Cellular Neural Network
Simulation and Modeling
Oroszi Balázs
2006.01.06.
Overview
Introduction: About the CNN in general
Basic characteristics of the CNN
Modeling and simulation of the CNN architecture
The functional model of the CNN architecture
–
Handling special cases to increase performance
From theory to practice: Realization of the CNN
simulator
Summary
Demonstration
About the CNN in general
In
1988
papers
from
Leon
O
.
Chua
introduced
the
concept
of
the
Cellular
Neural
Network
.
CNNs
can
be
defined
as
“
2
D
or
3
D
arrays
of
mainly
locally
connected
nonlinear
dynamical
systems
called
cells,
whose
dynamics
are
functionally
determined
by
a
small
set
of
parameters
which
control
the
cell
interconnection
strength
”
(Chua)
.
These
parameters
determine
the
connection
pattern,
and
are
collected
into
the
so

called
cloning
templates
,
which,
once
determined,
define
the
processing
of
the
whole
structure
.
Basic Characteristics of the CNN
The CNN can be defined as an
M x N
type array of identical cells
arranged in a rectangular grid. Each cell is locally connected to its 8
nearest surrounding neighbors.
Each cell is characterized by
u
ij
,
y
ij
and
x
ij
being the
input
, the
output
and the
state
variable of the cell respectively.
The
output
is related to the
state
by the nonlinear equation:
y
ij
=
f
(
x
ij
) = 0.5 (
x
ij
+ 1
–

x
ij
–
1)
The state transition of neuron (i, j) is governed by the following
differential equation:
Basic Characteristics of the CNN (2)
Where C(i,j) represents the
neuron
at column i, row j, S
r
(i,j) represents
the neurons in the
radius r
of the neuron C(i,j), and z
i,j
is the
threshold
(bias) of the cell C(i,j).
The coefficients
A
(
i
,
j,
k
,
l
) and
B
(
i
,
j,
k
,
l
) are known as the
cloning
templates
. In general, they are
nonlinear
,
time

and
space variant
operators
.
If they are considered
linear
,
time

and
space invariant
, they can
simply be represented by matrices.
Modeling and simulation of the CNN architecture
Simulation plays an important role in the design of the CNN cloning
templates.
Therefore, it has to be
fast
enough to allow the design phase of various
templates be accomplished in reasonable time.
At the same time, the simulation has to be
accurate
enough, to reflect
the behavior of the analog circuitry correctly.
In practice, the simulation of the CNN involves a trade

off between
accuracy and computation time.
Modeling and simulation of the CNN architecture (2)
The true processing capabilities of CNNs for high

speed parallel
processing are only fully exploited by dedicated VLSI hardware
realizations.
Typical CNN chips may contain up to 200 transistors per pixel.
At the same time, industrial applications require large enough grid
sizes (around 100 x 100).
Thus, CNN chip designers must confront complexity levels larger than
10
6
transistors, most of them operating in analogue mode.
Modeling and simulation of the CNN architecture (3)
On
the
one
hand,
high

level
simulation
,
which
is
focused
on
emulating
the
functional
behaviour,
cannot
reflect
realistically
the
underlying
electronic
circuitry
.
Their
lack
of
detail
makes
them
ill

suited
for
reliable
IC
simulation
.
On
the
other
hand,
the
SPICE

type
transistor

level
simulators
,
although
very
accurate,
are
barely
capable
of
handling
more
than
about
10
5
transistors
and
may
take
several
days
of
CPU
time
for
circuit
netlists
containing
about
10
6
transistors
.
Hence,
these
low

level
tools
are
ill

suited
for
simulating
large
CNN
chips
.
Therefore,
it
would
be
necessary
to
bridge
the
gap
between
these
approaches,
which
would
give
very
accurate
results
in
reasonable
(but
not
real

)
time
.
However,
our
main
concern
now
is
fast
simulation,
so
in
the
rest
of
this
presentation
we
shall
focus
on
the
functional
modeling
of
the
CNN
architecture
.
The functional model of the CNN architecture
The output of a CNN model simulation is the final state reached by the
network after evolving from an initial state under the influence of a
specific input and boundary conditions. The following block diagram
shows the state

transition and output of a single cell:
The functional model of the CNN architecture (2)
In
the
most
general
case,
the
final
state
of
one
cell
can
be
described
by
the
following
equation
:
As
a
closed
form
for
the
solution
of
the
above
equation
cannot
be
given,
it
must
be
integrated
numerically
.
For
the
simulation
of
such
equations
on
a
digital
computer,
they
must
be
mapped
into
a
discrete

time
system
that
–
emulates
the
continuous

time
behavior
,
–
has
similar
dynamics
–
and
converges
to
the
same
final
state
.
The
error
committed
by
this
emulation
depends
on
the
choice
of
the
method
of
integration,
i
.
e
.
the
way
in
which
the
integral
is
calculated
.
The functional model of the CNN architecture (3)
There is a wide variety of integration algorithms that can be used to
perform this task. However, only three of them are going to be considered
here. These methods are:
the
explicit
Euler’s
formula
:
the
predictor

corrector
algorithm
:
and the fourth

order Runge

Kutta method:
where
where
The functional model of the CNN architecture (4)
The Euler method is the
fastest
, but gives the
least accurate
convergence behaviour.
Runge

Kutta gives the
best results
, however,
much slower
. In this
case, four auxiliary components (k1

k4) are computed. These are
auxiliary values, which are then averaged. This makes it rather ill

suited for applications, that prefer speed over accuracy.
If the main goal would be accuracy and robustness, undoubtedly
Runge

Kutta would be the method of choice. In our case, however, as
the primary target is a fast, working implementation of a CNN
simulator as an image processor, we shall choose the Euler method.
Handling special cases for increasing performamce
What can be considered a special case from a programming point of
view?
–
special input
–
special templates (A, B)
To gain significant speed improvements, the case of special templates
should be examined.
It is not uncommon within templates that extract local properties of
the image (like edge detectors) to use a fully zero A template.
I have discovered, that revisiting the state equation when the A
template is fully zero, significant improvements in speed can be
achieved.
Handling special cases for increasing performamce (2)
Given A = 0, the state equation takes the following form:
(BU + Z) is constant during the process. Let: BU + Z = C
Using Euler integration:
The pattern can clearly be seen by now.
Handling special cases for increasing performamce (3)
In each new step
X
0
(1

Δt)
n
gets multiplied by
(1

Δt)
so it’s power index
increases. The remaining part is a geometric series, so the general
equation of calculating the n

th state is:
Using the general formula of calculating the sum of a geometric series:
So the state equation using this will be:
the sum of the above geometric series turns into:
Handling special cases for increasing performance (4)
This result is of utmost importance regarding speed, because:
–
the number of iterations that need to be performed to get to the n
th
state is reduced to 1
–
thus we can get to the final state
immediately
, given the U input,
the B template and Z bias
As multiple iterations through an image causes lots of non

cacheable
memory accesses (which is very slow), this improvement in the special
case of A = 0 gives a huge boost in speed.
From theory to practice:
Realization of the CNN simulator
Environment used: Avisynth (http://www.avisynth.org)
–
A powerful tool for video post

production.
–
Special programming language, designed specifically for video
processing.
–
It’s functions are implemented under

the

hood as C/C++ dynamic link
libraries (DLLs), which are called
plugins
.
–
Plugins expose an interface (functions) towards the scripting language,
from which these functions can be called.
–
The CNN simulator is realised as a plugin (DLL) for Avisynth, written in
C++.
Primary goal: speed
–
but also make sure it behaves according to the state

equation
Summary
CNN simulation:
–
functional modeling (mathematical calculation according
to the state

equation)
–
circuit

level modeling
Implementation:
–
based on functional model
–
using Avisynth (http://www.avisynth.org)
–
written in C++ programming language
–
available in my web

space at
http://digitus.itk.ppke.hu/~oroba
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