## 1 Introduction  
Modern widespread use of electronic devices and the popularity of social media allows people to easily obtain and share digital images in the manner they prefer. However, at the same time, the improvements in digital image processing software allows anyone to freely alter digital images. After an attacker has modified an image, the authenticity and integrity of the information in that image may be compromised. When an image is maliciously falsified, the top priority in the image processing field is to be able to restore the original image to the greatest extent. To solve this problem, digital image watermarking technology was invented. Digital watermarking capitalizes on the redundancy in digital images to embed hidden information, namely, a watermark, into the image. There are three kinds of watermarks: fragile watermark, semi-fragile watermark and robustness watermark. Among them, fragile watermark is often used in image tampering detection due to its sensitivity [1]. Thus far, scholars have carried out numerous studies in this field and achieved fruitful results, some of which are listed below. [PERSON] and [PERSON]. [1] proposed a fragile watermarking scheme in which the watermark is generated from the five most significant bits (MSBs) of each pixel and embedded into the three least significant bits (LSBs) corresponding to the mapped block. [PERSON] et al. [2] introduced a self-recovering fragile watermark scheme that embeds 10-bit of recovery data and 2-bit of authentication data into the LSBs of the corresponding mapped block. This method has a high capability for tampering detection and can recover an image effectively. However, it is not suitable for images containing random noise or that have undergone JPEG compression. [PERSON] and [PERSON] [3] proposed a multiple watermarking method that had an improved peak signal-to-noise ratio (PSNR) but that value was achieved at the cost of higher computational complexity. [PERSON] and [PERSON] [4] suggested an image authentication algorithm with recovery capabilities based on neural networks. The methods in References [1; 2; 3; 4] are all based on discrete cosine transforms (DCT). [PERSON] et al. [5] introduced a watermark scheme for authentication and self-recovery. Inverse halftoning techniques and median filtering were also used to improve the quality of the recovered image. However, to prevent image tampering attacks, both [4] and [5] take only JPEG compression into account. [PERSON] [6] proposed a semi-fragile watermarking technique based on a modified discrete wavelet transform quantization-based algorithm. The proposed method is suitable for grayscale image authentication and for tamper detection but was not tested for its ability to handle geometric attacks. [PERSON] et al. [7] developed a three-level strategy to improve tamper detection accuracy and introduced a new block classification scheme based on singular value decomposition (SVD). The length of the generated recovery watermark differs within different blocks. Compared to References [8; 9], this method achieved substantial improvements in tamper detection accuracy and recovery ability. Using self-recovery blocks, [PERSON] and [PERSON] [10] proposed a modified fragile watermarking scheme in which the detected tampered blocks were used to localize the erroneous regions of the tampered image, while the error-free blocks were used to recover the tampered blocks by applying block chaining. However, room still exists to improve image recovery. [PERSON] et al. [11] presented an SVD-based algorithm that used active watermarking that achieved both satisfactory self-recovery capability and was efficient at tamper detection. However, further research needs to be carried out for situations in which the recovery information is damaged by using the block-neighboring characteristic. Based on SVD, [PERSON] et al. [12] introduced a pixel-based fragile watermarking algorithm for image authentication.