## 3 Double Boomerang Watermarking
The embedding is performed in the DCT (discrete cosine transform) domain according to the following:  
\[F_{i}^{(w)}\left[k_{1},k_{2}\right]\!=\!\left\{\!\!\begin{array}{ll}F\!\left[ k_{1},k_{2}\right]\!+\!\beta\!\cdot\!w_{i}^{(s)}\left[k_{1},k_{2}\right]\!,&\!\!\!\!\!\!\!\!\!  
where \(F_{i}[k_{1},\,k_{2}]=DCT\{f_{i}[k_{1},\,k_{2}]\}\) is the DCT transform of the \(i\)-th frame; \(\Phi\) is the region of middle-high frequencies of the image in the DCT domain, while \(\beta\) determines the watermark strength and \(F_{i}^{\langle w\rangle}[k_{1},\,k_{2}]\) is the DCT of the \(i\)-th watermarked frame. By increasing the value of \(\beta\), the mark becomes more evident and a visual degradation of the image (or video) occurs. On the contrary, by diminishing its value, the mark can be easily removed by the coder and/or channel's errors. Therefore, in the application scenario of our simulation trials, the scaling factor \(\beta\) has been chosen in such a way to compromise between the two aforementioned requirements. The \(i\)-th watermarked frame is then obtained by performing the IDCT (inverse DCT) of \(F_{i}^{\langle w\rangle}[k_{1},\,k_{2}]\), finally the whole sequence is MPEG-coded and then transmitted through a noisy channel.  
The receiver implements video decoding as well as watermark detection. In fact, at the same time after decoding of the video-stream, a matched filter extracts the (known) watermark from the DCT of each \(n\)-th received I-frame of the sequence. The estimated watermark is matched to the reference one (despread with the known PN matrix). The matched filter is tuned to the particular embedding procedure, so that it can be matched to the randomly spread watermark only. It is assumed that the receiver knows the initial spatial application point of the mark in the DCT domain. The QoS is evaluated by comparing the extracted watermark with respect to the original one. In particular, its mean-square-error (MSE) is evaluated as an index of the effective degradation of the provided QoS. A possible index of the degradation is simply obtained by calculating the mean of the error energy:  
\[MSE=\frac{1}{M}\sum_{n=1}^{M}\biggl{[}\sum_{k_{1}=1}^{K_{1}}\sum_{k_{2}=1}^{K _{2}}(w_{n}[k_{1},k_{2}]-\widehat{w}_{n}[k_{1},k_{2}])^{2}\,\biggr{]} \tag{2}\]  
where \(\mathrm{w}_{n}[k_{1},k_{2}]\) and \(\,\hat{\mathrm{w}}_{n}[k_{1},k_{2}]\), represent the original and the extract watermark respectively, and \(n=1\),..., \(M\) is the current frame index. It is worth noting that the metric (2), which is evaluated using the estimated watermarks over the \(M\) transmitted frames, is employed to provide a quality assessment of the received video after the MPEG coding/transmission process. In particular, such a QoS index can be usefully employed for a number of different purposes in mobile multimedia communications such as: control feedback to the sending user on the effective quality of the link; detailed information to the operator for billing purposes and diagnostic information to the operator about the communication link status. Although sophisticated video quality perceptual metrics could be used for quality assessment purposes in multimedia communications (see, for instance, [21]), the MSE between the estimated watermark and the original one is used in this contribution as a proof of concept of the objective cooperative method proposed here, and shown by extensive simulation trials in Section 4.