89
(8-1)
C
= − X + ∑
(A
,
Y + B
,
U )
( , )∈
( , )
+ I
Where
the output equation Y
ij
can be written as
(8-2)
𝑌 = 𝑓 𝑋(𝑖, 𝑗) = (|𝑋 + 1| − |𝑋 − 1|)
The mathematical equation mentioned in Equation 8-1 is representing
the model of the
Continuous Time CNN (CT-CNN). In the equation,
C
is a linear capacitor and R is a resistor.
Y
kl
is the output state of each cell. U
kl
is the input of each cell. A
ij
and B
ij
are the template
elements. X
ij
represents the initial state and I represent the threshold or bias for each cell.
The Equation 8-2 is the output equation of each iteration. This equation gives the
functional model for the calculation of each pixel element to the output. This model in not
very fast in the real time image processing. In order to overcome
the drawbacks of CT-CNN,
the concept of Discrete Time CNN (DT-CNN) is developed. The DT-CNN is defined by the
difference equations instead of differential equations used in the CNN [140]. The model of
DT-CNN is derived from the model of CT-
CNN using the Euler’s method. The DT
-CNN can
be described with the following equation [140].
(8-3)
𝑋
,
(𝑡 + 1) ≈ ∑
𝐴 (𝑖, 𝑗; 𝑘, 𝑙)𝑓 𝑋
,
(𝑡) + ∑
𝐵 (𝑖, 𝑗; 𝑘, 𝑙)
( , )∈ ( , )
( , )∈ ( , )
𝑈
,
+ 𝐼
From Equation 8-3, we can see that X
i,j
is the state of the cell C(i, j) and f(X
k,l
) is the output
of cell C(k, l) within the neighborhood N
r(i, j)
of C(i, j). U
k,l
is the input of each cell C(k, l)
within N
r(i, j)
, and
I
is the bias of cell. A and B are called the
feed-back and feed-forward
templates of the CNN respectively.
CNNs are widely used for real time image processing applications. Though the CNN, as a
concept is characterized by a strict locality operation,
the large scale digital
implementation has been far from trivial [90].
