It is noted that the construction of normal equations requires roughly M N2/2 multiplications, where M is the number of training pixels and N is the pre-diction order [35]. Numerically, the normal equations (2.7) can be solved by Cholesky decomposition or Singular Value Decomposition (SVD) depending on the rank of P in (2.6). When P is full-ranked, PTP is nonsingular and positive definite. For a positive definite matrix, the Cholesky decomposition can be used, and it requires only N3/6 multiplications to solve (2.7), which is about half the usual number of multiplications required by alternative meth-ods [35], [36]. Only when P is not full-ranked, SVD, which requires much higher computations, is needed. Fortunately, our experiments show that in most cases P has full rank. This can be seen in Table 4.6, where we have listed the percentage of pixels performing LS adaption and the number of
Table 4.7: Operation counts for edge detector in (2.4).
Operation Compare ADD/SUB MUL/DIV Square Edge detection £ (N+2) £ (4N-2) £ 7 £ (N+3)
ent orders. Indeed, this is because pixels around boundaries usually have a large variation in gray levels and thus the matrix P in (2.6) is seldom rank deficient. Therefore, most of the computations take place in forming the normal equations (2.7) rather than solving them. For this, an inclusion and exclusion method for fast construction of the PTP matrix is proposed in [25]. The algorithm in [25] utilize the overlapped training area between successive coding pixels. Therefore, the fast algorithm can only be used in a pixel-by-pixel adaptation manner and can not be applied in the proposed approach as we activate the LS adaptation process only when necessary.
For the proposed edge detector, the operation counts for each coding pixel in the edge detection process are listed in Table 4.7. It should be noted that there is no need to check both of the two inequalities in (2.4) for every pixel. Only when the first inequality holds then we check the second condition. Therefore, the actual computational cost is lower than what is listed in Table 4.7. Though edge detection incurs a slight increase in computations, the overall complexity is reduced significantly when compared with that of pixel-by-pixel adaptation approach. The proposed approach has achieved a very good trade-off between runtime performance and prediction efficiency.
4.10 Concluding Remarks
In this chapter, we have evaluated the performance of the proposed lossless image codec. Extensive experiments as well as comparisons to existing state-of-the-art predictors and coders are also given to demonstrate the usefulness of the proposed system. In summary, we can make the following concluding remarks:
1. We have investigated in this chapter the usefulness of the proposed edge detector. As can be seen in our experiments, the proposed edge detector is very effective in detecting edges and robust to images with moderate salt-and-pepper noise although only four causal pixels are used.
2. The effectiveness of the error compensation mechanism in regular mode is also evaluated in this chapter. Our experiments show this very useful that further improves the bit rates by, on average, 0.2bpp in test images.
3. The prediction order affects the coding gain. In our experiments, we find the compression ratio quickly saturates when the prediction order is greater than six. Moreover, the use of a higher prediction order also means an increase in computational complexity. Therefore, we have found the use of a sixth-order predictor is a proper choice.
4. With the proposed edge-look-ahead approach, we activate the LS adap-tation process only when an edge is detected or when the prediction error is beyond a predefined threshold. As can be seen in our exper-iments, the results obtained by using the proposed approach are very close to those with pixel-by-pixel LS adaptation, but with a
signifi-more feasible under limited resources.
5. In this chapter, we also compare the proposed system with existing state-of-the-art predictors and coders. As can be seen in Table 4.3 and Table 4.5, comparisons on first-order entropies and actual bit rates have demonstrated the superiority of the proposed system.
6. A detailed analysis on the computational complexity of the proposed system is also given in this chapter. For LS adaptation process, fast algorithms, like Cholesky decomposition, can be used in most of the cases depending on the defectiveness of the matrix P in (2.6). Besides, we also list in Table 4.7 the operation counts of the proposed edge detector. Though edge detection incurs a slight increase in computa-tions, the overall complexity is reduced significantly when compared with that of pixel-by-pixel adaptation approach.
Chapter 5
Enhancing the Predictive
Coding Efficiency with Control Technologies
There has been great interest in applying predictive coding to lossless com-pression of images. Predictive coding is very useful for removing statistical redundancy among pixels in slowly varying areas. However, there can be large prediction errors for pixels around boundaries. In this chapter, we in-troduce techniques commonly used in control systems to enhance the coding efficiency of predictive coding. Actually, the predictive coding system, which calculates the system output based on the texture context of the coding pixel, behaves just like a multi-input single-output system with the predictor itself can be taken as the system model. Besides, the prediction error is usually feedback for the adaptation of predictor coefficients so that the difference between the desired and the actual output, i.e., the so-called error signal, for consecutive pixels can be minimized. When compared with the purpose of a control system, which is to follow the system command as precisely as possible, we find the objective of both systems are the same. Moreover, an
in control systems. These observations lead to the idea of using control tech-nologies to improve prediction result for pixels around boundaries. To realize this idea, we use an adaptive Takagi-Sugeno fuzzy neural network (TS-FNN) as the predictor for its advantages of fast convergence and parallel computa-tion. Furthermore, the widely used proportional controller (P-controller) in control system is implemented implicitly in the consequent part of the network so that the prediction error can be further compensated for pixels around boundaries. We find in experiments that the proposed approach can have a very good prediction result even without using any online training area for network adaptation process. This makes the proposed system more feasible under limited resources, and a very good run time performance can be obtained. Finally, comparisons to existing state-of-the-art lossless pre-dictors and coders will be given to highlight the advantages of the proposed novel approach.
5.1 Introduction
The performance of predictive image coding scheme highly depends upon the effectiveness of the predictor used in the coding process. Most of the image predictors perform very well in slowly varying areas. However, large prediction errors can take place around edges and boundaries, and this has remained a major problem in predictive coding schemes so far. Intuitively, the prediction results can be improved if we can foresee the existence of an edge and then predict along the edge orientation. However, the design of a robust edge detector and the analysis of edge orientation are difficult problems themselves, let alone to predict along the edge orientation. Re-cently, some approaches propose the use of a linear predictor adapted by least squares (LS) optimization during the coding process [25]-[31]. Among which, the EDP [26] pointed out that the superiority of LS adaptation is in its edge-directed property. That is, the LS-based predictor can adjust the prediction support along the edge orientation automatically during the adaptation process. With the edge-directed property, LS-based adaptive pre-dictor performs very well for pixels around boundaries. On the other hand, we know that the normal equations provide the key for LS adaptation, and some fast algorithms, “Cholesky decomposition” for example, can be ap-plied in the LS adaptation process. Therefore, the complexity in solving the normal equations itself is not a problem. Nevertheless, the computational cost for the construction of normal equations is rather high. Thus, a pixel-by-pixel LS adaptation process for predictive image coding is regarded as prohibitive. For this, an edge-look-ahead approach is proposed [27], [28].
The edge-look-ahead approach proposes a causal edge detector and initiates the LS adaptation process only when the coding pixel is around an edge or
look-approach, the advantage of edge-directed property in LS-based predictor can be fully exploited. Moreover, a noticeable reduction in computational complexity can also be obtained when compared with that of pixel-by-pixel LS adaptation approach.
In order that the highly complex statistical redundancy can be removed more effectively, some of the results are obtained by using fuzzy logic or neural network as the nonlinear predictor [21]-[23]. Theoretically, the use of nonlinear predictors can get better prediction performance than that of ob-tained by using linear predictors. However, we find in literatures that fuzzy logic or neural network based nonlinear predictors usually come along with a very high computational complexity for inherent nonlinear characteristics and the time-consuming online adaptation process. Furthermore, the im-provement on the prediction result obtained seems does not justify the use of such a highly complex nonlinear predictor when the run time performance is taken into consideration.
In this chapter, we propose an approach performing nonlinear predic-tion based on a four-layered Takagi-Sugeno fuzzy neural network (TS-FNN) [37]-[45]. The TS-FNN is applied in the proposed approach as the nonlinear predictor for its advantages in parallel computation, universal approximation and fast convergence [41]-[45]. With priori knowledge on the partitioning of input space, the network structure can be constructed with neurons provided with a set of Gaussian membership functions. Moreover, suitable fuzzy rules can also be derived and recruited by measuring the membership degrees so that a nonlinear prediction model is developed. To make the proposed sys-tem adapted to the varying statistics, the prediction error of the coding pixel is feedback to update the connection weights and the parameters in the mem-bership layer. In order that the high computational complexity incurred in
the network learning process can be avoided, the number of training pixels used during the network adaptation process is decreased substantially and even not used in the proposed TS-FNN approach. Moreover, all the oper-ations used in the proposed TS-FNN based predictor are linear except the the Gaussian membership degree measurement in second layer. This makes the proposed approach more feasible under limited resources and a very good run-time performance can be obtained.
Regarding to the problem of encountering a large prediction error for pix-els around boundaries in most of the predictive coding schemes, we propose in this chapter a novel idea based on commonly used control technologies. Ac-tually, the design of a controller is to follow the system command as precisely as possible, which has the same objective with predictive coding. Moreover, an edge or a boundary, i.e., an abrupt change among neighboring pixels, can be regarded as a step command in control systems. The above observations lead to the idea of enhancing the prediction result with control technologies.
In this chapter, both the horizontal and vertical deviations around the coding pixel are calculated such that we can know the existence of an edge. The cal-culated deviations are then used as the inputs of P-controller compensators which are implemented implicitly in the network weights connecting the rule layer and the output layer, i.e., the consequent part of a fuzzy rule. As we will see in the experiment that the proposed P-controller compensator is very useful in removing the inter-pixel redundancy and can further improve the entropies of prediction errors.
Novelty of the proposed approach
In this chapter, we propose the use of a TS-FNN based nonlinear predictor for lossless coding of images. In order a very good run-time performance can
be obtained, the number of training pixels for network adaptation process is decreased substantially and even not used in the proposed approach. For pixel around edges, we propose a novel approach by applying the commonly used P-controller in control system as a compensator so that the prediction error can be further reduced. It is noted that the conventional error model-ing technique, a statistical approach, can also be used for further reduction in entropies. With the proposed adaptive predictor and the P-controller compensation mechanism render the proposed approach very effective in re-moving statistical redundancy and feasible for real time applications.
The rest of the chapter is organized as follows. Section 5.2 gives an overview of the proposed TS-FNN based coding system. A detailed descrip-tion on the predicdescrip-tion and the network adaptadescrip-tion process are also addressed in this section. Besides, a brief introduction on the conditional entropy coder applied in the proposed coding system is given in Section 5.3. The exper-imental results are shown in section 5.4 to demonstrate the effectiveness of the proposed system. Finally, a concluding remark is given in Section 5.5.