Performance Comparison of MOI and COA in the Application of Image Library Indices

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

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) 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

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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.