# Hybrid Encoding Scheme for AMBTC Compressed Images Using Ternary Representation Technique
## 1 Introduction  
With the rapid development of imaging technology, digital images are perhaps the most widely used media of the Internet. Because digital images themselves contain significant amounts of spatial redundancy, an efficient lossy image compression technique is required for lower storage requirement and faster transmission. The Joint Photographic Experts Group (JPEG) [1, 2], vector quantization (VQ) [3, 4], and block truncation coding (BTC) [5, 6] are well-known lossy compression methods and have been extensively investigated in the literature. Among these techniques, BTC requires significantly less computation cost than others while offering acceptable image quality. BTC has been widely investigated in the disciplines of remote sensing and portable devices, in which computational costs are limited. BTC was firstly proposed by [PERSON] and [PERSON] [7]. This method partitions image into blocks, and each block is represented by two quantized values and a bitmap. Inspired by [7], [PERSON] and [PERSON] [8] propose a variant method called absolute moment block truncation coding (AMBTC), which offers a simpler computation than that of BTC. The applications of AMBTC are studied in video compression [9], image authentication [10, 11, 12], and image steganography [13, 14]. Moreover, some recoverable authentication methods adopt AMBTC codes as the recovery information to recover the tampered regions. Because the recovery codes have to be embedded into the host image, a more efficient coding of AMBTC is always desirable because the burden of the embedment can be reduced and the quality of the recovered regions can be enhanced. To improve the compression efficiency of the AMBTC method, several approaches, including bitmap omission [15], blockclassification [16; 17], and quantized value adjustment [18], are adopted to lower the bitrate while maintaining the image quality. For example, [PERSON] [15] recognizes that if the difference between two quantized values is smaller than a predefined threshold, the bitmap plays an insignificant role in reconstructed image quality. Therefore, [PERSON] employs the bitmap omission approach by neglecting the recording of a bitmap if a block is considered to be flat, and only uses block means to represent the flat block. [PERSON] et al. [17] adopt quadtree partitioning and propose a variable-rate AMBTC compression method for color images. The basic idea of [17] is to partition the image into blocks with various sizes according to their complexities. The AMBTC and bitmap omission technique are then employed to encode the image blocks. In some applications, such as data hiding or image authentication, bitmaps have to be altered to carry some required information, causing a degradation in image quality. [PERSON] [18] optimizes the quantized values so that the impact of bitmap alteration can be reduced. [PERSON] and [PERSON] [19] propose an adaptive AMBTC method based on edge quantization by considering human visual characteristics. This method separates image blocks into edge and non-edge blocks, and quantized values are calculated based on the edge information. Because the edge characteristics are considered, their method provides better image quality than other AMBTC variants. [PERSON] et al. [16] in 2019 proposed a dynamic multi-grouping scheme for AMBTC focusing on improving the reconstructed image quality and reducing the bitrate. Their method partitions an image into non-overlapping blocks. According to the block complexity, varied grouping techniques are designed. An indicator is employed to distinguish the grouping types. In addition, instead of recording the quantized values, the differences between them are recorded so as to reduce the bitrate. [PERSON] et al. 's method provides better compression performance than those of prior works. In [PERSON] et al. 's method, the number of pixel groups of an image block directly affects the reconstructed image quality and bitrate.