PERFORMANCE COMPARISON OF THE MOI, THE COA, AND THE MOM METHODS IN THE APPLICATION OF JPEG

C. J. Wu and A.H. Sung

Department of Computer Science

New Mexico Tech

Socorro, NM 87801

ABSTRACT

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.

1. INTRODUCTION

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.

2. THE MECHANISM OF 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

(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.

3. SIMULATION RESULTS

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.

4. CONCLUSIONS AND DISCUSSIONS

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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C.J. Wu and A.H. Sung, "Comparison of the MOI, the COA, and the MOM Methods in the Application of JPEG," Proc. of IEEE Int. Conference on Systems, Man and Cybernetics (Vancouver, British Columbia, Canada), pp. 2019-2024, Oct. 1995.