Study on the distribution of DCT residues and its application to R-D analysis of video coding

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Abstract

Quantization errors in discrete-cosine-transform (DCT) video compression are known as DCT residues. Knowledge on their distribution is essential in understanding rate-distortion (R-D) behaviors of generic video coding. Traditional R-D analysis adopted a simplified distortion model. Those distortion models took only quantization parameter into account. They lack adaptability to variation of video sources, as the distribution of coding errors also depends on the statistics of video source. Another common approach models the distribution of DCT residues by fitting experimental data from coded pictures to conjectured statistical distributions, but it did not provide insights into what gives rise to the distribution of DCT residues. This paper intends to quantify the distribution of DCT residues with respect to video source and with respect to the quantization strategy by understanding the quantization of DCT frequency components. Moreover, it is applied to derive an R-D model to show the advantage of the proposed distribution model.

Introduction

Block discrete-cosine-transform (DCT) is widely employed in a serial of digital video compression standards [13], [8], [9], [14], [10]. The dominant design of a video encoder is a three-stage structure, as shown in Fig. 1. Frames of raw video data are divided into non-overlapped blocks and then transformed to coefficients in the frequency domain by DCT. In the second stage, quantization is employed to considerably reduce the magnitude of these DCT coefficients. Lastly, techniques of entropy coding are used to further increase the compression ratio. Such a hybrid system can compress raw video data efficiently while suffering from distortion to some extent.

Quantization errors are introduced in the quantization of DCT coefficients. Those quantization errors in the base-layer/non-scalable video coding are usually known as DCT residues [20]. Their statistics implicitly define visual quality of compressed video. In a signal-to-noise ratio scalable coder [3], [20], they are the source of the enhancement-layer. Therefore, knowledge on the distribution of DCT residues is significant in understanding and analyzing the performance of a DCT-based video compression system. It also makes sense in applications regarding postprocessing of coded video [27] and so on.

This paper investigates the distribution of DCT residues and highlights its application to rate-distortion (R-D) modeling in generic video coding. Individual frequency components of DCT residues are studied on the basis of quantization theory and well-accepted statistical models for source DCT coefficients. The probability density functions (PDFs) of individual frequency components of DCT residues, as well as the overall PDF of DCT residues, are presented. The first major advantage of the proposed study is that it not only provides mathematical justification on the distribution of DCT residues but also quantifies its relationship to video source and the quantization strategy. Another important advantage is it can accurately predict the possible distribution of DCT residues under a certain quantization strategy prior to actual video coding. Hence, it is useful in the analysis of one-pass video coding. In order to show such advantages, the proposed distribution model is applied to develop an R-D model. Based on the R-D model, a rate control algorithm is developed to control an MPEG-4 video coder.

The rest of the paper is organized as follows. Section 2 discusses some limitations of existing modeling work for the distribution of DCT residues. Section 3 presents the proposed distribution model. Section 4 compares the goodness-of-fit of the proposed distribution model to the experimental distribution. Section 5 applies the proposed distribution model to R-D modeling for video coding. Section 6 shows experimental results of the proposed distortion model and the proposed rate control algorithm. A conclusion is given in Section 7. Supporting experimental results are shown in corresponding sections.

Section snippets

Background and problem description

Existing work on the distribution of DCT residues is reviewed first. Then, distortion models in typical R-D analysis work are presented by the analysis of their underlying idea dealing with the distribution of DCT residues.

Modeling the distribution of DCT residues

DCT residues are quantization errors in essence. Their distribution should depend on both the distribution of source DCT coefficients and the quantizer characteristics. This section models their relationship by studying the compression of individual frequency components of source DCT coefficients.

Goodness-of-fit test

This section investigates the goodness-of-fit of the proposed distribution model to the actual DCT residues by the KS test [4]. The KS test justifies whether or not the sample of data are consistent with a specified distribution model. The test statistic (tks) is defined by the maximal difference between the empirical distribution G(zi) and the specified distribution F(zi) at samples zi. A small value of tks indicates a good fit.

In the KS test, the empirical distribution function G(zi) is

Application to R-D modeling

In this section, a novel distortion model is proposed based on the proposed PDF for DCT residues. Subsequently, the distortion model is combined with the classic R-D function to obtain a rate model.

Experimental results of R-D modeling

In this section, the performance of the proposed distortion model at different QP is examined. Subsequently, the proposed rate model is applied to develop a rate control algorithm.

Conclusion

In this paper, the distribution of DCT residues is investigated. The PDF of DCT residues in individual frequencies, as well as the overall PDF of all DCT components, are modeled based on studying the quantization of individual DCT components. Goodness-of-fit test suggests the proposed PDF is more accurate in comparison to Laplacian and Gaussian distributions. Subsequently, the proposed PDF is applied to R-D modeling for generic DCT video coding. After a new R-D model is presented, the

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