# An Enhanced Adaptive Block Truncation Coding with Edge Quantization Scheme +
## 4 Experiment and Comparison
### Discussion  
We further discuss three important issues in-depth: (i) the visual quality of reconstructed image, i.e., PSNR, (ii) the size of a compressed rate, i.e., CR, and (iii) an appropriate way of using Scheme A, Scheme B, and Scheme C for applications. **(1) Visual Quality of Reconstructed Image:**  
In Schemes B and C, we adopt various quantization range for each cluster. Scheme B has three quantization ranges \(R_{i}\) for \(\delta_{i},0\leq i\leq 2\), while Scheme C has four quantization ranges \(R_{i}\) for \(\delta_{i},0\leq i\leq 3\). Consider Scheme C with four ranges \(R_{0}=(128\ 64\ 32\ 16\ 8\ 4)\), \(R_{1}=(64\ 32\ 16\ 8\ 4\ 2)\), \(R_{2}=(32\ 16\ 8\ 4\ 2\ 1)\), and \(R_{3}=(32\ 16\ 8\ 4\ 2\ 1)\). By the definition \(\mu_{0}^{\prime\prime}=\delta_{0}^{\prime}\), we have the quantization error \(\pm 2(\cdot\cdot\ R_{0}\) can represent the values 0, 4, 8, \(\ldots\), and 252). Because of \(\mu_{1}^{\prime\prime}=\delta_{0}^{\prime}+\delta_{1}^{\prime}=\mu_{0}^{\prime \prime}+\delta_{1}^{\prime}\) with minimum distortion,  
Figure 1: The reconstructed images for Lena: (**a**) original image (**b**) absolute moment BTC (AMBTC): 33.42 dB (**c**) modified BTC (MBTC): 33.87 dB (**d**) adaptive block truncation coding based on edge quantization (ABTC-EQ): 37.49 dB (**e**) Scheme A: 37.49 dB (**f**) Scheme B-I: 37.47 dB (**g**) Scheme B-II: 37.39 dB (**h**) Scheme B-III: 37.14 dB (**i**) Scheme B-IV: 35.96 dB (**j**) Scheme C: 39.62 dB. Figure 2: Selected portion of edge image for Cameraman: (**a**) original image (**b**) AMBTC (**c**) MBTC (**d**) ABTC-EQ (**e**) Scheme C.  
if the values of quantization range \(R_{1}\) can catch up the difference \(\delta_{1}^{\prime}\) then the quantization error of \(\mu_{1}^{\prime}\) is \(\pm 1(\cdot\cdot\)\(R_{1}\) can represent the values 0, 2, 4,..., and 126). By the same argument, we have no quantization errors for \(\mu_{2}^{\prime}\) and \(\mu_{3}^{\prime}\), because \(R_{2}\) and \(R_{3}\) can represent the values \(0,1,2,\dots,\) and 63. Therefore, when comparing with the original values of four clusters \((c_{0},c_{1},c_{2},c_{3})\), our recovered values have very small distortion. In Example 3, the original values are \((\mu_{0}^{\prime\prime},\mu_{1}^{\prime\prime},\mu_{2}^{\prime\prime},\)\(\mu_{3}^{\prime\prime})=(61,83,115,139)\), and the recovered values are (60, 82, 115, 139), almost the same to the original one. For this case, we still use 24 bits to represent four quantization values, which are the same to ABTC-EQ using 24 bits to represent three quantization values. By using the above analysis, we may derive quantization errors of \((\mu_{0}^{\prime},\mu_{1}^{\prime},\mu_{2}^{\prime})\) for Schemes B-I, B-II, B-III and B-IV as \((\pm 1,0,0)\), \((\pm 2,\pm 1,\pm 1)\), \((\pm 4,\pm 2,\pm 2)\), and \((\pm 8,\pm 4,\pm 4)\), respectively. For the original values \((\mu_{0}^{\prime},\mu_{1}^{\prime},\mu_{2}^{\prime})=(61,89,125)\), Example 2 demonstrates that the recovered values for Schemes B-I, B-II, B-III and B-IV are (60, 89, 125), (60, 88, 124), (64, 88, 124) and (64, 88, 128). These recovered results are consistent with the theoretical quantization errors. Now, we give an analysis on the probability of whether a quantization range can catch up the previous difference. For example, for the case using Scheme C, there are four clusters. On average, we may have four \((\mu_{0}^{\prime\prime},\mu_{1}^{\prime\prime},\mu_{2}^{\prime\prime},\mu_{3}^ {\prime\prime})=(51,102,153,204)\), and the first value \(\mu_{0}^{\prime}\) and the differences \((\delta_{1}^{\prime},\delta_{2}^{\prime},\delta_{3}^{\prime})\) are no larger than 51. The quantization ranges of Scheme C \(R_{0}:0\sim 252\), \(R_{1}:0\sim 126\), \(R_{2}:0\sim 63\), and \(R_{3}:0\sim 63\) satisfy the requirement.