International Conference on Information, Communications and Signal Processing ICICS '97 Singapore, 9-12 September 1997

Low Frequ cy Coefficient Prediction for Image Coding W.K. Cham and M.C. Au Yeung

Department of Electronic Engineering The Chinese University of Hong Kong

Shatin, N.T., Hong Kong wkcham 0 ee.cuhk.edu .hk

Abstract

It has been shown that about 80% of the DC coefficients of a transformed image can be discarded at the encoder and then restored at the decoder using a DC coefficient restoration (DCCR) algorithm. This approach can im- prove the compression rate of JPEG by over 10% at low bit rates. In this paper, we extend the DCCR algorithm not only to restore DC coefficients but also to predict some of the (0,l)th and ( 1 , O ) t h AC coefficients. The prediction thus allows these AC coefficients to be repre- sented using less bits. Experimental results show that, almost in all cases, images coded using the proposed algorithm have higher PSNR and better visual quality than those coded using only the DCCR.

1 Introduction

Transform coding is one of the most popular ap- proaches for image compression. It achieves compres- sion by compacting signal energy into a few low fre- quency coefficients using a transform and then allocat- ing more bits to the high energy low frequency coeffi- cients and less bits to low energy high frequency coef- ficients for quantization. While DC coefficients have the highest energy among all coefficients, Cham and Clarke [11 showed that DC coefficients of low activ- ity images could be discarded and then restored from the AC components by using an image model called minimum edge difference (MED). Experimental results show that in some cases reasonable visual image qual- ity can be maintained even if the values of the DC co- efficients restored using the MED model deviate quite substantially from the original. Hence, Me compression ability of a transform coding scheme, such as JPEG, may be improved by discarding the DC coefficients at the encoder and restoring them using a DC coefficient restoration @CCR) algorithm at the decoder.

However, all three DCCR algorithms proposed in 111 restore the DC coefficients with large errors at high activity regions of an image and, moreover, these

mcau@ee.cuhk.edu.hk

errors propagate to other parts of the image. Tse and Cham [21, by applying the MED criterion globally and retaining a small portion of the DC coefficients, successfully restored DC coefficients precisely with- out propagation errors. They went on showing that the same technique could be used to restore not only all DC coefficients but also a portion of the low frequency AC coefficients if the Walsh transform is used [31. In this paper, we propose a low frequency coefficient pre- diction (LFCP) algorithm, which can be used together with the DCCR, to predict the high energy (1,O)th and (0,l)th coefficients with good precision for the DCT- based JPEG. Hence, the (1,O)th and (0,l)th coeffi- cients can be represented using the prediction differ- ence using less bits and, as a result, higher compression rate can be achieved.

The paper is organized as follows. In section 2, the DCCR algorithms described in [ 1, 21 are reviewed. In section 3, the proposed LFCP algorithm is described. The simulation results and discussions are provided in section 4. Finally, section 5 is the conclusions.

2 The MED Model and DCCR

Consider an image which is divided into non- overlapping blocks each having n x n pixels. Let ai,j be the DC coefficient and [ui,j (p, q)] be the matrix of pix- els of the (i, j)th block, where (p, q) is the (p, q)th pixel in the matrix. The AC component of the block is

1 ... 11

The MED model suggests that in the (i, j) th block the value of a i j tends to minimize I Dl,i,j l2 + I Dz,i,j l 2 + I %i,j+l 1' + I D2,i+l,j 1' where Di,i,j, Da,i,j, D~,i,j+i and DZ,i+l,j are the Edge Difference vectors atthefouredgesofthe (i,j)thblockasshowninFig. 1. The k-th elements of theEdgeDifference vectors Dl,i,j

0-7803-3676-3/97/$10.00 0 1997 IEEE

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Therefore, given the AC components [w;,j (p, q)] and the DC coefficients of the four blocks surrounding the ( i , j) th block, we can obtain an estimate of ai,j using the MED model. If the absolute difference between the estimated and the original values of ai,j is larger than a threshold, then the MED condition is said to be not satisfied or the MED model is said to have failed.

A simple way to take advantage of the MED model is the Element Estimation [ 11 which estimates a i j from the estimated values of and the AC com- ponents. However, at the blocks where the MED model fails, large estimation errors will occur. Moreover, these estimation errors propagate along the direction of esti- mation and appear even in the blocks where the MED condition is satisfied. For a high activity image which contains many blocks not satisfying the MED condi- tion, Element Estimation cannot recover an image to one of reasonable visual quality.

This problem was tackled by a global minimum ap- proach which estimates all DC coefficients by minimiz- ing the mean-square magnitudes of all edge difference vectors [2]. Suppose there are N DC coefficients in an image. The global minimum can be found by solving a set of N linear equations with the N DC coefficients as variables. Hence, we need to find the inverse of a N x N matrix. For example, N is 4096 for an image of size 512 x 512 and block size 8 x 8. In [2], succes- sive over-relaxation (SOR) [4] is used to determine the solution of such a large system of linear equations. The estimated DC coefficient in the ( i , j ) th block after the m-th iteration is

(4)

where ( 5 )

. N-1 N-1

+ f r 4 , i j (k) + dZ,i,j(k) k=O k=O

N- 1 N-1

k=O k=O

D2,i+lj

Fig. 1 The four Edge Difference vectors.

The MED model estimates DC coefficients quite ac- curately in most cases but results in large estimation er- rors in some blocks. This problem was solved by retain- ing the locations where the MED model fails as well as the original values of the DC coefficients. The locations where the MED model fids can be found as follows:

1. The SOR method is applied to the image.

2. The location wheire the absolute difference be- tween the original and the estimated DC coeffi- cient is the largest is found.

3. If the absolute difference is larger than a thresh- old, say E, then the MED model is said to have failed at the location. Hence, go to (1) and use the DC coefficients at the location as a constraint in the SOR method. Otherwise end the iteration.

3 Low Frequency Coefficient Pre- diction

At low bit rates, both the PSNR and visual quality of an image are improved after using the DCCR algorithm. We propose to predict the (0,l)th and (1,O)th AC coef- ficients at some locations by means of the MED model.

In the DCCR as reported in [2], it is necessary to transmit some DC coefficients and a bit map which in- dicates the locations of tlhe transmitted DC coefficients. About 80% of DC coefficients can be restored accu- rately so the additional hits required by the bit map are significantly less than the: bits saved by dropping the DC coefficients. Having BO‘% of DC coefficients dropped, we expect most low frequency coefficients cannot be accurately restored. Also, any overhead data will likely consume all the bits salved by the restoration of the low frequency coefficients. Therefore, the following approach which does not require any overhead infor- mation is proposed. Let bi,j and ci,j be the (0,l)th and (1,O)th AC coefficients at the (i, j)th blockrespec- tively.

Encoding operation after JPEG quantization

1. Discard all DC coefficients to obtain the AC com- ponent as given by eqn (1).

2. Apply the DCCR. to obtain an image with r e stored DC Coefficients at some blocks.

3. Discard all b i j and then restore them using the MED criterion.

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4.

5.

6.

Compute I Dl,i,j I + I Dl,i,j+l I for each block. If I Dl,i,j I + I D ~ , i , j + ~ I> TI where TI is a threshold then bi,j is represented by the differ- ence between the quantized and the restored val- ues. Otherwise b i j is represented by its quan- tized value.

Discard all C i j and then restore them using the MED criterion.

Compute I D2,i,j 1 + I D2,i+1,j I for each block. If 1 D2,i,j I + I 0 2 , i + l , j I> T2 where T2 is a threshold then c i j is represented by the differ- ence between the quantized and the restored val- ues. Otherwise c i j is represented by its quan- tized value.

In summary, the proposed algorithm represents the DC component in the same way as the DCCR, i.e. by a small amount of DC coefficients and their loca- tions. The AC component is represented in JPEG for- mat. It differs from JPEG and DCCR in the represen- tation of the (0,l)th and (1,O)th coefficients at some blocks where the coefficients are represented by the dif- ferences between the quantized and the restored val- ues. Of course, the two thresholds Tl and Tz are also needed. The values of TI and T2 are image dependent. They are determined as follows. Let E1 , Ezl and ERI be the average values of I Dl,i,j I + 1 Dl,i,j+l I when the (0,l)th coefficients are the original ones, zero and the restored ones respectively. Tl is lirst chosen to be E l . If ER^ is larger than El, then Tl is chosen to be Ezl to limit the number of coefficients to be predicted using this algorithm. T2 is determined similarly.

At the decoder, the AC component is first decoded using JPEG. The DC cqmponent is decoded using DCCR from the decoded AC component and the re- tained DC coefficients. Repeat the encoding steps (3) to (6) to identify the locations and the restored values where bi,j and ~ , j are represented by the differences. The restored values will then be added to the differences to obtain the quantized values.

4 Results and Discussions

The proposed algorithm codes DC components using DCCR which can be set to restore arbitrary percentages of DC coefficients. DCCR attains the highest PSNR when about 30% of DC coefficients are transmitted. Better visual quality can be achieved by transmitting DC coefficients less than 30%. Experiments have been

carried out to investigate how the PSNR depends on the bit rate and the percentage of DC coefficients retained. Fig. 1 and Fig. 2 plot PSNR against the bit rate (BPP) for different percentages of DC coefficients retained in the DCCR for the 512 x 512 images Lena and Pep- per. The results show that LFCP (together with DCCR) achieves higher PSNRs than DCCR done for nearly d l cases except only 10% of DC coefficients are transmit- ted and the bit rate is close to 0.3 bpp for image Pepper. This is probably because there are large errors in the DC and AC components and so bi,j and cif cannot be predicted precisely.

Fig. 3 shows the coded images. While the image coded using DCCR at 0.33 bpp as shown in Fig. 3(c) has PSNR only 0.13 dB higher than that coded using JPEG as shown in Fig. 3(b), the visual quality of that coded using DCCR is obviously better. On the other hand, the image coded using the proposed algorithm as shown in Fig. 3(d) attains similar visual quality and PSNR as the DCCR at a lower bit rate.

5 Conclusions

This paper presents a way to apply the MED criterion to restore not only the DC coefficients but also the (0,l)th and (1,O)th AC coefficients for the DCT-based JPEG. The proposed LFCP algorithm does not require any overhead information. Experimental results show that both PSNR and image visual quality are improved in comparison to the DCCR algorithm.

References

[l] W. K. Cham and R. J. Clarke, “DC coefficient restoration in transform image coding,” IEE Pro- ceedings, PartF, pp. 709-713, Dec. 1984.

[2] F.W. Tse and W.K. Cham, “A DC coefficient esti- mation scheme for image coding,” in Proc. of IEE Fj2h International Conference on Image Process- ing and Its Applications, July 1995, pp. 569-573.

[3] W.K. Cham and F.W. Tse, “Restoration of low frequency coefficients,” in Proc. of 3rd Znterna- tional Con. on Signal Processing’96, Oct. 1996, pp. 1082-1085.

[4] G.H. Golub and C.F. van Loen, Matrix Computa- tions, The Johns Hopkins University Press, 1989.

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J P E ~ - DCCR-10% -+--- LFCR-10% -+---- DCCR-15yo ..a .......

LFCR- 15% DCCR-20% -A.-. LFCR-20% --*--- DCCR-30% --e--- LFCR-30% --+ - - - -

x' ,'/ ,;, .... ,'

., ..' ,' ,. .;.' "' ,/'

:;, x' ,,,, ,'I

,'/ ,<,' ,,;,

,I I ,$,' +, '

0.25 0.3 0.35 0.4 0.45 0.5 0.55 Bit rate(Bpp)

Figure 1 : PSNR vs bpp of DCCR and LFCP for image Lena.

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31 0.25 0.3 0.35 0.4 0.45 0.5 0.55

Bit rate(Bpp)

Figure 2: PSNR vs bpp of DCCR and LFCP for image Pepper.

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(a) Original image. (b) DCT-based JPEG, bit rate=O.33bpp, PSNR=32.37dB.

(c) DCCR with joint optimization, bit rate0.33bpp, PSNR=32.50dB.

(d) LFCP with joint optimization, bit rate4.3 1 bpp, PSNR=32.49dB.

Figure 3: Experimental results using Lena.

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