# Review of Image Forensic Techniques Based on Deep Learning
## 2 Image Forensic Techniques
### Active Forensics
#### 2.2.2 Performance Evaluation Metrics  
In a digital image watermarking algorithm, the most important three evaluation indicators are robustness, imperceptibility, and capacity.  
**Robustness:** Robustness is used to measure the ability of a watermark model to recover the original watermark after an image has been subjected to a series of intentional or unintentional image processing during electronic or non-electronic channel transmission. Bit error rate (BER) and normalized cross-correlation (NCC) are usually used as the objective evaluation metrics, which are expressed as Equations (7) and (8), respectively:  
\[E_{\text{BER}}(w,w^{{}^{\prime}})=\frac{1}{L}\sum\limits_{i=1}^{L}|w_{i}-w_{i} ^{{}^{\prime}}| \tag{7}\]  
\[E_{\text{NCC}}=\frac{\sum\limits_{i=1}^{L}\left(w_{i}-\bar{w}\right)(w_{i}{}^{ \prime}-\bar{w}^{\prime})}{\sqrt{\sum\limits_{i=1}^{L}\left(w_{i}-\bar{w} \right)^{2}\sum\limits_{i=1}^{L}\left(w_{i}{}^{\prime}-\bar{w}^{\prime}\right) ^{2}}} \tag{8}\]  
where \(w_{i}\) and \(\bar{w}\) represent the \(i\)th bit of the original watermark and the mean value of the original watermark, respectively; \(w_{i}{}^{\prime}\) and \(\bar{w}^{\prime}\) represent the \(i\) th bit of the extracted  
Figure 3: A basic framework of end-to-end robust watermarking based on deep learning.  
watermark and the mean value of the extracted watermark; \(L\) represents the length of the watermark.  
**Imperceptibility:** Imperceptibility is used to measure the sensory impact of the embedding point on the whole image after the model has completed watermark embedding (i.e., the watermarked image is guaranteed to be indistinguishable from the original image). Peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) are usually used as the objective evaluation metrics, which are expressed as Equations (9) and (10), respectively:  
\[E_{\text{PSNR}}(I,I_{\text{W}})=10\text{log}_{10}\frac{W\times H\times G_{\text {MAX}}^{2}}{\sum\limits_{i=0}^{W-1}\sum\limits_{j=0}^{H-1}\left[I(i,j)-I_{\text {W}}(i,j)\right]^{2}} \tag{9}\]  
\[E_{\text{SSIM}}(I,I_{\text{W}})=\frac{(2\mu_{I}\mu_{I_{\text{W}}}+C_{1})(2\sigma _{I,I_{\text{W}}}+C_{1})}{(\mu_{I}^{2}+\mu_{I_{\text{W}}}^{2}+C_{1})(\sigma_{I} ^{2}+\sigma_{I_{\text{W}}}^{2}+C_{2})} \tag{10}\]  
where \(W\) denotes the width of the original image; \(H\) denotes the length of the original image; \(G_{\text{MAX}}\) denotes the maximum gray level of the original image; \(\mu_{I}\) and \(\mu_{I_{\text{W}}}\) denote the mean value of the gray value of the original image and the watermarked image, respectively; \(\sigma_{I}^{2}\) and \(\sigma_{I_{\text{W}}}^{2}\) denote the variance of the gray value of the original image and the watermarked image, respectively; \(l(i,j)\) and \(I_{\text{W}}(i,j)\) denote the \((i,j)\) original image and watermarked image; \(\sigma_{I,I_{\text{W}}}\) denotes the covariance of the original image and the watermark image; \(C_{1}\) and \(C_{2}\) are constant in range \([10^{-4},\,9\times 10^{-4}]\).  
**Capacity:** The capacity represents the maximum watermark embedding bits of the model while maintaining the established required imperceptibility and robustness metrics. It is mutually constrained with imperceptibility and robustness. Increasing the watermark embedding capacity, imperceptibility and robustness decrease, and vice versa. The number of embedded watermark bits per pixel (bpp) is usually used to measure the capacity metric of the model, which is expressed as Equation (11):  
\[E_{\text{bpp}}=\frac{W_{\text{num}}}{I_{\text{num}}} \tag{11}\]  
where \(W_{\text{num}}\) denotes the number of watermark bits, and \(I_{\text{num}}\) denotes the total number of original image pixels.