In this work, the performance of the MOI (mean-of-inversion) method with respect to accuracy and convergence speed is compared with that of the COA (center-of-area) and the MOM (mean-of-maximum) methods in the application of JPEG. The feature of the MOI method is that it defuzzifies each fired rule individually instead of superimposing fired rules before defuzzification. As a result, the MOI method is less sensitive to the contents of decision tables. The simulation results based on 15 USC pictures indicate that the MOI method has the best performance on accuracy and ties the COA method in the aspect of convergence speed. The MOM method has the worst performance on both aspects of accuracy and convergence speed.

Fuzzy logic was introduced by Zadeh in [1-2] and used by Mamdani and Assilian to control the dynamic system in [3-4]. Since then, fuzzy logic has been successfully applied to many automatic control applications [5]. Although the effect of fuzzy controller components on overall performance of controller is not entirely clear, the design of fuzzy controllers is well known for its simplicity as mentioned in [6].

Although the lossy JPEG (Joint Photographic Experts
Group) model is one of the most widely used digital image compression
techniques for both grayscale and color still images [7], the
compression results are not predictable in advance due to wide
variety of image data. For a picture of *MxN*
pixels
with
*P* grayscale
levels,
there
are
up
to
possible
combinations. In other words, the compression and distortion ratios
depend heavily on the source image data.

Figure 1 shows the relation of the JPEG
input parameter *Quality*
and
the
distortion
ratio
AGE
(average
grayscale
error)
on
15
USC
pictures
using
the
JPEG
model
developed
by
Independent
JPEG
Group.
These
15
USC
pictures,
which
are
widely
used
in
image
compression
experiments,
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).
The
ratio
of
AGE
is
defined
by

(1)

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

(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.
As
seen
in
Fig.
1,
each
image
has
its
own
unique
curve
which
can
only
be
known
via
simulations.
Therefore,
our
aim
is
to
design
a
fuzzy
controller
that
can
deal
with
a
large
collection
of
monotone
functions
so
it
is
not
necessary
to
change
the
controller's
configuration
from
function
to
function.

For seeking better trade-off between distortion and compression ratios, incorporating a fuzzy controller with the JPEG model to achieve the automatic control during compression process on source side was proposed in [8-9]. The relation of JPEG (including coder and decoder) and the fuzzy controller on source side is depicted in Figure 2.

Section 2 describes the structure of the designed
fuzzy controller. The comparison of the MOI, the MOM, and the
COA methods in terms of accuracy and convergence speed in the
application of JPEG is given in Section 3. Conclusions and discussions
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

(3)

(4)

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

(5)

(6)

(7)

where C, L, and R are the central value, the left boundary, and the right boundary, respectively, and A={NM, NS, ZE, PS, PM}. For example, the fuzzy subsets of the form (L, C, R) used for the input and the entries of decision tables are

*NB*(-100, -50, -25), *NM*(-40,
-25, -10), *NS*(-20, -10, 0), *ZE*(-5, 0, 5), *PS*(0,
10, 20), *PM*(10, 25, 40), *PB*(25, 50, 100)

As for the input, the fuzzy subsets used are

*NB*(-100, -6, -3), *NM*(-5,
-3, -1), *NS*(-2, -1, 0),

*ZE*(-0.5, 0, 0.5),

*PS*(0, 1, 2), *PM*(1,
3, 5), *PB*(3, 6, 100)

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

(8)

where
is membership functions defined in Eqs. 5-7.

In the literature [5, 10], the two most
popular defuzzification methods are the mean-of-maximum (MOM)
and the center-of-area (COA) methods [10]. For the MOM method,
the crisp output is
the
mean
value
of
all
points
whose
membership values are maximum. In the
case of the discrete universal set *W*,
the
MOM
method
is
defined
as

(**MOM**)(9)

where
and *n*
is
the
number
of
such
support
values.
As
for
the
COA
method,
the
crisp
output
is the center of gravity of distribution
of membership function .
In the case of the discrete universal set *W*,
the COA method is defined as

(**COA**)(10)

where *n* is the number of quantization levels
of the fuzzy set *C*, and .
The comparison of these two defuzzification methods can be seen
in [5]. In addition, other defuzzification strategies
can be found in [11-14].

The basic feature of the MOI method is that it defuzzifies each fired rule individually instead of superimposing fired rules before defuzzification. As shown in Eq. 11, the result(s) of defuzzification for each rule are the corresponding inverse value(s) of the fired membership grade with respect to the membership function. The MOI method 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.

The JPEG package used in our experiments was developed
by the Independent JPEG Group with the fuzzy controller created
by the authors [8-9]. Experiments were performed on 15 USC images
which have been widely used in image compression experiments and
include eight 256x256 images and seven 512x512 images with 256
grayscales. In addition, ten random initial values (38, 58, 13,
15, 51, 27, 10, 19, 12, and 86) of *q* generated by the random
number generation function rand() are used to test the accuracy
of the designed fuzzy controller. The function used to describe
the behavior of JPEG is

where g is the generated AGE and *q*
is
the
value
of
parameter
Quality.
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.

Initially, Table 1 is used as the decision table. For the target grayscale error (TGE) 1.5, the simulation results of the configuration above on those 15 USC images for the MOI, the MOM, and the COA methods are given in Table 2. As shown in Table 2, based on those 15 images, only the fuzzy controller using the MOI method has no failure for all 150 trials; thus, the MOI method has the best performance on accuracy. The unsuccessful trials (UST) of the MOM and the COA methods are 134 and 41 (out of 150), respectively.

As for the convergence speed, in the case of TGE=1.5,
the average loops per trial of the MOI, the MOM, and the COA methods
are 8.26, 2.62, and 5.293, respectively. Clearly, the slow convergence
of the MOI method is the price paid for its accuracy. However,
this disadvantage of the MOI method is dramatically offset when
a small amount of tolerance is used, as shown in Table 3. More
specifically, for TGE=1.5 with tolerance 0.025, the average loops
per trial of the MOI and the COA methods are 5.38 and 4.7, respectively.
Although the COA method has slightly faster convergence, it has
27 unsuccessful trials out of 150 while the MOI method has no
failure. Therefore, based on Tables 2 and 3, the MOI method is
the best choice if the accuracy is the primary concern.

By closely examining the failure of
the COA method on accuracy, we hypothesize that arise the failure
coming from the combined effects of the numerator using a sum
(or an integral in the case of continuous distribution) in Eq.
10 and too many fuzzy subset ZE's in Table 1. As a result, sometimes
the numerator becomes 0, and the adjustment process will be forced
to stop when *q*
gets
close
to
the
target.

In
order
to
overcome
this
problem
for
the
COA
method,
Table
4
is
used
to
rerun
the
simulations
to
test
our
hypothesis.
As
seen
in
Table
4,
those
fuzzy
subset
ZE's
in
the
upper
part
of
column
3
in
Table
1
are
replaced
with
the
fuzzy
subset
PS's
while
those
in
the
lower
part
of
column
4
are
replaced
with
the
fuzzy
subset
NS's.

As can be seen in Table 5, there is
no failure for the COA method using Table 4 at TGE=1.5 with tolerance
0. This example clearly shows that decision tables can affect
the controller performance and are closely related to defuzzification
methods. Note that the MOI method also has no failure when it
uses Table 4. This indicates that the MOI method is less sensitive
to the design of decision tables than the COA method.

The convergence speed of the MOI method (tolerance 0.025) is 5.380 average loops per trial in the cases of TGE=1.5, as shown in Table 6. For the COA method with tolerance 0.025, the average loops per trial is 5.447 for TGE=1.5 .

Although the revised decision table
is able to help the COA method achieve 100% accuracy as shown
in Table 5 in the case of TGE=1.5, it has little impact on the
MOM method in terms of accuracy, as shown in Table 7. The MOM
method has 132 unsuccessful trials in the case of TGE=1.5. This
suggests that the MOM method is not suitable in the application
of JPEG.

JPEG is currently one of widely used image compression techniques for grayscale and color still pictures; however, the large variety of image source data makes the prediction of control parameters vis-a-viz reconstructed image quality very difficult for lossy models. To lower the cost and achieve efficiency, some kind of automatic control mechanism must be incorporated into JPEG.

Since we are dealing with a collection of functions, it is worthwhile to point out that the following two special design restrictions are needed to simultaneously achieve the guaranteed convergence and good performance in this application of JPEG.

(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 the minimum subtraction unit
*u *is
1
in
this
application.

The first criterion contributes to good performance while the second guarantees convergence.

The summary of the comparison among the MOI, the COA, and the MOM methods in the application of JPEG (based on those 15 USC images) is as follows:

(1) There is a close relation between decision tables and defuzzification methods; however, the degree of dependency varies from defuzzification method to defuzzification method. Clearly, the less sensitive method is preferred if accuracy is the primary concern. The MOI method can yield the more accurate results since it is the least sensitive defuzzification method of these three defuzzification schemes to the contents of decision tables.

(2) The MOM method performed badly in terms of accuracy, it does not seem suitable for this application, JPEG.

(3) When the COA method is used, the fuzzy set ZE should be used judiciously because the COA method is sensitive to the ZE's in the decision table.

(4) When the MOI method is used, a small amount of tolerance should be used to improve the convergence speed. For example, when tolerance 0.025 is used, the convergence speed is faster up to 34.87% (comparing with tolerance 0) in the case of TGE=1.5 (see Tables 2 and 3). From our experience, we know the difference of 0.025 of AGE, which is equal to 0.0638 of MAE, will have very little impact on reconstructed image quality in most of the cases.

(5) The COA method usually has a slightly faster convergence than the MOI method (when the same decision table is used), but the difference will be insignificant if a small amount of tolerance is used, as shown in Table 6. Actually, according to Table 6, the MOI method is little faster than the COA method (5.38 to 5.447 in the case of TGE=1.5) when a small amount of tolerance is used. Note that the MOI method goes with the decision table Table 1 while the COA method uses Table 4.

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1965,
pp.
338-353.

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to
the
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of
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