There have been great advances in lossless image coding recently [1]-[31].
Some of which are based on reversible wavelet transformation using lifting structure [6]-[10]. By using integer wavelet transformation, lossless to near-lossless compression as well as progressive reconstruction of image data can be achieved [6]-[10]. However the compression results obtained with the use of integer wavelet transformation are typically inferior to that of obtained by predictively encoded techniques [17].
The predictive coding scheme, known as the differential pulse code mod-ulation (DPCM), is used in a wide variety of applications such as image and speech compression for ease of implementation [1]. Due to the high correlation between successive image samples, the differential encoding tech-nique removes the inter-pixel redundancy by encoding the difference between successive image samples rather than the samples themselves. Since the dif-ference between samples is expected to be smaller than the actual sampled amplitudes, fewer bits are required to represent the difference. Thus, the differential encoding removes the inter-pixel redundancies and encodes only
For lossless compression of images, we show in Fig. 1.1 the basic block diagram of a differential encoding system (predictive coding system). As can be seen in Fig. 1.1, the lossless predictive coding system is composed of two major blocks; the predictor and the entropy coder. In order that a lower first-order entropy and hence a lower actual bit rate can be obtained, many researches on the design of an effective and efficient predictor for removing the statistical redundancy among coding pixels have been proposed. Among which, adaptive predictors with context modeling are often used to accom-modate the varying statistics of coding images [11]-[31]. Besides, adaptive prediction is achieved in most of the coders by using multi-predictor struc-tures [11]-[22]. Among which, the CALIC coding system [14], a state-of-the-art lossless coder proposed for JPEG-LS, uses a gradient adjusted predictor (GAP). Based on the gradient of neighboring pixels, one out of a set of seven predictors is chosen. The LOCO-I coder [15], an algorithm motivated by CALIC [14] and standardized into JPEG-LS, uses a median edge detec-tor (MED) to choose one of three predicdetec-tors for current prediction. In [16], adaptive prediction is achieved by choosing one out of a set of predictors that minimizes the energy of prediction errors in a specified cluster of causal pix-els, and the predictor coefficients of the selected predictor are then updated by applying gradient descent rule.
In [17]-[20], multi-pass prediction is introduced. With multiple passes, progressive transmission of lossless and near-lossless coding of image data can be achieved. Besides, the encoder can form a 360 degree prediction [19]
or perform a global image analysis [20] by using multi-pass prediction. A highly complex two-pass coder called TMW has been proposed in [20]. Using multiple linear predictors and global image analysis, the TMW system can achieve lower bit rates than existing coders for most images. While achieving
very low bit rates, the computational cost is regarded as prohibitive in TMW [20]. Recently, a fuzzy logic-based adaptive DPCM algorithm called FMP [21]
is proposed. The FMP presents a competitive, and in some cases superior result than TMW but with a lower computational cost. Though FMP is effective in removing the statistical redundancy, it still takes minutes.
In the context of optimal predictors, the minimum mean square error estimate of Y given observations X1, X2,· · · , Xn is E{Y |X1, X2,· · · , Xn}, generally a nonlinear function. Therefore, there have been many results using neural networks as nonlinear estimators [22]-[24]. Neural network based predictors perform well in slowly varying areas. However, there can be large prediction error around boundaries [32]. The result can be improved using additional hidden layers or hidden neurons, but this incurs a drastic increase in complexity [23], [33].
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 become a major problem to be conquered 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 de-tector and the analysis of edge orientation are difficult problems themselves, let alone to predict along the edge orientation. Recently, linear predictors adapted by least-squares (LS) optimization have been proposed as an effi-cient approach to accommodate varying statistics of coding images [25]-[31].
Among which, the EDP [26] pointed out that the superiority of LS adap-tation 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 predictor performs very well for pixels around boundaries. For complexity consideration, performing the LS adaptation process in a pixel-by-pixel man-ner is regarded as prohibitive. Therefore, the EDP [26] proposed initiating the LS optimization process only when the prediction error is beyond a pre-selected threshold such that the computational complexity can be reduced.
The EDP [26] has made a noticeable improvement over the state-of-the-art lossless coder CALIC [14]. On the other hand, we know that the normal equa-tions provide the key for LS adaptation, and some fast algorithms, Cholesky decomposition for example, can be applied 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 nor-mal equations is rather high. Thus, an algorithm for the fast construction of normal equations has been proposed in [25] so that the computational cost for LS adaptation process can be reduced significantly.