# An Enhanced Adaptive Block Truncation Coding with Edge Quantization Scheme +
## 3 The Proposed ABTC-EQ  
### Design Concept  
The advantage of ABTC-EQ [4] is to improve the quality of the reconstructed image because it can represent edge and non-edge blocks. However, it is impossible that BTC and its variants cannot represent edge block, because they only use two quantization levels, \((\mu_{0},\mu_{1},M)\), to represent a block. However, ABTC-EQ enhances the visual quality of the reconstructed image as well as reduces the CR due to using an extra flag bit \(f\) and extra quantization value \(\mu^{\prime}_{2}\). To enhance BTC-like approaches, we obviously should improve the CR as well as with a high PSNR. As a result of studying the ABTC-EQ, it was found that the PSNR improvement of the reconstructed image was due to the tri-clustering approach to the edge block. The weakness of this approach is that the CR decreases due to the increase of the additional bits to represent the edge block. Some in-depth observations on ABTC-EQ are listed below.  
**Observation 1**.: _It is not necessary to use two bits to represent a pixel (i.e., '00', '01', and '10') in bit map \(M_{e}\) for edge block._  
As shown in Equation (5), we use (00), (01), and (10) for three clusters \(c_{0}\), \(c_{1}\), and \(c_{2}\), respectively. However, we may use Huffman code, a variable-length code, to represent three clusters, by (0), (10), and (11) with average length 5/3 bits for clusters \(c_{0}\), \(c_{1}\), and \(c_{2}\). Note: the Huffman code can be uniquely decoded. By this approach, the size of bit map \(M_{e}\) is reduced from \(2k^{2}\) to \((5/3)/k^{2}\). Finally, the CR can be enhanced.  
**Observation 2**.: _Consider two quantization values (say \(\mu^{\prime}_{i}\) and \(\mu^{\prime}_{j}\), where \(\mu^{\prime}_{i}<\mu^{\prime}_{j}\)). The difference \((\mu^{\prime}_{j}-\mu^{\prime}_{i})\) between two quantization values is often smaller than the quantization values \(\mu^{\prime}_{i}\) and \(\mu^{\prime}_{j}\) themselves._  
By observing quantization value, we herein use a homologous way to describe the difference between two quantization values with the help of the coder in converting voice. A well-known coder, differential pulse code modulation (DPCM), is described as follows: obtain the pulse of analog signals by sampling and then convert the difference of pulses into binary sequences using the non-uniform coding scale. This property is also true for the quantization levels in ABTC-EQ, i.e., the large difference of quantization values does not occur frequently. Thus, we could carefully design our quantization ranges for the small difference between two quantization values.  
**Observation 3**.: _More clusters may enhance the PSNR of the reconstructed images._  
In previous BTC-like schemes, all blocks adopt bi-clustering, i.e., using two quantization ranges for each block. [PERSON] and [PERSON]'s ABTC-EQ performed a tri-clustering approach on edge blocks. Because there are three values \(\mu_{0}\), \(\mu_{1}\), and \(\mu_{2}\) to approximate the pixel grayscale values, it can reduce the mean square error. We may use more clusters (say four clusters) to more precisely approximate pixel values.