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An improved EZBC algorithm based on block bit length
EURASIP Journal on Advances in Signal Processing volume 2011, Article number: 84 (2011)
Abstract
Embedded ZeroBlock Coding and context modeling (EZBC) algorithm has high compression performance. However, it consumes large amounts of memory space because an Amplitude Quadtree of wavelet coefficients and other two link lists would be built during the encoding process. This is one of the big challenges for EZBC to be used in real time or hardware applications. An improved EZBC algorithm based on bit length of coefficients was brought forward in this article. It uses Bit Length Quadtree to complete the coding process and output the context for Arithmetic Coder. It can achieve the same compression performance as EZBC and save more than 75% memory space required in the encoding process. As Bit Length Quadtree can quickly locate the wavelet coefficients and judge their significance, the improved algorithm can dramatically accelerate the encoding speed. These improvements are also beneficial for hardware.
PACS: 42.30.Va, 42.30.Wb
1. Introduction
At present, the typical embedded waveletbased image coding methods includes Embedded Zerotree Wavelet coder (EZW) [1] proposed by Shapiro, Set Partitioned in Hierachical Tree (SPIHT) [2] proposed by Said etc., Set Partition Embedded block algorithm (SPECK) [3] proposed by Asad Islam, Embedded Block Coding with Optimal Truncation (EBCOT) [4] used in JPEG2000 [5], and Embedded ZeroBlock Coding and context modeling (EZBC) [6] proposed by ShihTa Hsiang etc.
Among these algorithms, EZW and SPIHT were based on the local characteristic in spatialfrequency domain of wavelet transform. They utilized the similarity of wavelet coefficients of the same spatial location in different subbands. SPECK was based on the characteristic that the energy concentrated in lowfrequency subband and the insignificant coefficients mainly concentrated in highfrequency subbands after wavelet transform, it utilized the correlation of insignificant coefficients in the same subband. EBCOT used the correlation of wavelet coefficients in the same subband to build the high efficiency context and adopted the Arithmetic Coder. EZBC utilized the correlation of coefficients in the same subband and coefficients in different subbands at the same time and adopted a simple and efficient quadtree coding structure and contextbased bitplane encoding method [6, 7], so it can achieve higher compression performance than SPIHT and EBCOT. EZBC algorithm is widely used in digital image compression and scalable video coding.
However, during the encoding process EZBC built an Amplitude Quadtree Q_{ k }[l](i, j) and two link lists: the insignificant nodes link list LIN and the significant pixels link list LSP. Both Amplitude Quadtree Q_{ k }[l](i, j) and the link lists consumed a large amounts of memory space and this was one of the big challenges for the EZBC to be used in real time or hardware applications. As it has a large amount of operations of link lists such as adding and deleting nodes during encoding, this made its coding efficiency decline greatly when good image quality is required. The algorithm proposed in reference [8] presented the significance statetable of Quadtree coefficients to record the significance state change of all the QuadTree corresponding coefficients in the coding of Quadtree bitplane, which removed the LIN and LSP. Some memory was saved but Q_{ k }[l](i, j) was still adopted in the algorithm. Instead of making any radical change, it increased the complexity of the algorithm on the contrary.
In order to overcome these disadvantages, an improved EZBC algorithm, Bit Length EZBC (BLEZBC) based on bit length of coefficients, was put forward in this article. It used Bit Length Quadtree instead of Amplitude Quadtree of wavelet coefficients and other two link lists, so it can save more than 75% memory space required in the encoding process. As Bit Length Quadtree can quickly locate the wavelet coefficients, judge their significance, and avoid a large amount operations of link lists used in EZBC, this algorithm can accelerate the encoding speed effectively. The test results indicates that this algorithm achieves the same signal to noise ratio as EZBC and gains much higher encoding speed, saves 75% memory usage than EZBC.
2. Bit Length Quadtree
The bitplane coding process in BLEZBC begins with establishment of the Bit Length Quadtree representations for the individual subbands. The bit length here refers to the significant bit length of the absolute value of quantized wavelet coefficient. The value of the Bit Length Quadtree node B_{ k }[l](i, j) at position (i, j), quadtree level l and subband k is defined by
where C_{ k }(i, j) is the quantized subband coefficient at position (i, j). x indicates the absolute value of x; \u230ax\u230b indicates the rounding operation on x. For example, if x=\lfloor 3.8\rfloor, then x ≡ 3 where " = " means assignment and "≡" indicates the equality test.
Below is an example for B_{ k }[l](i, j).
If C_{ k }(i, j) = (± 9), then B_{ k } [0](i, j) = 4. That is, the significant bit length of ± 9 is equal to 4.
Each bottom quadtree node is assigned to the bit length of the subband coefficient at the same position. The quadtree node at the next level is then set to the maximum of the four corresponding nodes at the current level, as illustrated in Figure 1a. The top quadtree node is just equal to the maximal bit length of all subband coefficients. Similar to the conventional bitplane coders, we progressively encode subband coefficients from the MSB toward the LSB. In the Bit Length Quadtree, a node is significant if the value of the node B_{ k }[l](i, j) is great than the index of the bitplane. A significant pixel (coefficient) is located by the testing and splitting operation recursively performed on the significant nodes up to the pixel (bottom) level of a quadtree, as shown in Figure 1b.
Instead of using Amplitude Quadtree Q_{ k }[l](i, j) and two link lists in EZBC, the bit length of the Quadtree in BLEZBC is built up with the bit length of the absolute value of the subband coefficients. It corresponds to the index of the bitplane. It means that Bit Length Quadtree can quickly locate the wavelet coefficients and judge their significance. Thus, it can accelerate the encoding speed effectively. And the lists of insignificant nodes LIN and significant nodes LSP required in EZBC are not necessary in BLEZBC. What iss more, the usual bit length of wavelet coefficients is 16. That is, each node of Amplitude Quadtree Q_{ k }[l](i, j) uses 16 bit length memory. However, in BLEZBC, each node of the Bit Length Quadtree B_{ k }[l](i, j) uses 4 bit length memory. For 4 bit length, the max value it can represent is 15. It means that the corresponding max absolute value of wavelet coefficients is 2^{15}  1 = 32767. Generally, the absolute value of the wavelet coefficients is less than 32767. That is, 4 bit length of each node of Bit Length Quadtree can represent the bit length of wavelet coefficients. This saves a large amount of memory space required in the coding process.
3. The coding process of BLEZBC
First, define parameters below,

C_{ k }(i, j): the quantized wavelet coefficient of subband k at position (i, j).

B_{ k }[l](i, j): Bit Length Quadtree B representation for the bit length of coefficients from the same subband with node of B_{ k }[l](i, j) corresponding to a quadtree node at position (i, j), Suband k, and level l. Its value is defined by (1).

D_{ k }: depth of the quadtree of the subband k.

D_{ max }: the maximum quadtree depth among all subbands.

K: total number of subbands.

X_{ k }: Horizontal offset of the subband k referring to the original image, lefttoright as a positive direction.

Y_{ k }: Vertical offset of the subband k referring to the original image, toptodown as a positive direction.

CodeBL(k, l): Function for coding the insignificant node of level l in Bit Length Quadtree of subband k. Its parent node should be signficant in the last bitplane if it exists.

CodeDescendant(k, l, i, j): function for processing all the descendent nodes of B_{ k }[l](i, j) after it just tested significant against the current threshold.

CodeLSP(k): function for refinement of the coefficients of subband k.
The coding process,
Initialization,
n=\underset{\left(k\right)}{max}\left\{{B}_{k}\left[{D}_{k}1\right]\left(0,0\right)\right\};
Coding the highest bitplane,
for k = 0: K1, Code BL(k, D_{ k }1);
n;
Coding the remaining bitplanes.
for (;n > 0; n){
for l = 0: D_{max}1
for k = 0: K1, Code BL(k, l);
for k = 0: K1, Code LSP(k);
}
Below are the functions in pseudocode,
CodeBL (k, l)
{

if (l < D_{ k } 1)
* for all (i, j) in quadtree level l+ 1, subband k. That is, all the nodes in level l+1 of Bit Length Quadtree of subband k
⋆ if (B_{ k }[l+ 1](i, j) > n), CodeDescendant (k, l+ 1, i, j);

else if (l ≡ D_{ k } 1)
* if (B_{ k }[l](0, 0) < n), output 0;
* else if (B_{ k }[l](0, 0) ≡ n){
output 1;
⋆ if (l ≡ 0)

output the sign of C_{ k } (0+X_{ k }, 0+Y_{ k });
⋆ else

CodeDescendant (k, l, 0, 0);
}
}
CodeDescendant (k, l, i, j)
{

for (x, y)∈{(2i, 2j), (2i, 2j+ 1), (2i+ 1, 2j), (2i+ 1, 2j+ 1)}. That is, the four child nodes in level l1 that mapping to the node (i, j) in level l, subband k
* if (B_{ k }[l 1](x, y) < n), output 0;
* else if (B_{ k }[l 1](x, y) ≡ n){
output 1;
⋆ if (l ≡ 1)

output the sign of C_{ k } (x+X_{ k }, y+Y_{ k });
⋆ else

CodeDescendant (k, l 1, x, y);
}
}
CodeLSP (k)
{

for all (i, j) in quadtree level 0, subband k. That is, all the nodes in level 0 of Bit Length Quadtree of subband k, scanning first in row then column.
* if (B_{ k }[0](i, j) > n), output the value of  C_{ k } (I + X _{ k }, j + Y _{ k })  in bitplane n.
}
4. Experimental results and analysis
The improved algorithm BLEZBC and the original EZBC were verified and compared in Pentium(R) D CPU 2.80 GHz computer with 512 × 512 × 8 bits Standard grayscale images of Lena, Goldhill and Barbara. 9/7 Wavelet filters boundary symmetrical extension was used. The context model was the same as which used in EZBC algorithm, the algorithm data in reference [8] were quoted here. Table 1 listed out the PSNR comparison. Table 2 shows the memory usage comparison during coding. Table 3 shows the coding time comparison with different thresholds of BLEZBC and EZBC.
From Table 1, the same PSNR was achieved by using BLEZBC, algorithm in reference [8] and EZBC. However, EZBC used Amplitude Quadtree Q_{ k }[l](i, j) and two link lists (LIN and LSP) to finish the coding process and utilized the significance of the neighbor nodes and the node in parent subband to construct the context, so a large amount of memory was required to store Amplitude Quadtree Q_{ k }[l](i, j) and two link lists. Therefore, more memory was required for more complex image and higher coding bit rate. These increased the complexity of the hardware. Although linked lists LIN and LSP were removed from the algorithm in reference [8], Amplitude Quadtree Q_{ k }[l](i, j) and significance statetable of Quadtree coefficients were still adopted, which also occupied a lot of memory and increased the algorithm complexity. Bit Length Quadtree was presented to replace the Amplitude Quadtree in the improved algorithm, which was used to complete the coding process and construct the context, so the memory usage was greatly reduced during the coding. As shown in Table 2, by using the improved algorithm BLEZBC, more than 75% memory was saved compared to the original EZBC algorithm. Therefore, the memory usage was significantly reduced.
As EZBC used a large amount of operations of link lists such as adding and deleting nodes during encoding, this made its coding efficiency decline greatly when good image quality was required. Instead of using Amplitude Quadtree Q_{ k }[l](i, j) and two link lists in EZBC, the bit length of the Quadtree in BLEZBC is built up with the bit length of the absolute value of the subband coefficients. It corresponds to the index of the bitplane. It means that Bit Length Quadtree can quickly locate the wavelet coefficients, judge their significance. Thus, it can accelerate the encoding speed effectively. As shown in Table 3, if the threshold less than 32, the coding time of BLEZBC was significantly less than EZBC. If the threshold is great than or equal to 32, the coding time of BLEZBC was less than EZBC too. However, it was not so obvious. A larger threshold means less bitplanes would be scanned, the size of link lists was smaller, and the operations of link lists were also fewer. As the larger threshold means the worse image quality, in most situations in order to achieve better image quality the threshold was not so large. When good image quality was required, the coding speed of BLEZBC is significantly faster than EZBC.
5. Conclusion
An improved algorithm BLEZBC based on EZBC was proposed in this article. A new model Bit Length Quadtree was used to complete the coding process and construct the context. It can achieve the same compression performance as EZBC but the memory usage was greatly reduced during the coding process. Bit Length Quadtree can quickly locate the wavelet coefficients, judge their significance, and avoid a large amount operations of link lists used in EZBC. Thus, it can accelerate the encoding speed effectively. These improvements are also beneficial for the hardware.
Abbreviations
 EBCOT:

Embedded Block Coding with Optimal Truncation
 EZBC:

Embedded ZeroBlock Coding and context modeling
 EZW:

Embedded Zerotree Wavelet coder
 SPECK:

Set Partition Embedded block algorithm
 SPIHT:

Set Partitioned in Hierachical Tree.
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Acknowledgements
This work was supported in part by the Shenzhen Double Hundred Person Project and the Shenzhen University Dr startup fund research (000201) of China.
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Wang, R., Ruan, S., Liu, C. et al. An improved EZBC algorithm based on block bit length. EURASIP J. Adv. Signal Process. 2011, 84 (2011). https://doi.org/10.1186/16876180201184
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DOI: https://doi.org/10.1186/16876180201184
Keywords
 block bitlength
 zeroblock
 EZBC
 Quadtree
 DWT