Arithmetic coding for image compression with adaptive weight-context classification

Signal Processing: Image Communication 28 (2013) 727–735

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Signal Processing: Image Communication journal homepage: www.elsevier.com/locate/image

Arithmetic coding for image compression with adaptive weight-context classification Jiaji Wu a,n,1, Zhenzhen Xu a, Gwanggil Jeon b,n,2, Xiangrong Zhang a,n,1, Licheng Jiao a a Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China, Xidian University, Xi'an 710071, China b Department of Embedded Systems Engineering, Incheon National University, Incheon 406-772, Republic of Korea

a r t i c l e i n f o

abstract

Article history: Received 30 August 2012 Received in revised form 9 April 2013 Accepted 10 April 2013 Available online 25 April 2013

In this paper, a new binary arithmetic coding strategy with adaptive-weight context classification is introduced to solve the context dilution and context quantization problems for bitplane coding. In our method, the weight, obtained using a regressive– prediction algorithm, represents the degree of importance of the current coefficient/block in the wavelet transform domain. Regarding the weights as contexts, the coder reduces the context number by classifying the weights using the Lloyd–Max algorithm, such that high-order is approximated as low-order context arithmetic coding. The experimental results show that our method effectively improves the arithmetic coding performance and outperforms the compression performances of SPECK, SPIHT and JPEG2000. & 2013 Elsevier B.V. All rights reserved.

Keyword: Context-based arithmetic coding Adaptive Regressive–prediction Weight Context classification

1. Introduction Arithmetic coding [1], introduced by Rissanen and Pasco and generalized by Langdon and Rissanen, is a powerful entropy coding method for data compression. Compared with the well-known Huffman coding algorithm, arithmetic coding benefits from the fact that it approximates the source entropy using fraction bits, while the Huffman algorithm is limited to integer bits. Hence, the adaptive arithmetic coding can achieve better compression performance. An essential element of arithmetic coding is the probability model, which reflects the statistical properties of the input signal and so affects compression performance. The coder can get a shorter average code length if the

n

Corresponding authors. E-mail addresses: [email protected] (J. Wu), [email protected] (Z. Xu), [email protected] (G. Jeon), [email protected] (X. Zhang), [email protected] (L. Jiao). 1 Tel.: +86 29 88202279. 2 Tel.: +82 32 835 8946. 0923-5965/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.image.2013.04.004

probability model it uses is very close to reality, that is, the more accurately the estimated probabilities of the incoming symbols are estimated, the higher the compression performance that is achieved. For arithmetic coding, establishing the probability model is called the context modeling procedure [2]. One well-known context modeling method is Rissanen's context modeling for universal data coding. This can theoretically approach the lower bound for the code length, due to the concept of stochastic complexity [2]. Most image/ video compression algorithms choose context models with fixed complexity that are based on domain knowledge such as the correlation structure of the coefficients [3]. For example, the JPEG2000 image compression standard is a mainstream algorithm that adopts context-based arithmetic coding as a key technology; this is analyzed in detail in [4]. Usually, an ordinary 8-site neighborhood template employed in JPEG2000 for binary arithmetic coding brings 28 contexts. In JPEG2000, this context model is fixed and is available for the coding process. When estimating the probability model using such already-coded samples, severe complexity problems result when the order of the

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context is high. However, context orders that are too low may not be able to capture the rich, local statistical properties of the input signal. In order to resolve the trade-off between performance and complexity, in this paper we employ a context clustering scheme to categorize the coefficients/blocks through the degree of importance of the coefficient/block, which is defined as weight. This weight is obtained through regressive–prediction [5] with a large context template consisting of 14 neighboring coefficients/blocks placed into 16 contexts. Here, a classifier can be regarded as a quantizer, a common tool often used to solve the context dilution problem [6]. One such quantizer is the JPEG2000 entropy coding algorithm EBCOT [7], which quantizes the context into a relatively small number of conditioning states. In this study, we performed the probability statistics and classification procedure using an adaptive-weights context model and using a fixed context model. The experimental results show that the proposed method using adaptive weighting achieves higher compression performance. This paper is organized as follows. In Section 2, an overview of high-order entropy coding is given. Section 3 introduces two different context models, describes the regressive–prediction algorithm, which is used to capture various intra- and inter-subband correlations among the wavelet coefficients, and provides some prediction coefficient results. In Section 4, we describe how optimal classification can be obtained using Lloyd–Max quantization instead of an exhaustive search. Also, the results of our method using two different strategies are illustrated by graphs in this section. Experimental results are reported and analyzed in Section 5. Finally, we provide conclusions for the paper in Section 6.

2. High-order entropy coding Arithmetic coding acts as a tool to compress the code stream in bitplane coders such as SPECK, SPIHT, or JPEG2000. Suppose all bitplanes of a set of quantized wavelet coefficients are coded into a sequence of binary symbols: x1 ; x2 ; x3 ; …; xn , xi ∈f0; 1g. When entropy coding such a sequence of symbols, the minimum coded length is given by n

−log2 ∏ pðxi jxi−1 Þ;

ð1Þ

i¼1

where pðxi jxi−1 Þ denotes the conditional probability of xi given xi−1 , and xi−1 denotes the sequence fxi−1 ; xi−2 ; …; x1 g [8]. At each time instant i−1, with the coder having coded the past data xi−1 ¼ fx1 ; x2 ; x3 ; …; xi−1 g, the next coefficient value xi can be predicted by assigning xi a conditional probability distribution pðxi jxi−1 Þ. Here, xi−1 contains the known information about xi, so it is called the context of xi. From formula (1), we see that the higher the conditional probability, the shorter the average code length. Therefore, to achieve good compression efficiency and conditional probability accuracy, our proposed method estimates the conditional probability based on weights whose magnitudes are much smaller than 2n .

In [8], the conditional probability distributions are trained with a sufficiently large set of images. The compression result may vary with different training sets. Contrast to Ref. [8], the training of context in proposed method only depends on the image to be coded, which guarantees the robustness of encoding. It should be noted that the method in this paper mainly focuses on the coding of significance. Signs are directly outputted. For comparison, this paper presents two different context-determination strategies: off-line and on-the-fly. The off-line strategy obtains the conditional probability through a pre-coding procedure, followed by context classification and arithmetic coding with a stable and accurate model. The on-the-fly strategy performs adaptively, that is, the conditional probability is calculated and updated continually, followed by a real-time process of context classification and arithmetic coding. This paper presents and compares the performance results of two context models: the conventional context model used in JPEG2000 and a new model described in [9].

3. Context models and regressive–prediction algorithm Wavelet theory indicates that wavelet coefficients are sparse with only a small number of large coefficients and a large number of small coefficients, and wavelet coefficients were assumed to be independent in early wavelet coders. However, a careful check of the wavelet transform coefficients reveals that there are various redundancies among coefficients in different subbands [4]. Specifically, most wavelet coefficients in the high-frequency subbands have strong correlations in the edge areas, as shown in Fig. 1. Coefficients in the HL subbands have strong correlations in the vertical direction, such as the coefficients along the edge of the hat which all have high magnitudes; also, horizontal correlations and diagonal correlations can be found in the LH subbands and HH subbands. Due to these correlations among the coefficients, it is necessary to employ a proper context model which can well capture the correlations, as well as an algorithm which can utilize the correlations within the context model. It is said in [6] that a larger neighbor template seems to lead to better performance in arithmetic coding. However, a larger neighbor template is more memory intensive and can cause complexity problem. As in [6], three different templates contain 9  9 contexts, respectively, with at least 80 neighbors used for entropy coding. In this paper, a context model with 14 neighbors is adopted to ensure that the complexity of our method is lower than [6] while the compression results are close to [6]. Fig. 2(a) shows the smaller context model used in JPEG2000, and Fig. 2(b) shows the larger context template: 12 spatially adjacent coefficients/blocks in the current subband, one parent coefficient/block in the lower subband, and one child coefficient/block in the higher subband [10]. In order to obtain the correlations of the wavelet coefficients, we adopt the regressive–prediction algorithm. In [9], W is defined as representing the degree of

J. Wu et al. / Signal Processing: Image Communication 28 (2013) 727–735

importance of the current coefficient/block X and calculated by n

W ¼ ∑ αi Si ;

ð2Þ

i¼0

where Si ði ¼ 0; 1; 2; …; nÞ denotes the degrees importance of neighbors of X, where n is the number of coefficients in the context template. If a particular coefficient/block is found to be important, that is, if a particular coefficient/ block is greater than current threshold T, then Si ¼1; else, Si ¼ 0. The bitplane coding starts at nth ðn ¼ ⌊log2 C max ⌋Þ plane. Here Cmax means the maximum coefficient of whole image. The threshold T is defined as T ¼ 2n . After each plane coding, n will be decreased by 1 (provided that n 4 0) and the threshold will be divided by 2, which means the coding going on to the next plane. Here, regressive– prediction is used to obtain αi ði ¼ 0; 1; 2; …; nÞ, which is the set of prediction coefficients for the wavelet coefficients, which represent the degrees of neighbors’ contributions to the current wavelet coefficient/block. In our paper, the bitplane coding can be substituted for Trellis Coded SpaceFrequency Quantization (TCSFQ) [11] and a better performance can be achieved. However, the complexity of TCSFQ is much higher than bitplane coding. According to Eq. (2) and Fig. 2, the weight W can be determined as W ¼ ∑ni¼ 0 αi Si , where n¼13. To calculate above formula, prediction coefficients αi ði ¼ 0; 1; 2; 3; …; 13Þ should be trained. All the prediction coefficients αi are obtained using the least mean squares method. First, in a specific window, a group of predictions should be determined using a set of wavelet coefficients/blocks. The prediction value b x i of an arbitrary wavelet coefficient with 14 neighbors is 11

b x i ¼ ∑ αi N i þ α12 P 0 þ α13 C 0 :

ð3Þ

i¼0

where Ni, P0, and C0 are parameters of the neighbors. Therefore, the prediction coefficients are a set of values that minimize the sum of mean square errors of all

729

coefficients min∑∥xi −b x i ∥:

ð4Þ

i

We test using a 512  512 Lena image to demonstrate that correlations do exist in wavelet coefficients, within-subband and cross-subband. The prediction coefficients in different subbands of the context templates are shown in Fig. 3.

Fig. 2. Context model used in JPEG2000 and the neighbors in the larger context template: (a) Context model in JPEG2000. (b) Larger context model. Notes: X denotes current coefficient/block; symbols such as NN and SS denote the spatially adjacent coefficients/blocks of X, P represents the parent coefficient/block of X and C represents the child coefficient/ block of X which is calculated by the average of the four child coefficients/ blocks of X.

Fig. 1. Correlations within the wavelet transformed coefficients: (a) Original 512  512 Lena image. (b) Wavelet transformed coefficients after three levels of wavelet decomposition using 9/7 filters.

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current subband are significant. On the other hand, if the wavelet coefficients in the child subband are estimated to be significant, then there is a large probability that the corresponding wavelet coefficients/blocks in the current subband are also significant. All the prediction coefficients αi are obtained through the off-line method and used as known results in the encoding and decoding processes. The weight W is obtained according to W ¼ ∑ni¼ 0 αi Si (n ¼13), and W is used as the context to be classified. This weight W needs to be transmitted to the receiver, which will cost 0.0018 bpp for 512  512 size images and 0.0004 bpp for 1024  1024 size images. The rate cost of weight is small enough, and thus is negligible.

4. Context classification using Lloyd–Max quantization 4.1. Idea of context classification Through regressive–prediction, we obtain the prediction coefficients αi . In order to avoid the context dilution problem and to reduce complexity, we quantize the coefficients/blocks, whose weight has the same or similar conditional probability, into integers from 1 to 30. The number of quantization levels was determined empirically. According to formula (1) and with the quantized prediction coefficients αi , we can obtain the value of weight W. In our experiment, we find that the largest value of W is less than 100. Thus, using weight W as the context can significantly reduce the complexity, since the largest value of weight W is smaller than 2n (n equals the size of the template. For example, if we use the template shown in Fig. 2(a), n ¼8; or the template in Fig. 2(b), n¼ 14). Using the importance weight W as the context, the estimated conditional importance probability is b ðxi jwi Þ: p

Fig. 3. Prediction coefficients in LH, HL, and HH subbands: (a) Prediction coefficients in LH subband. (b) Prediction coefficients in HL subband. (c) Prediction coefficients in HH subband.

As shown in Fig. 3, the prediction coefficients have strong correlations in the vertical direction, since the coefficients in the vertical direction have larger magnitudes than the others in the HL subband. The same observations can be made in the HH subband. However, for the LH subband, an unexpected finding is that the correlation strength of prediction coefficients in the horizontal direction is weak. It is worth noticing that the prediction coefficient of the child subband is greater than any other prediction coefficients in the template. This situation does not always occur, for the wavelet coefficients/blocks in the child subband may not be significant when its corresponding wavelet coefficients/blocks in the

ð5Þ

Though the number of contexts will be reduced by the replacement with weight W, the resultant number of contexts still results in a context dilution problem. Literature [4] proposes a principle of mutual information to solve this. It is pointed out in [4] that the performance will not be improved unless the conditional probabilities of the combined contexts are all the same. In practice, some of the conditional probability of different weights W are very close or the same. Accordingly, we try to combine the weights which have similar conditional probabilities to further reduce the context number while maintaining the compression performance. Fig. 4 illustrates the distribution of the conditional probabilities of weight W for the 14-neighbor model. The horizontal axis represents the weight W, and the vertical axis represents the probability of the current coefficient/block being significant. This probability is calculated based on the condition that the coded coefficients/blocks under current weight W are significant. As shown in Fig. 4, the four images are quite similar. (Here, the weights with probability 0 indicate that there are no samples with those weights, and we do not consider them in context-classification.) The weights are compact except those with probability 0.

J. Wu et al. / Signal Processing: Image Communication 28 (2013) 727–735

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Fig. 4. Different distributions of conditional probability with different weights.

To classify the contexts with similar probabilities into one context, we employ the Lloyd–Max algorithm [6]. 4.2. Lloyd–Max algorithm The Lloyd–Max algorithm is used widely in clustering algorithms and quantization [6]. The main principle of the Lloyd–Max algorithm is to classify a data set into various clusters through an iterative method, to make the data in each cluster compact. In our method, the weights used in classification are continuous, so the weight set W ¼ fw1 ; w2 ; …; wn g is classified into K clusters using the Lloyd–Max quantizer, represented by Q(w). The conditional b ðxi jwi Þ of the weights in each cluster approxiprobabilities p mately the same. Thus, the estimated conditional probability becomes b ðxi jQ ðwi ÞÞ: p

ð6Þ

The algorithm is composed of the following steps:

1. Place K centroids into the space. The coefficients/ blocks being clustered represent the space. Let pi ði ¼ 1; 2; …; KÞ represent the ith centroid.

2. Assign each coefficient/block to the group whose centroid is closest to the coefficient/block, that is, minimize b ðxi jwi Þ−pj ∥2 , the objective function ∑ni¼ 1 minj∈ð1;2;…;KÞ ∥p where pj represents the jth centroid. 3. When all coefficients/blocks have been assigned, recalculate k new centroids as barycenters of the clusters from the previous step, then calculate the new cluster centroids using formula (7) pi ¼

 1 b ðxi wi Þ; ∑ p n pðwi Þ∈C i

ði ¼ 0; 1; 2; …; KÞ;

ð7Þ

where n is the number of coefficient/block in cluster Ci. 4. Repeat Steps 2 and 3 until the centroids no longer change. The ultimate set of cluster centroids p i is then used as the quantized context Q ðwi Þ.

Although it can be proved that the procedure will always terminate in convergence, the Lloyd–Max algorithm does not necessarily find the most optimal classification, because the algorithm is significantly sensitive to the initial selected cluster centers. In order to reduce the instability of initialization, we initialize the cluster centers, as follows. Based on the characteristics of the probability distribution discussed in

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Section 4.1, we initialize the centroids with formula (8) b b b ðxi jwi Þ þ maxP ðxi jwi Þ−minP ðxi jwi Þ : pi ¼ minP 2K

ð8Þ

The purpose of Eq. (8) is to limit the initial centroids such

Fig. 5. Schematic drawing of weight clustering.

that they fall within the range of each class, making initialization more stable. The context clustering method presented in this paper is shown in Fig. 5. The horizontal axis represents the weight W of the current coefficient/block, and the vertical axis represents the conditional probability. The interval 0–1 on the vertical axis is divided into K subintervals based on the principles of the Lloyd–Max quantizer. Then, the points in each subinterval are combined into one context. We next seek to determine the best number of clusters K and whether the adaptive or fixed context classification scheme achieves better performance. Fig. 6 shows the results for different context numbers obtained through on-the-fly and off-line context classification. We also list the results using the SPECK-AC algorithm for comparison, since it also employs context classification [12], with wavelet coefficients, blocks, refinement, and sign-in LSP (list of significant pixels) each having their own contexts in SPECK's low-order arithmetic coding. The experimental curve is plotted for rates of 0.5 bpp and 1 bpp with context numbers ranging from 1 to 30. From Fig. 6, we see that different context numbers do achieve different compression performances for both onthe-fly and off-line context classification. For the on-the-

Fig. 6. Results of different classifications at different rates: (a) Results of different context numbers both on the fly and off-line classification at 0.5 bpp. (b) Results of different context numbers both on the fly and off-line classification at 1 bpp.

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Fig. 7. Results of two templates at different rates: (a) Results of different context numbers both eight-neighborhood model template and 14-neighborhood model template at 0.5 bpp. (b) Results of different context numbers both eight-neighborhood model template and 14-neighborhood model template at 1 bpp.

fly strategy, as the context number increases, the PSNR curves first move up to the max point of 22, and then begins to decrease for most cases. The PSNR curves for the off-line strategy are the same except that the max point is 14. Therefore, in our method the optimum context number is 22 for the on-the-fly and 14 for the off-line strategy. The difference in PSNR from the optimal to the worst is 0.03 dB for the goldhill image, and it can approach 0.02 dB for other images. However, the performance is almost the same for the on-the-fly and off-line context classification schemes, with off-line being a little higher in most cases. The better results of the off-line method are because it utilizes the global probability distribution which contains more samples than the local probability distribution used by the on-the-fly method, and also because an insufficient number of symbols when using the adaptive statistical process can result in unstable conditional probabilities, especially in the initiation phase. 5. Experimental results In this section, experimental procedures and results are described. Different test images were applied in the

experiments, including 10 images of natural objects from the USC image database and one satellite image from the CCSDS image database. These are frequently used in image compression papers to facilitate comparison. Two of the 10 natural images had 512  512 pixels, and the rest had 1024  1024 pixels. All test images were in gray-scale. The entropy coder employed in our method was the arithmetic coder, and the Daubechies 9/7 filters were used with five-level dyadic wavelet decompositions. In this paper, the SPECK algorithm is used for bitplane coding. The context number was selected as 22 for the on-the-fly strategy and 14 for the off-line strategy. In order to demonstrate that larger context modeling with 14 neighbors is feasible, the experiment was conducted using context modeling with an eight-site neighborhood template, which is employed in current mainstream compression standard JPEG2000, and a 14-site neighborhood template in the on-the-fly mode. Fig. 7 shows the different results for the smaller and larger contexts. As we can see, by virtue of its prediction exploiting more image orientation information, the larger context model usually provides higher compression performance.

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To compare the results using our method with those using the SPECK-AC, SPHIT-AC and JPEG2000 algorithms, we report the peak signal-to-noise ratio (PSNR) for each image. Table 1 provides the results comparing the different compression methods. The images were compressed at different rates. Table 1 demonstrates that our method can improve the PSNR of the image up to 0.55 dB compared with SPECK-AC and

Table 1 Comparison of different compression methods. (A) Bpp. (B) SPECK-AC. (C) SPIHT-AC. (D) JPEG2000. (E) Our-method on-the-fly. (F) Our-method off-line. Images 512  512 Lena

Baboon

Goldhill

Barbara

Girl

Aerial

Boat

Finger

1024  1024 Man

Airport

Pentagon

A

B

C

D

E

F

0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1

34.03 37.1 40.25 23.1 25.4 28.9 30.5 33.03 36.36 27.76 31.54 36.49 28.75 32.04 36.92 25.53 28.58 33.04 29.89 33.11 36.54 24.44 27.9 31.51

34.12 37.22 40.42 23.26 25.64 29.17 30.56 33.13 36.55 27.58 31.4 36.41 29.06 32.29 37.26 25.56 28.74 33.17 30.09 33.22 36.57 24.25 27.67 31.35

34.15 37.28 40.35 23.1 25.52 29.02 30.54 33.25 36.6 28.27 32.15 37.11 28.77 32.29 37.27 25.44 28.7 33.18 30.1 33.34 36.72 24.37 27.86 31.6

34.3 37.4 40.53 23.32 25.71 29.32 30.67 33.32 36.73 28.18 32.06 37.05 29.05 32.48 37.49 25.7 28.96 33.39 30.31 33.43 36.61 24.49 27.99 31.91

34.34 37.42 40.53 23.32 25.72 29.33 30.68 33.35 36.74 28.19 32.09 37.05 29.08 32.5 37.5 25.73 28.97 33.4 30.32 33.43 36.62 24.49 27.99 31.92

0.25 0.5 1 0.25 0.5 1 0.25 0.5 1

31.26 34.14 37.24 28.16 30.27 33.11 28.81 31.14 34.12

31.36 34.25 37.37 28.24 30.38 33.27 29.02 31.34 34.31

31.22 34.18 37.36 28.28 30.41 33.17 28.99 31.3 34.21

31.47 34.35 37.44 28.36 30.5 33.35 29.03 31.39 34.35

31.48 34.36 37.44 28.36 30.51 33.35 29.04 31.4 34.37

0.69 dB compared with SPIHT-AC. In most cases, greater enhancement of PSNR emerges occurred at the rate of 1 bpp, since a higher bit rate can yield more samples for the probability statistics, which makes regression forecasting and classification more accurate. For the satellite image Pentagon, the proposed method can adequately exploit the correlation between complicated textures, which is unmatched by many other coding schemes. Our proposed algorithm is superior to JPEG2000 in most cases. An exception occurred for the Barbara image due to the strong direction performance of the image area segment. This indicates that a larger context model does contribute to higher performance, except for images with rich textures in small regions. To analyze the time complexity of the method proposed in this paper, we report the runtime for Lena.raw, girl.raw and boat.raw. Table 2 provides the running times of our method, JPEG2000 (Jasper) and SPECK-AC. From Table 2 we can see, for encoding process, the maximum runtime difference between our method and JPEG2000 at 0.25 bpp is less than 0.13 s, 0.5 bpp 0.2 s, and 1 bpp 0.3 s. Compared to SPECK-AC, the running time of our method at 0.25 bpp is 0.21 s longer than that of SPECKAC at most, at 0.5 bpp 0.29 s, and at 1.0 bpp 0.36 s. For decoding process, the running time of our method can be up to 0.13 s shorter than that of JPEG2000 for all three rates, and 0.38 s longer at most than SPECK-AC for 0.25 bpp, 0.5 bpp and 1.0 bpp. Although SPECK-AC has a lower time complexity, it is short on compression result. The price of performance enhancements is the increase of complexity. The table shows that the running times of proposed method and that of JPEG2000 are similar. That is to say, the performance boost of our method is traded by a lower complexity added and would not give rise to serious complexity problem. From the analysis above, it can be seen that the proposed approach is an effective one.

6. Conclusions In this paper, we propose a new context-based arithmetic coding algorithm. The proposed algorithm effectively exploits the correlations of the wavelet coefficients using a large-context 14-site neighborhood template. The

Table 2 Running time of our methods, JPEG2000 and SPECK-AC. Images

Method

Our method

JPEG2000

SPECK-AC

Rate (bpp)

Encode(s)

Decode(s)

Encode(s)

Decode(s)

Encode(s)

Decode(s)

Lena

0.25 0.5 1

0.27 0.38 0.58

0.36 0.45 0.53

0.17 0.23 0.29

0.37 0.37 0.36

0.11 0.14 0.22

0.11 0.14 0.22

Girl

0.25 0.5 1

0.29 0.39 0.58

0.37 0.42 0.5

0.17 0.25 0.3

0.51 0.53 0.51

0.09 0.12 0.2

0.09 0.14 0.22

Boat

0.25 0.5 1

0.3 0.41 0.56

0.37 0.45 0.58

0.17 0.23 0.3

0.4 0.5 0.52

0.09 0.12 0.2

0.09 0.12 0.2

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weights W are calculated using regressive prediction and are regarded as contexts, the order of which is then reduced through context quantization. Thus, high-order context-based arithmetic coding can be approximated as low-order. Different context models and context quantization strategies have been implemented to seek optimal performance. Experimental results show that the off-line strategy provides better performance than the on-the-fly strategy, and a larger context template can also lead to better results. The proposed approach can also be applied to satellite images; however, it is not suitable for images with a great deal of high-contrast regions, such as the Barbara image. The comparison results demonstrate that our proposed method achieves better compression performance for most images compared with state-of-the-art compression methods. Acknowledgments This work is supported by the National Natural Science Foundation of China (Nos. 61077009, 61001100, 61001206) and the Ocean Public Welfare Scientific Research Project, State Oceanic Administration People's Republic of China (No. 201005017). References [1] D. Marpe, H. Schwarz, T. Wiegand, Context-based adaptive binary arithmetic coding in the H2.64/AVC video compression standard, IEEE Transactions on Image Processing 13 (July 7) (2003) 620–636.

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