**In this work, the performance comparison of fuzzy
controllers using the MOI (mean-of-inversion) defuzzification
method and conventional proportional controllers with respect
to accuracy and convergence speed in the application of image
compression (JPEG) is given. The simulation results based on 15
USC pictures and various test points indicate that the fuzzy controller
with MOI performs is less sensitive to its configurations than
conventional proportional controllers. In additional, the largest
performance difference between these two kinds of controllers
is in the steep slope area. **

Fig. 1 shows the relation of the JPEG (Joint Photographic
Experts Group) input parameter *Quality*
and
the
distortion
ratio
AGE
on
15
USC
pictures
using
the
JPEG
model
developed
by
Independent
JPEG
Group.
The
distortion
ratio
AGE
(average
grayscale
error)
is
defined
as

(1)

where *G* is the maximum grayscale levels of
pictures and MAE (mean-absolute-error) is

(2)

where and
are the corresponding frames in the original and reconstructed
pictures, respectively, *N* is the dimension of the input
vector *I* (or frame *M)*, and *F* is the total
number of frames. Those 15 USC pictures, which are widely used
in image compression experiments, include eight 256x256 pictures
and seven 512x512 pictures.

As seen in Fig. 1, each image has its own unique curve which can only be known via simulations. Thus, although the lossy JPEG model is one of the most widely used digital image compression techniques for both grayscale and color still images [1], the compression results are not predictable in advance because of wide variety of image data. Consequently, a controller which can deal with a large collection of monotone functions so it is not necessary to change the controller's configuration from image to image is needed.

In [2-3], incorporating a fuzzy logic controller
using the MOI method with the JPEG model to achieve the automatic
control in the application of image compression was proposed.
The relation of JPEG and the fuzzy controller (or proportional
controllers) is depicted in Fig. 2. Fuzzy logic was introduced
by Zadeh in [4-5] and used by Mamdani and Assilian to control
the dynamic system in [6-7]. Since then, fuzzy logic has been
successfully applied to many automatic control applications [8].

As mentioned in [9], the proportional-integral-derivative (PID) controller is widely used in industry to deal with non-linear systems. It consists of three terms and can be characterized as [10]

(3)

where is the feedback from
controller(s) to controlled object(s), *t* is a positive
integer which represents the time index, and
is the difference between the target and the measured variable,
i.e.,

(4)

As seen in Eq. 3, the first term is the proportion of the error , the second terms is proportional to the integral of , and the third term is a proportion of the derivative . As mention in [9-12], the values of those three coefficients must be carefully chosen such that the stability and a good overall performance can be obtained, and trial-and-error is usually used to search proper values for those three coefficients in Eq. 3. As shown in Fig. 1, since basically we are dealing with monotone functions here, only P controllers (the first part of PID controllers) is adopted in this work.

As seen in [13], the mechanism used to force the convergence of fuzzy controllers is applied to PID controllers as well, that is:

(1) The present adjustment is no larger than the previous adjustment except in the case of overshooting, that is, .

(2) The present adjustment is smaller than the previous
adjustment in the case of overshooting, that is,
where *u* is 1 in this work.

The rest of this work is organized as follows. Section 2 describes the structure of the designed fuzzy controller. The comparison of the fuzzy controller with MOI and conventional proportional controllers in terms of accuracy and convergence speed is given in Section 3. Conclusions are presented in Section 4.

The internal structure of the designed fuzzy controller
is depicted in Figure 3. The two inputs,
and , to the fuzzy controller at the time
*t* are defined as

(5)

(6)

where TGE (target grayscale error) is the desired
output and AGE is defined in Eq. 1. The universal set of ,
*E*, is the interval of real numbers
while the universal set of , *Q*,
is the interval of integer numbers.

In this work, seven fuzzy subsets are used: PB (positive big), PM (positive medium), PS (positive small), ZE (zero), NS (negative small), NM (negative medium), and NB (negative big).

For an input crisp value *x*, the membership
value of a fuzzy subset is decided by

(7)

(8)

(9)

where C, L, and R are the central value, the left boundary, and the right boundary, respectively, and A={NM, NS, ZE, PS, PM}.

The following fuzzy subsets are used for the input and the entries of decision tables:

(1) **Group Q**:

*NB*(-100,
-50,
-20),
*NM*(-45,
-25,
-15),
*NS*(-20,
-10,
0),

*ZE*(-5,
0,
5),

*PS*(0,
10,
20),
*PM*(5,
25,
45),
*PB*(20,
50,
100)

As for the input , there are three groups of fuzzy subsets used as follows:

(1) **Group 1**.

NB(-100, -6, -2.5) NM(-5.25, -3, -0.75) NS(-2, -1, 0)

ZE(-0.5, 0, 0.5)

PS(0, 1, 2) PM(0.75, 3, 5.25) PB(2.5, 6, 100)

(2) **Group 2**:

NB(-100, -3, -1.25) NM(-2.6, -1.5, -0.4) NS(-1, -0.5, 0)

ZE(-0.25, 0, 0.25)

PS(0, 0.5, 1) PM(0.4, 1.5, 2.6) PB(1.25, 3, 100)

(3) **Group 3**:

NB(-100, -12, -5) NM(-11, -6, -1) NS(-4, -2, 0)

ZE(-1, 0, 1)

PS(0, 2, 4) PM(1, 6, 11) PB(5, 12, 100)

The relation among Groups 1, 2, and 3 of fuzzy subsets is as follows:

(1) Comparing to Group 1, the fuzzy subsets of Group 2 are moved close to the center of input distribution 0. In addition, the bases of those triangular fuzzy subsets are half of the widths as the corresponding one in the Group 1 of fuzzy subsets.

(2) The fuzzy subsets of Group 2 are gone toward opposite direction, i.e., moved away from the center 0. In addition, the bases of those triangular fuzzy subsets are twice as large as the corresponding one in the Group 1.

Table 1 is used as the decision table in this work.
The confidence value of decision table entry *ij*,
, is calculated using the fuzzy-min (or
intersection) operator

(10)

where is membership functions
defined in Eqs. 7-9.

The MOI method [3] for (the feedback from the controller to JPEG) is calculated as

(**MOI**)(11)

where K and L are the number of rows and columns in the decision table, respectively. is decided by the following algorithm:

Step 1. Calculate the corresponding crisp values
*T* of the fuzzy membership value .

where is the inverse function
of and *A* is the corresponding
fuzzy subset of . Note that there might
be more than one value generated by .

Step 2. Adjust each inverse value *T* for guaranteed
convergence.

Step 3. Pick the largest absolute value of *.*
If , ; otherwise,
.

For guaranteed convergence,
will be no larger than . Therefore, at
step *t*, is adjusted (for the MOI,
the COA, and the MOM methods) as follows:

(i) ,

(1) Overshooting occurred:

(2) No overshooting:

(ii) ,

where *m* is a positive integer, *u* is
the measure unit of *c* , and in
this work.

At last, the new input is calculated as

(12)

where and
are the maximum and minimum values in the input domain *C*.
Whenever , a special signal **STOP**
is generated to stop the adjusting process.

Experiments were performed on 15 USC images which
have been widely used in image compression experiments. Those
15 USC pictures include eight 256x256 pictures (Girl-1, Couple,
Lady, Girl-II, House, Tree, Ball-I, and Ball-II) and seven 512x512
pictures (Splash, Girl-III, Baboon, Lena, F16, Park, and Pepper).
Ten random initial values (38, 58, 13, 15, 51, 27, 10, 19, 12,
and 86) of *q* generated
by
C's
rand()
were
used
to
test
the
accuracy
of
the
designed
fuzzy
controller
where
*q* is
a
control
variable
used
to
adjust
reconstructed
images.
The
function
used
to
describe
the
behavior
of
JPEG
is

where *g* is the generated AGE. To measure the
performance of the fuzzy controller, we now define successful
trials and unsuccessful trials.

Definition 1. *Successful Trials*: We say the
fuzzy controller succeeds at a trial if either of two conditions
is met:

(1) The final value is
in the target range, that is, where *T*
is the TGE and is the tolerance.

(2) If no value of is inside
the interval of target range, and the closest values on both sides
of target range are and ,
respectively, then either or
where is the final value of *q*
after convergence.

Definition 2. *Unsuccessful Trials*: Any trial
which does not satisfy the conditions above.

In order to test the performance of controllers in
different regions of curves as shown in Fig. 1, three test points
are selected. Two of them, TGE=1.0 and TGE=1.5, are located in
flat area while TGE=5.0 is in the steep slope region. The simulation
results of the fuzzy controller with three different groups of
fuzzy subsets and the proportional controller with two different
coefficients, -10 and -30, at *TGE=1.0, TGE=1.5,*
and
*TGE=5.0* are
given
in
Tables
2,
3,
and
4,
respectively.
Note
that
accuracy
is
measured
in
terms
of
successful
trials
as
defined
above.

The large variety of image source data makes the prediction of control parameters vis-a-viz reconstructed image quality very difficult for the lossy model of JPEG. To lower the cost and achieve efficiency, some kinds of automatic control mechanism must be incorporated into JPEG.

The observations based on Tables 2, 3, and 4 are as follows:

(1) The performance of proportional controllers is
more sensitive to its coefficients than that of fuzzy controllers.
The biggest difference between two coefficients of the proportional
controllers is *(14.693-8.513=)6.18*
loops
while
the
biggest
difference
among
three
groups
of
fuzzy
subsets
is
3.927
loops.
In
addition,
the
proportional
controller
has
larger
difference
between
various
configurations
than
the
fuzzy
controller
at
all
three
test
points
as
shown
in
Tables
2-4.

(2)
According
to
the
simulation
results,
the
proportional
controller
with
the
coefficient
-30
has
better
performance
at
*TGE=1.5* while
the
fuzzy
controller
with
the
MOI
method
performs
much
better
in
the
steep
area
such
as
*TGE=5.0*.

(3)
As
proved
in
[13],
the
fuzzy
controller
with
MOI
defuzzification
method
can
achieve
guaranteed
accuracy;
therefore,
the
single
most
important
advantage
of
conventional
PID
controllers
is
no
longer
unmatched,
at
least
for
monotone
functions.
On
the
other
hands,
the
simulation
results
indicates
that
fuzzy
controllers
has
the
advantage
of
less
sensitive
to
their
configurations
which
makes
fuzzy
controllers
be
a
better
choice
in
this
application.

(4) It seems that the performance of these two kinds of controllers is heavily dependent on source images, locations of targets, coefficients (or configurations) of controllers, and the initial input value(s).

The author would like to thanks Drs. A.H. Sung and S. Mazumdar for their valuable suggestions.

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pp.
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C.J. Wu, "Performance Comparison of
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