## 2 Related Works
### Watermarking Algorithm in the Transform Domain  
The transform domain watermarking technique can help improve the imperceptibility and robustness of watermarking, and many watermarking schemes, such as DWT [14; 15], DFT [16], DCT [17; 18], quaternion discrete Fourier transform (QDFT) [19; 20; 21] and quaternion Hadamard transform (QHD) [22], have been proposed. Discrete cosine transform (DCT) is a widely used transform domain watermarking technique that modifies the image from the spatial domain to the frequency domain. DCT divides the image into different frequency bands, namely, high-, medium- and low-frequency bands, to select frequency bands easily when embedding the watermark. The middle-frequency band is mainly selected in this work because it can balance the relationship between robustness and imperceptibility and avoid diffusion to high-frequency components against compression and noise attacks to improve image robustness [2]. Watermarking algorithms for different carriers can be divided into two categories in the frequency domain, namely, single- (SCI) and multi-channel images (MCI).  
Watermarking-algorithm-based SCI. [PERSON] et al. [23] designed a robust digital image watermarking algorithm that can resist image attacks and distortions and embed the watermark into the two-level DCT coefficients of the image using certain rules. A robust watermarking scheme based on the wavelet transform domain was proposed to resist attacks for single-channel images effectively [24]. A novel image watermarking scheme based on singular value decomposition (SVD) was proposed [25] to overcome the FPP problem and meet the watermarking requirements using integer wavelet transform (IWT). Multi-objective ant colony optimization (MOACO) is utilized to balance robustness and imperceptibility and find the optimal scaling factor while completely avoiding the FPP problem. [PERSON] et al. [26] embedded a gray-scale watermark into single-channel images by blending singular value decomposition and the steerable pyramid wavelet transform domain. A robust watermarking method with SVD and DWT [27] was put forward to embed the watermark into the singular values of DWT subbands of the gray image.  
Watermarking algorithm-based MCI. Watermarking schemes for color image have continuously emerged because color images are widely used in network transmission and storage [15; 19; 21; 28; 29; 30; 31; 32; 33; 34; 35]. [PERSON] and [PERSON] [2] established a new color-image watermarking algorithm based on wavelet transform and significant difference, as well as embedded the maximum watermark information under imperceptible distortion. A link was established between the QDFT coefficient, and the color channel's DFT and QDFT coefficients were modulated to embed the watermark in the transform domain [19]. [PERSON] et al. [21] embedded a watermark into the cover image by adaptively modulating the real coefficients of the quaternion Fourier transform while using LS-SVM to correct the attacked images. [PERSON] et al. [28] developed a local QDFT watermarking method for color images that utilize the advantage of QDFT to improve the robustness and imperceptibility of watermarking. [PERSON] et al. [29] developed an image hashing algorithm dependent on the analysis of local geometric changes of multi-channel images via gradient measurement for image tempering detection. A geometrically corrected robust color image watermarking method based on the quaternion exponential moments was proposed [30]. [PERSON] et al. [33] combined Tucker decomposition with A-LQIM for a robust image watermarking approach with satisfactory performance. [PERSON] et al. [35] proposed a watermarking method based on the Neyman-Pearson rule for multi-channel images to reduce the error rate when the watermark is nonexistent and to conduct blind image watermarking for gray-scale images in the DCT transform domain.