**Abstract - In this paper, the application of a
Fuzzy ART (adaptive resonance theory) network with a fuzzy controller
to (grayscale) image data compression is presented. The unique
feature of a vigilance parameter of ART allows the direct control
of trade-off between compression ratio and image quality; the
fuzzy controller is used to adjust vigilance to seek a better
compromise automatically. Therefore, the network is insensitive
to the given initial vigilance values. Simulations are performed
and the results indicate that this fuzzy-control-equipped Fuzzy
ART network provides a promising technique for image data compression.**

The large amount of time and storage required to transmit pictorial data brings about the need of image data compression. In [1], the application of Adaptive Resonance Theory 1 (ART1) [2, 3], to image data compression was studied, and it showed that ART1 networks can be a promising alternative. In this work, another important issue for image data compression in real-time environment is studied. Since compression and distortion ratios depend on the source pictures and techniques used in the compression process; it is usually difficult to achieve the balance between those two ratios for various pictures in real-time environment. Trial-and-error may be used to search for the best tradeoff between distortion and compression ratios.

Fuzzy logic has been successfully used in many applications to control the process automatically. In this work, the incorporation of fuzzy controllers in the design of Fuzzy ART networks is investigated. The Fuzzy ART networks [4] has several advantages over the ART1 networks including less implementation cost and processing time, and the ability to handle the grayscale image. The rest of the paper is organized as follows. Section II describes the architecture and the learning algorithms of Fuzzy ART networks. The mechanism of fuzzy controller is introduced in section III. Simulation results are given in Section IV. At last, conclusions and discussions are given in section V.

The advantage of using adaptive resonance theory (ART) in image data compression is that it has an external control mechanism--the vigilance parameter--to control directly the trade-off between compression and distortion ratios. The Fuzzy ART has two advantages. One is the ability to handle both binary and analog vectors and the other is less implementation cost and time. In this work, we proposed a modified Fuzzy ART (MFART) network which is the hybrid of ART1 and Fuzzy ART networks. The MFART counts the grayscale difference between the input and category and picks up the one has the minimum difference instead of using fuzzy min operator in Fuzzy ART.

*The Learning Algorithm Of Modified Fuzzy ART
[4]*

Step 1: Initialize the vigilance parameter and weight
vector of each uncommitted node *j *as
follows:

(1)

(2)

where and
are the *2N*-dimensional weight vector and vigilance parameter
respectively, and *N* is the dimension of input vector before
transformation.

Step 2: Transform the *N*-dimensional input
vector , whose components are in the interval
[0,1], to *2N*-dimensional vector
as follow before presenting it to .

(3)

(4)

Step 3. The winning node (or category), say *j*,
is the node with the weight vector ()
most similar to input in terms of the
minimum difference of grayscale value between the input
and category *j*, that is, , in layer
where *2N* is the dimension of .
In case of tie, one of them is to be selected arbitrarily.

Step 3: The selected category *j* is said to
meet the vigilance criterion if the following inequality stand.

(5)

Step 4: If resonance, go to Step 5; otherwise, the
reset occurs and a new category (node) is added to layer *C*
unless there is no new node available. In that case, the operation
terminates.

Step 5: Update only the weight vectors associated
with the selected category *J *(either the winner
or new added category) as follows.

(6)

where is the fuzzy min operator.

Step 6: If no new input vector, terminate the process; otherwise, get the next input vector and go back to Step 2.

*The Measure Criteria*

The following criteria are used in this work mean absolute error (MAE), mean square error (MSE), single-to-noise-ratio (SNR), bit-per-pixel (BPP, compression ratio). They are defined as follows:

(7)

where and
are the subimage in the original and reconstructed pictures respectively,
*N*
is the dimension of the input vector *I* or subimages *M*,
and *F* is the total number of subimages

(8)

(9)

where *G*
is
the
maximum
grayscale
value
of
picture.

(10)

where *B* is the bits per pixel in the original
picture and *C* is the total number of categories formed
during training.

Fuzzy logic has been successfully applied to automatic control [4]. Therefore, we propose to incorporate a fuzzy controller into the MFART to achieve the automatic control during the compression process. The relation of MFART and fuzzy controller is depicted in Figure 2. The two inputs, and , to fuzzy controller are defined as follows:

(11)

where G is the maximum grayscale value.

(12)

(13)

In general, the membership function of a fuzzy set
*A, *, is defined as follows: [5]

(14)

where *X*
is
the
universal
set.
In
this
paper,
seven
fuzzy
sets
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
output
of
fuzzy
controller.

Their membership functions are defined in Eq. 15-17 and in the form of (left boundary, central value, right boundary). (15)

(16)

(17)

where C, L, and R are the central value, left boundary, and right boundary respectively and A={NM, NS, ZE, PS, PM}. For the input , we choose the membership functions of the fuzzy sets to be as follows:

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)

For the input and output , the membership functions of fuzzy sets are defined as follows:

NB (-1, -0.25, -0.1) NM (-0.15, -0.1, -0.05) NS (-0.1, -0.05, 0)

ZE (-0.025, 0, 0.025)

PS (0, 0.05, 0.1) PM (0.05, 0.1, 0.15) PB (0.1, 0.25, 1)

The decision table used by fuzzy controller is designed as follow.

The feedback from controller to MFART is decided as follow: [6]

(19)

where K and L are the number of rows and columns
in the decision table respectively, and
is the value of entry *ij*
in
the
decision
table.
is decided by the following algorithm
to guarantee that will be smaller than
in terms of crisp value to force the
convergence, that is, reaches 0 and the
process will terminates.

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

(20)

where is the inverse function
of and *A*
is
the
corresponding
fuzzy
set
of
. Please note there might have more than
one value generated by .

Step 2. Pick up the biggest *T,*
say
,* *
whose
absolute
value----is
smaller than , that is, .
Otherwise, .

Experiment is performed on Lena, of
pixels with 256 grayscale using the Modified Fuzzy ART network
with fuzzy controller. It is well known that the presented order
of subimage will affect the system performance. To eliminate this
effect, each subimage is reassigned to a cluster which has the
minimum grayscale difference with this subimage after the initial
categories are formed. Simulation results with parameters--,
initial , and *4x4*
frames--are
listed
in
Table
2.

Since we can not predict the proper vigilance value in advance, with the help of fuzzy controller, the ART1 network can be insensitive to a given initial vigilance values. Nevertheless, our computer simulations show that the final vigilance will converge to the target range. In addition, the Modified Fuzzy ART has two advantages over the ART1--less processing time and implementation cost. Hence, the image data compression using MFART with fuzzy controller in the real-time environment is promising. The on-going research is to tune-up the fuzzy controller to keep the final vigilance as close as possible to the target value.

[1] C. J. Wu, A. H. Sung, and H. S.
Soliman, "Image Data Compression Using ART Networks,"
in *Proc. Artificial Neural Networks In Engineering*, 1993,
pp. 417-422.

[2]
S.
Grossberg,
"Adaptive
Pattern
Classification
and
Universal
Recording:
I.
Parallel
Development
and
Coding
of
Neural
Feature
Detectors,"
*Biological Cybernetics*,
vol.
23,
pp.
121-134,
1976.

[3] G. A. Carpenter and S. Grossberg,
"A Massively Parallel Architecture for a Self-Organizing
Neural Pattern Recognition Machine," *Computer Vision,
Graphics, and Image Processing*, vol. 37, pp. 54-115, 1987.

[4] G. A. Carpenter, S. Grossberg, and D. B. Rosen, "Fuzzy ART: An Adaptive Resonance Algorithm for Rapid, Stable Classification of Analog Patterns," pp. 411-416, 1991.

[4] M. Sugeno, *Industrial Applications
of Fuzzy Control*. North Holland, Amsterdam, 1985.

[5] G. J. Klir and T. A. Folger, *Fuzzy
Sets, Uncertainty, and Information*, Englewood Cliffs, NJ:
Prentice- Hall, 1988.

[6] B. Kosko, *Neural Networks and
Fuzzy Systems: A Dynamical Systems Approach to machine Intelligence*,
Englewood Cliffs, NJ: Prentice-Hall, 1992.

C.J. Wu, A.H. Sung, and H.S. Soliman, "A
Fuzzy ART Network with Fuzzy Control for Image Data Compression,"
Proc. of IASTED Int. Conf. on Modeling, Simulation and Control
in the Process Industry, (Otiwa, Canada), pp. 95-98, May 1994.

**Figure 3: The reconstructed image
of Lena
**