# An Enhanced Adaptive Block Truncation Coding with Edge Quantization Scheme +
## 4 Experiment and Comparison
### Discussion
For a \((\delta_{0}^{\prime},\delta_{1}^{\prime},\delta_{2}^{\prime},\delta_{3}^{ \prime})\), the difference \(\delta_{1}^{\prime}\) is bounded with the value \((255-s)\), where \(s=(\mu_{0}^{\prime\prime}+\delta_{2}^{\prime}+\delta_{3}^{\prime})=(\delta_{0} ^{\prime}+\delta_{1}^{\prime}+\delta_{2}^{\prime})\). Because the quantization range \(R_{1}:0\sim 126\) is used for \(\delta_{1}^{\prime}\), for an extreme case which \(s\) is very small, the value of \((255-s)\) may be larger than 126, out of range \(R_{1}\). This may give rise to large error of recovered value. As we know, k-means clustering algorithm is a good _quantization_ algorithm. Therefore, for most data, we may recover the original \((\mu_{0}^{\prime\prime},\mu_{1}^{\prime\prime},\mu_{2}^{\prime\prime},\mu_{3}^ {\prime\prime})\) with small distortion. Experimental results in Table 1 also confirms the above statement. **(2) Size of Compressed Rates (CRs):**  
Here, we deal with the enhancement of three modified ABTC-EQ schemes such as Scheme A, Scheme B, and Scheme C. Except the Scheme C (using two bits to represent four clusters), the other two schemes enhance the CR. As we know about compression technology, the CR is the most important key property. Better CR implies that compression technology has a better performance. Therefore, we showed a theoretical analysis of the estimated CRs in Theorems 1, 2, and 3. In addition, to prove the accuracy for theorem, we show a comparison of the simulation results (Table 2) and the estimated CRs derived by the Theorems. That is when \(k\) is [4,6,8] for all values of the given \(P_{e}\), the expected compression ratio of the proposed method is shown in Table 2.  
Moreover, the average of CRs for five test images (Lena, Butterfly, Cameraman, Lake, Pepper) are listed. Consider the CRs using Scheme A. For the case \(k=4\), the average CR of experimental values is 3.23 near the theoretical CR = 3.24 (\(p_{e}=0.35\)). The average CR of experimental values is 4.27 (respectively, 4.78) for \(k=6\) (respectively, \(k=8\)), which is near the theoretical CR = 4.27 (\(p_{e}=0.45\)) (respectively, CR = 4.78 (\(p_{e}=0.50\))). This result consists with the increment of \(p_{e}\) for a large \(k\). As we know, if the edge values \(e_{i},1\leq i\leq k^{2}\), in E is \"1\" and not all the edge values are \"1\", then the image block is defined as an edge block. The number of edge blocks is increased when the value of \(k\) is increased, and thus the probability is increase for the large \(k\). **(3) Time Complexity:**  
The time complexity of all the proposed methods is an important criterion for the performance evaluation of compression algorithms. The suggested method fits this criterion very well. Because the only difference between Scheme A and the original ABTC-EQ is that Scheme A adopts Huffman code for representing three clusters, namely using (0), (10), and (11). This is a very simple Huffman code. In fact, we do not need encoding/decoding for this Huffman code in Scheme A. From another viewpoint, the original ABTC-EQ uses (00), (01), and (10) for representing three clusters, while Scheme A uses (0), (10), and (11) instead. The Huffman code in Scheme A is only used to represent the index of clusters \(c_{0}\), \(c_{1}\), and \(c_{2}\), and we do not need encoding/decoding of Huffman code. At this time, the (0), (10), and (11) are to only assure the unique representation of the index of the cluster. Scheme B uses the same approach as Scheme A, represent clusters using Huffman code, and this approach combines various quantization ranges. Quantized pixels are used as representative pixels (8 bits) of each block when the image is decompressed. We use Scheme B-I as an example for describing the quantization representation is the same as a binary representation. For a pixel with 8 bits \((b_{8},b_{7},\ldots,b_{1})\), its pixel value is \(\sum_{i=1}^{8}(b_{i}\times 2^{i-1})\).