# An Enhanced Adaptive Block Truncation Coding with Edge Quantization Scheme +
## 3 The Proposed ABTC-EQ
### The Proposed Schemes
Moreover, as explained in Observation 2, a new compressed quantization levels \((\delta_{0},\delta_{1},\delta_{2})\) is exploited. This is derived from the idea that compression performance can be improved by exploiting the difference between the two quantization levels. For this, we define a way of classifying the range of quantization levels into four categories. That is, for the tri-cluster \(\delta_{i}\), \(0\leq i\leq 2\), we use \(n_{i}\) bits with a radix \(R_{i}:(r_{n_{i}},r_{n_{i}-1},\ldots,r_{1})\) to represent its quantization levels. \[\left\{ \begin{array}{l}(I)(n_{0}=7,n_{1}=7,n_{2}=7)\text{ with }\left\{\begin{array}{l}R_{0}:(r_{7},r_{6},\ldots,r_{1})=(128 \;64\;32\;16\;8\;4\;2),\\ R_{1}:(r_{7},r_{6},\ldots,r_{1})=(64\;32\;16\;8\;4\;2\;1),\\ R_{2}:(r_{7},r_{6},\ldots,r_{1})=(64\;32\;16\;8\;4\;2\;1).\end{array}\right.\\ \\ (II)(n_{0}=6,n_{1}=6,n_{2}=6)\text{ with }\left\{\begin{array}{l}R_{0}:(r_{6},r_{5}, \ldots,r_{1})=(128\;64\;32\;16\;8\;4),\\ R_{1}:(r_{6},r_{5},\ldots,r_{1})=(64\;32\;16\;8\;4\;2),\\ R_{2}:(r_{6},r_{5},\ldots,r_{1})=(64\;32\;16\;8\;4\;2).\end{array}\right.\\ \\ (III)(n_{0}=5,n_{1}=5,n_{2}=5)\text{ with }\left\{\begin{array}{l}R_{0}:(r_{5},r_{4}, \ldots,r_{1})=(128\;64\;32\;16\;8),\\ R_{1}:(r_{5},r_{4},\ldots,r_{1})=(64\;32\;16\;8\;4),\\ R_{2}:(r_{5},r_{4},\ldots,r_{1})=(64\;32\;16\;8\;4).\end{array}\right.\\ \\ (IV)(n_{0}=4,n_{1}=4,n_{2}=4)\text{ with }\left\{\begin{array}{l}R_{0}:(r_{4},r_{3}, \ldots,r_{1})=(128\;64\;32\;16),\\ R_{1}:(r_{4},r_{3},\ldots,r_{1})=(64\;32\;16\;8),\\ R_{2}:(r_{4},r_{3},\ldots,r_{1})=(64\;32\;16\;8).\end{array}\right.\end{array} \right. \tag{10}\]  
The new quantization levels due to the tri-cluster, \(\delta_{i}\), \(0\leq i\leq 2\), is then iteratively determined based on radix \(R_{i}\) with the equation \(min\{\mu_{i}^{\prime}-\sum_{j=0}^{i}\delta_{j}\}\). Finally, we obtain new quantization levels \((f=0,\delta_{0},\delta_{1},\delta_{2},M_{e}^{\prime})\) for an edge block. This process can greatly reduce the bits of the quantization level. **Theorem 2**. : _Suppose that the percentages of edge blocks and non-edge blocks in an image be \(p_{e}\) and \(p_{n}\), where \(p_{e}+p_{n}=1\), when dealing with \((k\times k)\)-pixel block in an image. Scheme B has the \(CR_{B}\)\(=\frac{8k^{2}}{(17+k^{2})+(\sum_{i=1}^{3}n_{i}(2k^{2}/3))\times p_{e}}\), and \(CR_{B\text{-IV}}>CR_{B\text{-III}}>CR_{B\text{-II}}>CR_{A}\)._  
Proof. : By the compression data formats of Scheme B \((f=1,\mu_{0},\mu_{1},M_{n})\) and \((f=0,\delta_{0},\delta_{1},\delta_{2},M_{e}^{\prime})\), we may derive in Equation (11). \[\left\{ \begin{aligned} &\text{CR}_{\text{B}}=\underbrace{\frac{8k^{2}}{(1 +2\times 8+k^{2})\times p_{n}}+\underbrace{(1+\sum_{i=1}^{3}n_{i}+(2/3)\times k^{2}) \times p_{e}}}_{\text{edge block}}\\ &\hskip 142.26378 pt=\frac{8k^{2}}{(17+k^{2})+(\sum_{i=1}^{3}n_{i} -16+(2k^{2}/3))\times p_{e}}\end{aligned} \right. \tag{11}\]  
Via Equation (11) and four quantization ranges in Equation (10), we have compression ratios for these four quantization ranges \(\text{CR}_{\text{B-I}}=\frac{8k^{2}}{(17+k^{2})+(5+(2k^{2}/3))\times p_{e}}\), \(\text{CR}_{\text{B-II}}=\frac{8k^{2}}{(17+k^{2})+(2+(2k^{2}/3))\times p_{e}}\), \(\text{CR}_{\text{B-III}}=\frac{8k^{2}}{(17+k^{2})+((2k^{2}/3))\times p_{e}}\), and \(\text{CR}_{\text{B-IV}}=\frac{8k^{2}}{(17+k^{2})+((2k^{2}/3)-4)\times p_{e}}\). From the above and Equation (9), we have \(\text{CR}_{\text{B-IV}}>\text{CR}_{\text{B-III}}>\text{CR}_{\text{B-II}}>\text{ CR}_{\text{B-I}}>\text{CR}_{\text{A}}\). All compression ratios of Scheme B are larger than \(\text{CR}_{\text{MN}}\) (ABTC-EQ). By Observation 3, we may obtain the approximated \((\mu_{0}^{\prime},\mu_{1}^{\prime},\mu_{2}^{\prime})\) with a tolerant distortion from \((\delta_{0},\delta_{1},\delta_{2})\). Moreover, Scheme B may have the higher PSNR than those of AMBTC and MBTC. **(3) Scheme C**: As the number of clusters for a block increases, the PSNR of an image increases proportionally like the case of Observation 3.