## 1 Introduction
It is effective because the detection is based on a hierarchical structure so that the accuracy of tamper localization can be ensured. The drawback is that it is not possible to check whether the location of the error is an error of a lower block of the current block or an error of a lower block within the same block. Therefore, the scheme [21] proposed a new mechanism to facilitate recovery with a higher probability by inserting a double copy of the watermark into two different blocks. [PERSON] and [PERSON] (2008) [22] proposed a new vector coding-based fragile watermarking method that can recover the tampered area without error, as long as it is not too serious. For restoration, recovery and authentication, bits are compressed losslessly and then are embedded in the cover image. The mean PSNR is about 28.70 dB. Moreover, if the tampering rate is less than 3.2%, the tampered area may be totally recovered. To improve the tampering rate, [PERSON] et al. (2009) [23] proposed another fragile watermarking scheme that restores the content of the original image. While the reference bits are embedded into the entire cover image, the hash bits are embedded into the local blocks. [PERSON] et al. [24] proposed a fragile watermarking method for high-quality restoration. They first categorized the image into one of six types depending on the degree of smoothness. Complex blocks were compressed into more bits for recovery and smooth blocks were needed fewer bits for recovery. Finally, the recovery bit and authentication bit were embedded into the three number of LSBs of every pixel for the image. [PERSON] et al. [25] proposed a self-contained watermarking scheme for digital images. The host image was converted into a halftone image using a digital halftone technique, and the converted pixels were used as recovery bits. Halftone can preserve the characteristics of the host image in the most compressed type. The halftoning watermark was used for tamper detection, and the tampered area can be approximately recovered using the extracted watermark. They adopted a simple low-pass filtering approach for inverse halftoning. The reconstructed image based on halftone is not satisfactory from the perspective of the image quality. [PERSON] and [PERSON]'s study [26] used the degree of smoothness to distinguish the types of image blocks, and they employed different watermark embedding, tamper detection and recovery strategies for different block types to enhance hiding efficiency, authentication and recovery effects. [PERSON] & [PERSON] [27] proposed a method to detect and recover images tampered with by integrating [PERSON]'s watermarking method [28] and vector quantization (VQ). This integration also required a little extra cost, i.e., an increase in codebook size. However, with a codebook, the image is recovered by using VQ if the mapping information for recovery is lost. Due to the limitation of the quality of VQ, the restored image is not of high quality. In [29; 30; 31], they proposed a fragile watermarking technique based on Block Truncation Coding (BTC). In this method, the bitstream compressed with BTC [32] or AMBTC [33] of the original image was hidden in the LSB and 2 LSB of the cover image to store the features of the original image. When a part of the image has been tampered, the location is detected, and the information on the tampered area is recovered with AMBTC. [PERSON] et al. [29] adopted a method of improving the image quality through Gaussian filtering after using the reconstructed bits for image restoration. [PERSON] et al. [30] used a quantum chaos map to escape the tampering attack of the mark. The error rate of tamper detection using the XOR operation between the bitmap of each block and the binary random number may higher than that of using the decimal number. [PERSON] et al. [31] proposed a method to improve the quality of marked images by enhancing the compression performance of AMBTC encoding bits for the original image.