**Abstract: **A simple and
efficient fuzzy controller for image compression using the JPEG
lossy model is proposed. A new defuzzification method, the MOI
(mean-of-inversion) method, is also proposed. The performance
of the MOI method is compared with that of the COA (center-of-area)
method with respect to convergence speed. Unlike the COA method,
the MOI method defuzzifies each fired rule separately instead
of superimposing fired rules before defuzzification. Consequently,
the MOI method is less sensitive (than the COA method) to the
contents of decision tables (or rule bases). Simulation results
based on 15 pictures and 4 groups of fuzzy subsets also indicate
that, under certain conditions, the MOI method competes favorably
with the COA method in convergence speed.

**Index terms****:
MOI, COA, JPEG, Fuzzy Controller, Defuzzification.**

The lossy JPEG model is one of the most widely used
digital image compression techniques for both grayscale and color
still images [1]. However, that compression results are not predictable
in advance is a common problem shared by such lossy image data
compression techniques due to the large variety of image data.
For a picture of *M *x
*N* pixels
with
*P* grayscale
levels,
the
possible
combinations
are
up
to
. With this kind of variety, it is very
difficult for any lossy compression model to predicate the relation
of its parameters and compression results, such as mean square
error. In other words, the compression and distortion ratios may
depend heavily on the source image data.

The following ratio AGE (average grayscale error) is used in this work.

(1)

where *G*
is
the
maximum
grayscale
value
of
pictures.
MAE
is
the
mean
absolute
error
defined
as

(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.

The rest of paper is organized as follows. Section II describes the structure of the designed fuzzy controller. The various experimental results are given in section III. Lastly, conclusions are given in section IV.

In this work, we propose a fuzzy controller to be used in conjunction with the JPEG model to achieve the automatic control during compression process on source side. The relation of JPEG (including coder and decoder) on source side is depicted in Figure 1.

The two inputs, and ,
to the fuzzy controller at the time *t* are defined as

(3)

(4)

The universal set of , *E*,
is
the
interval
of real numbers, while the universal
set of , *Q*,
is
the
interval
of integer numbers.

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 those
inputs and the output of fuzzy controller. For an input crisp
value, the membership value of fuzzy subset *A*
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

**Group 0**:

*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 , Four fuzzy subsets used are

**Group 1**:

*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)

**Group 2**:

*NB(-100, -3, -1.5), NM(-2.5, -1.5, -0.5),
NS(-1, -0.5, 0), ZE(-0.25, 0, 0.25), PS(0, 0.5, 1), PM(0.5,
1.5, 2.5), PB(1.5, 3, 100).
*

**Group 3**:

*NB*(-100, -12, -6), *NM*(-10,
-6, -2), *NS*(-4, -2, 0), *ZE*(-1, 0, 1), *PS*(0,
2, 4), *PM*(2, 6, 10), *PB*(6, 12, 100)

**Group 4**:

*NB*(-100, -1.5, -0.75),
*NM*(-1.25, -0.75, -0.25), *NS*(-0.5, -0.25, 0), *ZE*(-0.125,
0, 0.125), *PS*(0, 0.25, 0.5), *PM*(0.25, 0.75, 1.25),
*PB(0.75, 1.5, 100).*

Two decision tables are used in this work, Table
1 for the MOI method (defined in Eq. 10 below) and Table 2 for
the COA method (defined in Eq. 9 below). 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 [2, 3], the center-of-area (COA)
method is one of the most commonly
used defuzzification methods. 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 by [4]

(**COA**)
(9)

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

The MOI method 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
fuzzy
membership
value
.

(11)

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.

(12)

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

For guaranteed convergence, will be no larger than . In addition, in the case of overshooting, will be smaller than . The proof of guaranteed convergence can be seen in [5].

Experiments were performed on 15 USC images which
include eight 256x256 images and seven 512x512 images with 256
grayscales. The eight 256x256 pictures are Girl-I, Couple, Lady,
Girl-II, House, Tree, Ball-I, and Ball-II. The seven 512x512 pictures
are Splash, Girl-III, Baboon, Lena, F16, Park, and Pepper. The
JPEG model is developed by Independent JPEG Group with the fuzzy
controller created by the authors [6]. In addition, ten random
initial values, 38, 58, 13, 15, 51, 27, 10, 19, 12, and 86, of
*q* generated
by
C's
rand()
are
used
to
test
the
performance
of
the
designed
fuzzy
controller.
The
performance
comparison
of
the
MOI
and
the
COA
methods
in
terms
of
accuracy
under
these
four
groups
can
be
seen
in
[5].

The simulation results are given in Tables 3 and 4 for the cases of tolerance 0 and 0.025, respectively. As seen in Tables 3 and 4 , the convergence speed is affected by the groups of fuzzy subsets used and tolerance significantly.

Unlike the COA methods, the MOI method defuzzifies each fired rule individually. As shown in Eq. 11, the result of defuzzification for each rule is the corresponding inverse value of fired membership grade with respect to the membership function.

As observed in Tables 3 and 4, for the MOI method, the convergence speed is greatly improved when a small nonzero tolerance is used. Usually, the COA method converges faster than the MOI method when tolerance 0 is used, as indicated in Table 3. However, as Table 4 shows, the MOI method could outperform the COA method when a small amount of tolerance, e.g., 0.025 of AGE, is used. Also note that the convergence speed depends heavily on images and initial input values, as can be seen in Tables 3 and 4.

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

[2]
R.M.
Tong,
"A
Control
Engineering
Review
of
Fuzzy
Systems,
"
*Automatica*,
vol.
13,
1977,
pp.
559-568.

[3]
C.C.
Lee,
"Fuzzy
Logic
in
Control
Systems:
Fuzzy
Logic
Controller,"
*IEEE Trans. Syst. Man. Cybern.*,
vol.
20,
no.
2,
1990,
pp.
404-435.

[4] H.R. Berenji, "Fuzzy Logic
Controllers," in *An Introduction to Fuzzy Logic Applicatons
in Intelligent Systems*, R.R. Yager and L.A. Zadeh, Boston:
Kluwer Academic Publishers, 1992, pp. 69-96.

[5] C.J. Wu, "A General Purpose Fuzzy Controller for JPEG and Monotone functions," PH.D. Thesis, New Mexico Tech., Socorro, New Mexico.

[6]
C.J.
Wu
and
A.H.
Sung,
"The
Application
of
Fuzzy
Logic
to
JPEG,"
*IEEE Trans. Consumer Electronics*,
vol.
40,
no.
4,
1994,
pp.
976-984.