# A Brief, In-Depth Survey of Deep Learning-Based Image Watermarking
## 2 Preliminaries
### Typical Metrics and Factors  
Although there can be a large number of evaluation metrics and factors considered in image watermarking based on different applications, certain metrics are ubiquitously employed across both traditional and deep learning-based watermarking in the extant literature. In this context, we briefly discuss the typical factors of imperceptibility and robustness, which are integral to assessing the efficacy of watermarking techniques. **Imperceptibility. ** The ability of the watermark to be embedded into the image data in a way that is invisible to human vision is referred to as imperceptibility. The imperceptibility helps ensure that the watermark does not interfere with the quality of the image. One most frequently applied evaluation metric is the peak signal-to-noise ratio (\(PSNR\)):  
\[PSNR=10\times log_{10}(\frac{max(c)^{2}}{\frac{1}{RC}\sum_{i=1}^{R}\sum_{j=1}^ {C}(c_{ij}-m_{ij})^{2}}), \tag{1}\]  
where \(max(c)\) is the largest possible pixel value for the cover image \(c\), which is 255 if we use 8 bits for each grayscale value, and \(\mathcal{R}\) and \(\mathcal{C}\) denote the height and width of images \(c\) and \(m\). Notably, apart from the extensively utilized \(PSNR\), the structural similarity index measure (\(SSIM\)) [24] is also commonly employed to assess imperceptibility, incorporating evaluations of luminance, contrast, and structural disparities. An essential augmentation to visual imperceptibility is security [25]. This entails ensuring that the embedded watermark is not only invisible to the human eye but also resistant to detection through computational analysis, a criterion of paramount importance in secure watermarking applications, exemplified in domains like smart city planning and digital forensics. **Robustness. ** Robustness characterizes the watermark's resilience, denoting its capacity to be reliably extracted amidst attacks on the watermarked image, such as compression, filtering, or cropping. The assessment of robustness involves calculating the disparity between the extracted and original watermark post-attack. When the watermark is binary, the bit error rate (BER) serves as a prevalent metric, computed by dividing the number of erroneous bits by the total number of bits embedded. In instances where the watermark takes the form of a two-dimensional matrix, its resilience is often assessed via the normalized cross-correlation (\(NC\)), measuring the similarity between the original watermark \(w\) and the extracted watermark \(w^{\prime}\):  
\[NC=\frac{\sum_{i=1}^{H}\sum_{j=1}^{C}(w_{ij}\cdot w_{ij}^{\prime})}{\sqrt{ \sum_{i=1}^{H}\sum_{j=1}^{C}(w_{ij})^{2}}\sqrt{\sum_{i=1}^{H}\sum_{j=1}^{C}( w_{ij}^{\prime})^{2}}}, \tag{2}\]where \(\mathcal{H}\) and \(\mathcal{L}\) define the height and width of the watermark. **Blindness. ** Blind and semi-blind watermarking represent two prominent approaches in the domain of image watermarking. Blind watermarking is characterized by its ability to detect and extract watermarks without the necessity of referring to the original cover image nor the original watermark (while a key may be required). On the other hand, semi-blind watermarking, while also not requiring the original cover image for watermark extraction, requires the original watermark (and the key) for extraction. **Capacity. ** Capacity denotes the upper limit of watermark data that can be embedded into an image, giving rise to two primary types of image watermarking: (1) Zero-bit watermarking, which focuses on detecting the presence of a watermark in an image rather than extracting data, and (2) multi-bit watermarking, which involves embedding and extracting a watermark comprised of multiple bits. Zero-bit watermarking serves as a signature for authenticating image data, without the provision to incorporate additional embedded data. This form of watermarking is effective for verification purposes, ensuring the integrity and authenticity of the image content.