Our approaches can match the common encoding setting and eliminate the huge computations in selecting the key coding parameters (such as the temporal prediction type and the block partition mode). However, we do not address the determinations of the subordinate coding parameters that may further reduce the encoding time. Moreover, our proposed schemes may not perform well in some specfic encoding scenario. Those inadequate considerations are listed as follows.
In the encoding process, H.264/AVC [4] and H.264/SVC [2] perform macroblock-wise Lagrangian optimization to determine the optimal coding parameters of the current block.
Firstly, the temporal prediction type of each block partition is selected by Eq. (2.19). Then, the best partition mode and the preferable transform size (that is, ABT) are chosen simultaneously. When choosing these two coding parameters, the selection process adopts Eq. (2.21) to obtain the optimal parameters by competing and , where
denotes the rate-distortion cost applying the transform size of . Hence, determining these two coding parameters needs to (1) calculate the transform coding, quantization, and the entropy coding to get the actual bits, and (2) perform the reconstruction process so that the distortion (SSD) can be computed. If we can make
decision on the transform size in advance, the computations of is reduced by half.
Although our algorithms are specifically designed for the combined CGS and dyadic temporal scalability, they can also be used for the spatial and the non-dyadic temporal scalability. However, these tools must be adjusted properly to fit into the special scalability structure. For instance, two important issues need to be addressed for the spatial scalability:
(1) the change in statistical data collection due to the multiple-to-one macroblock mapping from a spatial enhancement layer to its base layer and (2) the aliasing effect due to the interpolation of residual and motion signals. The former may decrease the dependency of the enhancement-layer coding mode/type on its base-layer counterpart, and the latter may affect the probability of the base-layer motion parameters. In general, the extension of our schemes to the non-dyadic temporal scalability is straightforward. It is expected that the statistics in the non-dyadic case are similar to those in the dyadic one. In practice, the non-dyadic temporal scalability is seldom used.
Appendix
Distribution of the Approximated Distortion
In Subsection 3.2.3.3, we have derived the upper bound of , as shown in Eq. (A.1).
(A.1)
In addition, we further study its probability distribution in this appendix.
First, we try to find out the distribution of the prediction error . The probability distribution of the transform coefficients in the image and/or video coding system have been investigated in the literature [93]–[99]. The earlier studies [93]–[96] found that the transform coefficients of images or video prediction residuals have Laplacian distribution as identified by the goodness-of-fit tests. Later, Lam and Goodman [97] analytically derived this model. A few other research reports [98][99] used more complex probability density functions, such the generalized Gaussian distribution and the mixture of several probability density functions, to improve the modeling accuracy, but these complicated distributions are not widely used in practice because they are difficult for mathematical analysis. Generally, the Laplacian distribution is the most popular one for practical use.
Ideally the FW and BW operations can find the shifted version of the current block if there are no quantization error and noise in the reference frames. As a result, most correlations between frames can be removed by the inter prediction, except for the prediction error term, composed of the
quantization error and noise. Typically, the prediction error from FW and BW is nearly Laplacian distributed, as reported in [100]. Hence, we assume that the ’s inside a block have the i.i.d.
Laplacian distribution, and so do the ’s. Also, we assume the data in one block have the same statistical parameters such as mean and variance.
Next, for a specific location , we like to show that the and pair is jointly Laplacian. Note that although two random variables and are marginally Laplacian distributed, it does not imply the pair is jointly Laplacian. We adopt two popular goodness-of-fit tests to examine the distribution of . They are the Kolmogorov-Smirnov test (KS-test) and the Pearson’s -test.
Kolmogorov-Smirnov (KS) test: The one-sample KS-test is a non-parametric test, which compares the empirical cumulative probability function (ECDF) with the given model CDF. The KS statistic defined in [101] and by Eq. (A.2) quantifies a distance between the ECDF of the data sample and the candidate CDF.
(A.2)
Moreover, the KS statistic ranges from 0 to 1.
Chi-square test: The -test divides the data range into mutually exclusive and exhaustive intervals (events), denoted by . The -test statistic is defined as [102]
(A.3)
(number of samples) of the event , and is the expected value of the event ( is the model probability of event ). Essentially, the -test statistic shows the difference between the empirical frequencies and the model-derived mean values.
These two tests measure the similarity between the collected observations and a chosen model distribution. We pick up the following two bivariate distributions to match our collected data.
Bivariate Gaussian distribution: The commonly used bivariate Gaussian distribution is defined as
(A.4)
where two random variables and form the vector , is the expected value of , the covariance matrix , and is the correlation between and . Bivariate Laplacian distribution: The bivariate Laplacian distribution has heavier tails than the bivariate Gaussian distribution and its PDF is defined by [103]
(A.5)
where is the modified Bessel function of the second kind.
In the data fitting process, we need to decide two parameters and by adopting the approach of method of moments [104][105]. Again, the distribution parameters of each block are calculated individually because they may vary from block to block. After the parameters of these two bivariate PDF are determined, we evaluate how well they match the empirical data of pair .
Table A-1. The average Kolmogorov-Smirnov test-statistic values for temporal enhancement layer
Table A-2. The average test-statistic value for temporal enhancement layer Test
15 30 45 60 75
Fig. A-1 The average test-statistic value for individual hierarchical-B frame
We examine these two goodness-of-fit tests in two distinct block sizes, 16x16 and 8x8. The empirical data are evaluated against these two selected model distributions. In Table A-1, the reference model
is better than the other model in terms of the KS test-statistic value . The value of usually varies from 0.062 to 0.083, but the value is about 0.11 on average. The -test in Table A-2 show similar results, in which averagely is much smaller than . We thus conclude that the collected data are closer to the bivariate Laplacian distribution. Furthermore, Fig. A-1 shows the test-statistic value of each hierarchical-B frame. The value typically ranges from 20 to 40, while the value is often more than 40. For the MOBILE, it can be up to 100. Moreover, small partitions usually match the mode distributions better. The test metrics in the small 8x8 partition are slightly smaller than that in the large 16x16 block size.
After we use the jointly Laplacian distribution to model is, we can derive the
distribution of ; namely, the distribution of . According
to the property of Laplacian distribution [103], a linear combination is one-dimensional Laplacian distribution, if and are jointly Laplacian. Hence, the term
(A.6)
is also Laplacian distributed. Moreover, from the probability theory, the absolute value of a Laplacian distribution is exponentially distributed and the sum of i.i.d. exponential distributions forms a Gamma distribution, as shown below [106][107].
(A.7)
Hence, should have a Gamma distribution , where is the shape parameter and is the scale parameter.
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