PERFORMANCE COMPARISON OF FUZZY AND PROPORTIONAL CONTROLLERS IN THE APPLICATION OF IMAGE COMPRESSION

C. J. Wu

Department of Library & Information Science

Fu-Jen Univ.

Hsin-Chuang, Taiwan, ROC 242

ABSTRACT

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.

1. INTRODUCTION

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.

2. THE FUZZY CONTROLLER

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.

3. SIMULATION RESULTS

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.

4. CONCLUSIONS AND DISCUSSIONS

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).

ACKNOWLEDGMENTS

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

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C.J. Wu, "Performance Comparison of Fuzzy and Proportional Controllers in the Application of Image Compression," Proc. of IEEE Int. Conference on Systems, Man and Cybernetics (Beijing, PRC), Oct. 1996, to appear.