Modified Quantization Constraint Set

for Deblocking Transform Coded Images

ZHANG Song, LUAN SujuanCollege of Communication Engineering

Hangzhou Dianzi UniversityHangzhou, Zhejiang, China

luansujuangl26.com

Abstract At low bit rates, the block discrete cosine transform(BDCT) compression methods often give rise to blockingartifacts. In both enhancement and restoration approaches tothe problem, quantization constraint set (QCS) is widely usedas a priori information about the coding process. In this paper,we propose a modified constraint set based on the statisticalproperty of quantization errors. The experimental resultsshow that the extension can be effective. While combined withother algorithms, the proposed QCS outperforms traditionalQCS and produces results comparable with the state of the artin the literature.

I. INTRODUCTIONAmong various transforms, the discrete cosine

transform (DCT) is one of the most popular for multimediacommunications due to its high coding performance closeto that of the optimal Karhunen-Loeve transform (KLT).For practical applications, as in international image andvideo coding standards, the DCT is usually implemented inblock-wise way as a tradeoff between coding efficiency andcomputational complexity. Therefore the original image isfirst divided into nonoverlapping uniform blocks, and thentransformed, quantized and entropy coded independently toproduce bitstreams for transmission. The cost paid for theimplementation ease is that at low bit rates with coarsequantization noticeable blocking artifacts arise in decodedimages which severely deteriorate the subjective imagequality.

Until now, the most popular approach to reducingblocking artifacts is post-processing which does not dictatefundamental modifications to coding systems. There aretwo major classes of methods for post-processingdeblocking. One is based on image enhancement [1], [2],where linear or nonlinear, global or adaptive low-passfilterings are used to smooth out block boundary while notincurring much loss to image details. The other kindincludes image restoration techniques, which can be cast inthe framework ofPOCS [3], [4] or MAP estimation [5].

In both enhancement and restoration approaches todeblocking, quantization constraint set (QCS) is frequentlyused as important a priori information about the codingprocess. While incorporated in post-processing algorithms,it helps improve the final performance. In this paper, wepropose a modified quantization constraint set based on thestatistical property of quantization errors. In Section II, thetraditional QCS is briefly reviewed. We introduce the newQCS in Section III. The property of the proposed QCS isalso given and its projector derived. In section IV we givesimulations results on typical still images, which prove theeffectiveness of the modified QCS. Finally, in section V,we present our conclusions.

II. TRADITIONAL QCSThroughout this paper, we discuss problems in the

framework of JPEG coding systems, although the proposedideas can also be applied to other block transform codingschemes. So the image is partitioned into N blocks withequal size of 8x8 and coded nearly independently. For eachblock i, it is DCT transformed and scalar uniform quantizedbased on the quantization table. This process can beexpressed as

Coek'qkI= round( ), k,I= 0,1,...,7,

Qk,l(1)

where Coe', is the k,lth transform coefficient of the ithDCT block, Qk,l is the corresponding quntization stepsizeand round( ) stands for rounding operation. The resultantquantization indices q%7s are then grouped and entropycoded to produce the bitstream. At the receiver side, thequantization indices qk7s are first entropy decoded from thebitsream. With the knowledge of the quantization table, itcan be inferred from q ,, that the original DCT coefficientCoe', must fall in the following quantization interval:

Coe,',, E [(q ,I-2 )Qk,' (qk,2+ 2)Qk,l]2 (2)

This work has been supported by research project from ZhejiangProvincial Department of Education (No. 20040462).

which is just the traditional quantization constraint set [3].

The quntization constraint set is usually exploited in thedeblocking algorithms by a projection operation. Let d%,denote the BDCT of the intermediate post-processed image,the QCS projection operation updates d, by the followingrule:

I(q'1 9 Qk.1 if dkl< (q1l -Qk

PQCS

dikl

1+ -)Qkl12 (3)

otherwise.

The above projector was first proposed in the context ofPOCS-based restoration [3] but can be equally well appliedas a final rectification step in iteration-free enhancement-based deblocking methods, where convergence becomes nolonger a concern.

III. MODIFIED QCSDeblocking can be readily formulated as an image

restoration problem, or more precisely an image denoisingproblem. From the discussion in the last section, JPEGdecoder simply takes qQ,,Qk, as the estimate of the originalCoe',l. Therefore the degradation model can be written as

Y =X+n, (4)where X and Y are the BDCT of the original and JPEGdecoded images respectively and n denotes quantizationnoise. Deblocking aims to obtain a better estimation ofXgiven the data Y.

It is well known that the key to the success of imagerestoration algorithms lies in exploiting both the signal andnoise property as ingeniously as possible. From this point ofview, the traditional QCS can also be thought of as a prioriknowledge on the property of the quantization noise, namelythe noise amplitude bound. While this information is helpfulfor deblocking, it is no doubt the most trivial. Moremeaningful statistical property of quantization noise can beused to complement the simple QCS. An example is noisevariance, which is widely used in least squares imagerestoration methods [6]. Although the task here is clearlytougher, where the noise is signal dependent and far fromi.i.d. gaussian, the idea can be fruitfully applied todeblocking, which will be confirmed by our experiments.

Since the DCT coefficients at different frequencypositions have different statistical properties andquantization stepsizes, we take them separately, or onemodified QCS for one frequency position. In the following,we give the definition of the modified QCS for typicalfrequency location, the case for other frequencies beingsimilar. Referring to section II, there are NDCT coefficientsat the same k,lth frequency position. As commonly used in

the literature [7], the constraint on the quantization noisevariance is formulated as

||2 = Xk 1 2 < I(5)

where the subscript denotes the subvector composed only ofcoefficients at the k,lth frequency position. For completespecification of the modified QCS, it remains to determinethe bound ek, 1. We first derive the bound under the idealcondition and introduce a global correction factor to accountfor the deviation later. If the quantization noise is assumed tobe uniformly distributed within interval [-0.5Qk,,, 0.5Qk, 1],its variance is Vk, / = (Qk, 1)2/12. So for simplicity(correctedafterwards as mentioned above), we just take the Nquantization errors at the k,lth frequency position as i.i.drandom vector, each element being gaussian with zero meanand variance of Vk, 1. From elementary statistics theory, thevariance of i.i.d. gaussian vector with zero mean andvariance 1 obey x distribution which is relatively complexto compute. But for large N, which is just the case for ourproblem, we can make use of asymptotic results to simplifythe evaluation. It is well-known [8] that when N approachesinfinity, I2%2 asymptotically obey normal distribution withmean tI2N 1 and variance 1. The results are summarized inour notation as

N (Coek, q,lQk,l )2il

k

ki G(12N, 1), (6)

where to avoid confusion, we use G to denote normaldistribution and Vk, I in denominator for normalization. Using3 arule, we have that with the confidence level of 99%

N (Coek{1 qklQk)12N-1-3< 2Z < 12N 1+3. (7)

Only the upper bound is retained to ensure convexity, whichleads to the following set:

|N

Z(Coe1k q IQkI )2 < O0.5 *Vkl ( N + 3). (8)

As pointed out before, these results are obtained underthe ideal assumptions. The real situation in deblockingcontext must be much more complicated if we consideradditional factors like correlation between neighboringblocks, local variation of image features and non-uniformityof the noise. Due to the difficulty of accurate analysis, wetake a simple experimental approach. We introduce acorrection factor A to accommodate the deviation from theideal case, which finally gives the definition of the modifiedquantization constraint set:

INZ(Coek' -q IQI) <2 V0.5 * Vk>I(12N-I + 3)i=l

k,l = 0,1, ...,17.(9)

2 is global and applies to all frequency. So it also reflects themixing among different frequencies. The value of 2 isoptimally selected by experiments.

It is easy to see that the proposed QCS is both closed andconvex and represents a ball in RN. This geometricinterpretation make the corresponding projector intuitivelyclear. Referring to Fig. 1, the projection operation is derivedas

K1 Qk,, ±a(d= q,Qk.,1), if DklI > r, I

dk I' otherwise,

where

IN

Dk,I=/(dklI-qk Qk,l) '(1

r,,l 2J0.5*Vkl (12N-I +3),

and a= rk,l Dk,,.As a final remark, for implementation in Section IV we

combine the traditional and modified QCS, i.e. theintersection of two convex sets, whose projection isperformed by alternating projections.

Fig. 1. Projection onto ball

IV. SIMULATION RESULTS

In this section, simulation results are given to evaluatethe performance of the modified quantization constraint set(MQCS). In the experiments, we substituted MQCS fortraditional QCS in existing algorithms [3], [9], [10] andcompared the results. Four 256x256 8b/pixel gray-scaleimages, namely "Lena", "Baboon", "Pepper" and "Bridge"were chosen as test images. The images were compressedusing a JPEG baseline codec with the second quantizationtable in [3].

While for comprehensive evaluation both subjective andobjective image quality should be considered, weconcentrate here on the latter due to space limitation. Thepeak signal-to-noise ratio (PSNR) is used as the soleobjective measure, which is defined in dB by

where g and h are respectively the original and processedNxN images with [0,255] gray-level range. The higher thePSNR, the greater the fidelity of the reconstructed image tothe original.

TABLE I summarize the main results. As shown, thealgorithms listed are named after the author. "+NQCS" and"+MQCS" refer to the original algorithm with the QCSreplaced by narrow QCS [11] and modified QCSrespectively. The PSNR results given are those obtainedafter one iteration because in Zakhor's and Paek's methodPSNR generally decrease stating from the second iteration.Note that Zou's algorithm employs adaptive NQCS. It isclearly seen that the proposed QCS is effective forimproving PSNR and gives comparable performance to thatof NQCS.

For both NQCS and MQCS, there is an importantscaling coefficient that impacts the final performance. Theyare both optimally determined by experiments to generatethe above results. The coefficient ,u corresponding to NQCSis optimally set to be 0.4 [12]. For completeness, Fig. 2gives the PSNR variation for MQCS with respect to scalingcoefficient 2 for several typical bit rates. In general thePSNR performance increases in proportion with 2 startingfrom 0 until an intermediate optimal value, after which thePSNR decreases.

For more thorough investigation, Fig. 3 shows thePSNR variation for QCS, NQCS and MQCS versus codingbit rate and number of iterations respectively. Only the datafor "Baboon" are given for illustration and Zakhor'sprocedure is chosen as basic algorithm. The results areconsistent with those in TABLE I.

26.5

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Oy 24.5zC')0 24

23.

23

22.550 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Scaling Coefficient A0.8 0.9 1

Fig. 2. Maximum PSNR versus scaling coefficients i for MQCS basedon Zakhor's algorithm. The curves correspond to "Baboon" image coded at0.20 bpp (O), 0.39 bpp (+), 0.45 bpp (EL), 0.53 bpp (X), 0.60 bpp (A).

PSNR =ogog, llg-Nh2 ((12)

TABLE. I. PSNR PERFORMANCE WITH VARIOUS ALGORITHMS

Algorithm PSNR~(dB)Lena(0.30bpp) Baboon(O.39bpp) Pepper(O.32bpp) Bridge(O.43bpp)

Zakhor[3] 29.500 24.658 29.679 25.485Zakhor+NQCS 29.643 24.728 29.890 25.628Zakhor+MQCS 29.594 24.752 29.807 25.623

Paek[9] 29.596 24.570 29.792 25.423Paek+MQCS 29.599 24.602 29.798 25.456

Zou[10] 29.116 24.457 29.292 25.268Zou+MQCS 28.944 24.540 29.114 25.350

25.55

25

m 24.5

zU) 240-

Zakhor

NQCS

MQCS

23.5

200 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Coding bit rate (bpp) of Baboon

(a)

1.823 _

o 10 15 20

Number of iterations25 30

(b)

Fig. 3. For "Baboon" image, (a) PSNR versus coding bit rate. (b) PSNR versus number ol iterations.

V. CONCLUSION

In this paper, we present a modified quantizationconstraint set (MQCS) based on the statistical properties ofquantization error. The experimental results demonstratethat the MQCS extension is effective for improvingdeblocking performance. Although we only provide theresults based on the POCS class of deblocking methods, theidea can be definitely combined with enhancement-basedmethods just as well.

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[2]

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[9] H. Paek, R.-C.Kim, and S. U. Lee, "On the POCS-basedpostprocessing technique to reduce the blocking artifacts in transformcoded images," IEEE Trans. Circuits Syst. Video Technol., vol. 8, no.

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[10] J. J. Zou and H. Yan, "A deblocking method for BDCT compressedimages based on adaptive projections," IEEE Trans. Circuits Syst.Video Technol., vol.15, no.3, pp. 430 - 435, Mar. 2005.

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