Block Effect Reduction by the 1-D Grey Polynomial InterpolationCheng-Hsiung Hsieh1 and Ren-Hsien Huang21Department of Computer Science and Information EngineeringChaoyang University of Technology2Mars Semiconductor CorporationE-mail: [email protected]—In this paper, the one-dimensional grey polynomial interpolation (1-D GPI) developed for image enlargement in[27] is applied to reduce the block effect in block discrete cosine transform (BDCT) image/video coders. Note that the blockeffect in BDCT coders results from insufficient bits for transform coefficients. An interpolation approach is proposed to relievethe block effect problem. The proposed approach consists of three stages: First, the input image is down sampled to reduce theamount of data. Second, the down sampled image is put into a BDCT coder, such as JPEG and MPEG-2. Finally, the 1-D GPIis applied to enlarge the decoded image. Examples with the JPEG and MPEG-2 BDCT coder are provided to verify theproposed approach. Simulation results indicate that the proposed scheme is able to effectively reduce the block effectsignificantly and therefore the subjective visual quality improved in the given examples.Index Terms—Grey 1-AGO, Polynomial Interpolation, Block Effect, Block Discrete Cosine Transform,Low Bit Rate CodingI. INTRODUCTIONowadays, image and video data are quite common in the network transmission. Generally, these data require largememory capacity and therefore large transmission bandwidth. To relieve the problem, image and video codingschemes are sought. Among them, the block discrete cosine transform (BDCT) is the most popular scheme forimage and video compression because of its near-optimum energy compaction and the availability of fast algorithms andhardwares. Consequently, the BDCT coder is extensively used by many current image/video standards, such as JPEG, ITU-TH.261/3/L and ISO MPEG-1/2/4. However, one of problems in the BDCT coder is the block effect in low bit rate or highcompression cases. To deal with the problem, several approaches have been reported recently.NFor the still image coding standard JPEG, several block effect reduction methods have been reported. In [3], a low-passfilter was applied to decoded image to smooth out the blockness. However, a blurred image resulted accordingly. In [ 4-9],several types of de-blocking filters and post-processing were proposed. Yet, the improvement seemed not significant. In [11-15], the block effect reduction was performed in the wavelet domain which requires high computational cost generally. In [16-20], by the POCS (Projection onto Convex Set) the visual quality was enhanced significantly. Nevertheless, the computationalcomplexity is quite high.As for the video coding standard MPEG-2, several approaches to reduce block effect have been reported. In [ 21], a low-passfilter with Gaussian-shaped impulse response was applied to the video sequence. In [22], it utilized information obtainedfrom the bitstream, including DCT coefficients and motion vectors. Based on the information, each block was classified anddetected for block effects. To remove the block effect, a filter was applied to the decoded images. In [ 23], the slope measurewas applied to the (macro block) MB boundaries to classify whether an MB was well compensated or not. If it was not, thenthe MB was adaptively quantized to reduce the block effect. In this paper, we propose a novel approach, based on one-dimension grey polynomial interpolation (1-D GPI), to reducethe block effect in the BDCT coding. When the JPEG and MPEG-2 are used in the proposed approach, the coding systems areabbreviated as JPEG-GPI and MPEG-GPI, respectively. This paper is organized as follows. In Section II, 1-D polynomialinterpolation is briefly reviewed. Then the 1-D grey polynomial interpolation is described. In Section III, the proposed JPEG-GPI and MPEG-GPI coding systems are introduced. In Section IV, simulations are given to justify the proposed codingsystems. Finally, conclusion is made in Section V.II. THE 1-D POLYNOMIAL INTERPOLATIONIn this section, 1-D polynomial interpolation (1-D PI) is briefly reviewed. For details, one may consult [24]. Given)(kx, the implementation steps for 1-D PI are described as follows.Step 1. Assume )(kx is an L-order polynomial as0111)( ckckckckx-LLLL (1)Step 2. Substitute 11 Lk into (1) asVcx (2)where elements of x, c, and V are )(kx, kc, andjLkjkv for 11,0 LkLj, respectively.Step 3. Find the interpolated data, )1(ˆMkx , as01)1()1()1(ˆcMkcMkcMkxLL (3)where kc are the coefficients found in (2).Step 4. Obtain the final interpolated data asMMkxMkx )1(ˆ)1(ˆ (4)where M denotes an up sampling factor and is assumed an integer without the loss of generality.III. THE 1-D GREY POLYNOMIAL INTERPOLATIONNote that randomness in data affects the performance of 1-D PI. Therefore the performance can be improved by reducingthe randomness in data. And it is known that the preprocessing scheme in the grey system [ 25], the first-order accumulatedgenerating operation (1-AGO), is able to reduce the randomness in data. Consequently, 1-AGO is incorporated into 1-D PI toimprove the interpolation performance. The new 1-D interpolation scheme is called 1-D grey polynomial interpolation (1-DGPI) which is described as follows.Given data 11,)( Lkkx, the 1-AGO converted data is found askiixkx1)1()()( (5)for 11 Lk. An example is depicted in Figure 1 where the original data )(kx is 3} 5, 2, 4, ,1{ and the 1-AGOconverted data )()1(kx is 15} 12, 7, 5, ,1{. It is easy to see the data after 1-AGO is smoother than theoriginal data. Thus itmay improve the interpolation performance of 1-D PI.In the 1-D GPI, the 1-AGO is applied to preprocess data. By the 1-D PI described in Section II, the1-AGO preprocessed data is interpolated. Next, interpolated pixels are found through the inverse of1-AGO, the first-order inverse accumulated generating operation (1-IAGO) which is defined as)1()()()1()1( kxkxkx (6)for 12 Lk. Finally, an filter is applied to interpolated pixels to further enhance the interpolationperformance. The 1-D GPI has been applied to image enlargement in [26]. With less
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