C. J. Wu A.H. Sung

Department of Library & Department of Computer Science

Information Science New Mexico Tech

Fu-Jen Univ. Socorro, NM 87801

Hsin-Chuang, Taiwan, ROC 242

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) method in the application
of image library indices in which only very rough images are needed
and high compression ratios are obtained. The simulation results
based on 15 USC pictures indicate that MOI performs better than
COA on convergence speed as target points located in the steep
slope region of curves.

**1. Introduction
**

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 [1], the compression results are not predictable in advance due to the wide variety of image data. In this work, the distortion ratio AGE (average grayscale error) is adopted and defined as

(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.

In the application of image library indices, only
very rough images are needed and high compression ratios can be
obtained. Fig. 1 shows the relation of the JPEG input parameter
*q* (which
is
defined
in
the
package
developed
by
the
Independent
JPEG
Group)
and
the
output
distortion
ratio
AGE
on
15
USC
pictures
using
the
JPEG
model
developed
by
the
Independent
JPEG
Group.
Those
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).
As
seen
in
Fig.
1,
each
image
has
its
own
unique
curve
which
can
only
be
known
via
simulations.
Therefore,
the
function
of
fuzzy
controllers
is
to
help
the
system
to
find
the
proper
input
value(s)
to
produce
the
desired
output
result(s),
i.e.,
the
compression
ratios
in
this
work.
In
addition,
we
need
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
image
to
image.

In [2-3], incorporating a fuzzy controller into the JPEG model to achieve the automatic control for image compression was proposed. Fuzzy logic was introduced by Zadeh in [4-5] and used by Mamdani and Assilian to control the dynamic systems in [6-7]. Since then, fuzzy logic has been successfully applied to many automatic control applications [8]. The relation of JPEG (including coder and decoder) and the fuzzy controller on the source side is depicted in Fig. 2.

In this work, a test point TGE (target grayscale
error) 5.0 is used because only rough reconstructed images are
needed in this application. In [9], the simulation results indicated
that COA and MOI had equivalent performance at test point TGE=1.5.
However, the reconstructed images generated under 1.5 are usually
too good to be used as image indices since it is waste of space.
Therefore, in this work the test point 5.0 is selected to produce
rough images and obtain high compression ratios, and the simulation
results will be used to compared with those in [9] to see the
effect of curve's slopes on accuracy and convergence speed of
fuzzy controllers. (As seen in Fig. 1, the test point 1.5 is in
the flat region of curves while the test point 5.0 is located
in the steep area.) Section 2 describes the structure of the designed
fuzzy controller. The comparison of MOI and COA in terms of accuracy
and convergence speed for TGE=5.0 is given in Section 3. Conclusions
are presented in Section 4.

**2. The fuzzy controller
**

The internal structure of the designed fuzzy controller
is depicted in Fig. 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}.

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

*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 , the following fuzzy subsets are used.

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)

In this work, Tables 1 and 2 are used by MOI and
COA, respectively. The difference between those two tables is
that some ZEs on the columns of NS and PS of Table 1 are replaced
with PSs and NSs, respectively, in Table 2. This is due to that
COA is quite sensitive to ZE. 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 [8, 10], one of the most popular
defuzzification methods is the center-of-area (COA) method [10].
For COA, the crisp output is the center
of gravity of distribution of membership function .
In the case of discrete universal set *W*, COA is defined
as

(**COA**) (9)

where *n* is the number of quantization levels
of the fuzzy set *C*, and .

In comparison with COA, MOI defuzzifies the output of each fired rule individually instead of superimposing fired rules before defuzzification, as shown in Eq. 10; thus, 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 (mean-of-inversion) method [2] for (the feedback from the controller to JPEG) is calculated as

(**MOI**) (10)

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 both
of MOI and COA) as follows:

(i) ,

(1) Overshooting occurred:

(2) No overshooting:

(ii) ,

where *m* is a positive integer and *u*
is the measure unit of *c*. (*u* equals 1 in this work.)

At last, the new input is calculated as

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.

** 3. Simulation results
**

Experiments were performed on 15 USC images which
have been widely used in image compression experiments. 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 this work the test point *TGE=5.0*,
which
is
located
in
the
steep
area,
is
used
to
test
the
performance
of
MOI
and
COA
on
both
aspects
of
accuracy
and
convergence
speed.
For
*TGE=5.0*,
the
simulation
results
of
MOI
and
COA
on
those
15
USC
images
with
tolerance
0
is
given
in
Table
3.

**4. Conclusions
**

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 kinds 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 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 between MOI and COA in the application of image indices using JPEG (based on those 15 USC images) is as follows: (Note more images may be needed to further verify the following conclusions.)

(i) As suggested in [9], COA usually has better performance
on convergence speed than MOI in the case of tolerance 0. However,
if the target point is located in the steep slope area, like *TGE=5.0*,
MOI
performs
much
better
than
COA
as
shown
in
Table
3.

(ii) As shown in Tables 3 and 4, the Comparison of fuzzy controller's performance at two test points TGE=1.5 and TGE=5.0 implies there is a close relation between fuzzy subset configurations and the gradient of the curves. However, this kind of relation may be application-dependent.

(iii)
Since
the
proposed
fuzzy
controller
has
the
quality
of
guaranteed
convergence
and
accuracy,
it
can
be
used
to
create
very
rough
images
as
indices
for
image
library
automatically.

**References
**

[1] G. K. Wallace, "The JPEG still Picture Compression
Standard," *Communications of the ACM*,
Vol.
34,
No.
4,
pp.
30-44,
1991.

[2]
C.J.
Wu
and
A.H.
Sung,
"The
Application
of
Fuzzy
Controller
to
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*Electronics Letters*,
Vol.
30,
No.
17,
pp.
1375-1376,
1994.

[3]
C.J.
Wu
and
A.H.
Sung,
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of
Fuzzy
Logic
to
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*IEEE Trans. Consumer Electronics*,
Vol.
40,
No.
4,
pp.
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C.C.
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[9]
C.J.
Wu
and
A.H.
Sung,
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of
the
MOI,
the
COA,
and
the
MOM
Methods
in
the
application
of
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Proc.
of
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(Vancouver,
British
Columbia,
Canada),
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[10]
H.R.
Berenji,
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to
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C.J. Wu and A.H. Sung, "Comparison
of MOI and COA in the Application of Image Library Indices,"
Proc. of IEEE Int. Conference on Fuzzy Systems (New Orleans, Louisiana,
USA), pp. 130-135, Sept. 1996.