# High-Quality Video Watermarking Based on Deep Neural Networks for Video with HEVC Compression +
## 3 Proposed Method
### Watermarking Learning Process
A more detailed description of the components of the training process follows from the objective function used:  
\[MSE_{EncCdr}=\text{MSE}\big{(}\text{Image}_{OriginalHrec},\text{Image}_{ EncoldHrec}\big{)} \tag{6}\]  
\[MSE_{EncHrec}=\text{MSE}\big{(}\text{Image}_{OriginalHrec},\text{Image}_{ EncoldHrec}\big{)} \tag{7}\]  
\[MSE_{EncOriginal}=\text{MSE}\big{(}\text{Image}_{Original},\text{Image}_{ Encold}\big{)} \tag{8}\]  
\[MSE_{DecCdr}=\text{MSE}\big{(}\text{Image}_{TplHitLitE},\text{Image}_{ DecoldHrec}\big{)} \tag{9}\]  
\[MSE_{DecCdrIrec}=\text{MSE}(\text{Image}_{TplHitLitE},\text{Image}_{ DecoldHrec}) \tag{10}\]  
\[MSE_{DecOriginal}=\text{MSE}(\text{Image}_{TplHitLitE},\text{Image}_{ Decold}) \tag{11}\]  
\[\text{EpochChange}(i)=i\cdot 0.001 \tag{12}\]  
\[L_{EntLoss}(i)=\big{(}\big{(}(1.01-\text{EpochChange}(i))\cdot MSE_{EncOriginal} \big{)}+((0.50+\text{EpochChange}(i))\cdot MSE_{EncHrec})+(0.50\cdot MSE_{ EncCdr})\big{)}\ i\ e(0;250) \tag{13}\]  
\[L_{EntLoss}(i)=\big{(}\big{(}(1.01-\text{EpochChange}(i))\cdot MSE_{EncOriginal} \big{)}+((0.50+\text{EpochChange}(i))\cdot MSE_{EncHrec})+(0.50\cdot MSE_{ EncCdr})\big{)}\ i\ e(251;500) \tag{14}\]  
\[L_{DecLoss}(i)=\big{(}\big{(}(0.75\cdot\text{EpochChange}(i))\cdot MSE_{DecOriginal} \big{)}+((0.50+\text{EpochChange}(i))\cdot MSE_{DecHex})+(0.50\cdot MSE_{ DecCdr})\big{)}\ i\ e(251;500\big{)} \tag{15}\]  
\[MSE_{EncHrec}=\text{MSE}\big{(}\text{Image}_{OriginalHrec},\text{Image}_{ EncoldHrec}\big{)} \tag{7}\]0  
\[MSE_{EncHrec}=\text{MSE}\big{(}\text{Image}_{OriginalHrec},\text{Image}_{ EncoldHrec}\big{)} \tag{7}\]1  
\[MSE_{EncHrec}=\text{MSE}\big{(}\text{Image}_{OriginalHrec},\text{Image}_{ EncoldHrec}\big{)} \tag{7}\]2  
\[MSE_{EncHrec}=\text{MSE}\big{(}\text{Image}_{OriginalHrec},\text{Image}_{ EncoldHrec}\big{)} \tag{7}\]3  
\[MSE_{EncHrec}=\text{MSE}\big{(}\text{Image}_{OriginalHrec},\text{Image}_{ EncoldHrec}\big{)} \tag{7}\]4  
where \(\text{Image}_{OriginalHrec}\) is the original image after HEVC channel compression, \(\text{Image}_{ Encold}\) is the image with an embedded watermark, \(\text{Image}_{ EncoldHrec}\) is the image with an embedded watermark after HEVC channel compression, \(\text{Image}_{ EncoldHrec}\) is the image with an embedded watermark after compression, \(\text{Image}_{TplHitE}\) is the hidden image constituting a watermark, \(\text{Image}_{TplHitErec}\) is the hidden image constituting a watermark after HEVC channel compression, \(\text{Image}_{ DecoldHrec}\) is the image recovered with a hidden watermark, \(\text{Image}_{ DecoldHrec}\) is the image recovered with the hidden watermark after HEVC channel compression, \(\text{Image}_{ DecoldHrec}\) is the image recovered with the hidden watermark after compression by the order DNN, \(MSE_{ EncOrder}\) is the mean square error of the image with the embedded watermark after compression of the HEVC channel, and the image with the embedded watermark after compression of the HEVC channel, \(MSE_{ EncHrec}\) is the mean square error of the original image, and the image with the embedded watermark after compression of the HEVC channel, \(MSE_{ DecOriginal}\) is the mean square error of the image with the hidden watermark after compression of the HEVC channel, and the image recovered with the hidden watermark after compression by the order, \(MSE_{ DecHeader}\) is the error of the image with a hidden watermark after compression of the HEVC channel, \(MSE_{ DecOriginal}\) is recovered with the hidden watermark after compression of the HEVC channel, \(MSE_{ DecOriginal}\) is the mean squared error of the image with a hidden watermark, and the image recovered with the hidden watermark, \(i\) is the training epoch number, \(\text{EpochChange}(i)\) is the value modifying the error function calculated in dependence of epoch number, \(L_{ EncLoss}(i)\) is the encoding error (embedding the watermark while maintaining transparency), \(L_{ DecLoss}(i)\) is the decoding error (of recovering the watermark), \(L_{ EncFinalLoss}(i)\) is the proportionally summed encoder error including the decoder error, \(L_{ DecFinalLoss}(i)\) is the proportionally summed encoder error taking into account the encoder error, \(Var_{ Enc}(i)\) is the tensor representing the weights of the decoder model, \(\
abla L_{ Enc}(i)\) is the gradient of the encoder error function, \(\
abla L_{ Dec}(i)\) is the gradient of the decoder error function.