performance Information

1

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