# An Enhanced Adaptive Block Truncation Coding with Edge Quantization Scheme +
## 3 The Proposed ABTC-EQ
### The Proposed Schemes  
We aim to achieve the high CR and the reasonable PSNR of the reconstructed image. For the purpose, we proposed three schemes: (1) Scheme A motivated from Observation 1, (2) Scheme B is based on Observations 1 and 2, and (3) Scheme C is based on Observations 2 and 3. Compared with [PERSON]'s ABTC-EQ [4], Scheme A has the same PSNR, while enhances the CR. However, Scheme C has the same CR, while enhances the PSNR. On the other hand, Scheme B further enhances the CR than Scheme A but still retain a reasonable PSNR compared with AMTC [2] and MBTC [3]. **(1) Scheme A**: The algorithm is the same as that of ABTC-EQ. If a block belongs to the edge block as described in Observation 1, the bitmap (\(M^{\prime}_{e}\)) is composed of {'0', '01', '10'} (see Equation (7)). In this case, we show the compression performance of the bitmap when a variable-length coding is applied to the bitmap. The proof is demonstrated by the Equations (8) and (9). Finally, we use \((f=0,\mu^{\prime}_{0},\mu^{\prime}_{1},\mu^{\prime}_{2},M^{\prime}_{e})\) for an edge block. \[b_{i}= \begin{cases}0,&\text{if }x_{i}\in c_{0},\\ 01,&\text{if }x_{i}\in c_{1},\\ 10,&\text{if }x_{i}\in c_{2}.\end{cases}  \tag{7}\]  
**Theorem 1**. : _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. The CR of ABTC-EQ is \(\text{CR}_{\text{MN}}\)\(=\frac{8k^{2}}{(17+k^{2})+(8+k^{2})\times p_{e}}\), and Scheme A has the \(\text{CR}_{A}=\frac{8k^{2}}{(17+k^{2})+(8+(2k^{2}/3))\times p_{e}}\), where \(\text{CR}_{A}>\text{CR}_{\text{MN}}\). Meanwhile, they have the same PSNR of reconstructed image, i.e., \(\text{PSNR}_{\text{MN}}=\text{PSNR}_{A}\)._  
**Proof. ** By the compression data formats \((f=1,\mu_{0},\mu_{1},M_{n})\) and \((f=0,\mu_{0}^{\prime},\mu_{1}^{\prime},\mu_{2}^{\prime},M_{e})\) of ABTC-EQ, and the formats of Scheme A \((f=1,\mu_{0},\mu_{1},M_{n})\) and \((f=0,\mu_{0}^{\prime},\mu_{1}^{\prime},\mu_{2}^{\prime},M_{e}^{\prime})\), we may easily derive \(\text{CR}_{\text{MN}}\) and \(\text{CR}_{\text{A}}\) in Equations (8) and (9), respectively. \[\text{CR}_{\text{MN}}=\left\{ \begin{array}{c}\frac{\frac{8 \times k^{2}}{(1+2\times 8+k^{2})\times p_{n}}+(1+3\times 8+2\times k^{2}) \times p_{e}}{\text{non-edge block}}\\ =\frac{8k^{2}}{(17+k^{2})+(8+k^{2})\times p_{e}}\end{array} \right. \tag{8}\]  
\[\text{CR}_{\text{A}}=\left\{ \begin{array}{c}\frac{\frac{8 \times k^{2}}{(1+2\times 8+k^{2})\times p_{n}}+(1+3\times 8+(5/3)\times k^{2}) \times p_{e}}}{\text{non-edge block}}\\ =\frac{8k^{2}}{(17+k^{2})+(8+(2k^{2}/3))\times p_{e}}\end{array} \right. \tag{9}\]  
By Equations (8) and (9), since \((8+(2k^{2}/3))<(8+k^{2}))\) we obviously have \(\text{CR}_{\text{A}}>\text{CR}_{\text{MN}}\). Except using different bit map \(M_{e}^{\prime}\) from \(M_{e}\), Scheme A uses the same approaches of ABTC-EQ. Thus, both schemes have the same PSNR, i.e., \(\text{PSNR}_{\text{MN}}=\text{PSNR}_{\text{A}}\). Notes: In Equations (8) and (9), \((1+2\times 8+k^{2})\) represents data format of the edge block, i.e., \"1\" is a flag bit, \"2 \(\times\) 8\" is 2 pixels \(\times\) 8 bits (two quantization levels), and \"\(k^{2}\)\" is \(k\times k\ \times 1\) bits (bitmap) in non-edge block. In each block, \((1+3\times 8+2\times k^{2})\) denotes the format of the non-edge block, i.e., \"3 \(\times\) 8\" is 3 pixels \(\times\) 8 bits and \"\(2\times k^{2}\)\" is \(k\times k\times 2\) bits in edge block. The 5/3 of Equation (9) indicates that the number of cluster is 3 and the total bit length is 5 when a block belongs to an edge block. By this approach, the size of bit map \(M_{e}\) is reduced from \(2k^{2}\) to \((5/3)/k^{2}\). Finally, we proved that the CR performance is enhanced by Scheme A. **(2) Scheme B**: Basically, this scheme is based on the compressed bitmap \(M_{e}^{\prime}=[b_{i}]\) derived from Observation 1.