# Screen-Cam Robust Image Watermarking with Feature-Based Synchronization
## 3 Proposed Watermarking Scheme
### Local Square Feature Region Construction
#### 3.1.2 Modified Harris-Laplace Detector  
The Harris-Laplace detector proposed by [PERSON] and [PERSON] [48; 49] has been extensively employed in different watermarking schemes. Therefore, here we give a brief description of it, and the modified part will be explained in detail.  
First, Harris points are detected in the scale space. To obtain the invariance to scale variation, we built a scale-space representation with the Harris function for preselected scales. The Harris detector is based on a specific image descriptor, which is referred to as the second moment matrix, and reflects the local distribution of gradient directions in the image [50]. To make the matrix independent of the image resolution, the scale-adapted second moment matrix is defined by:  
\[M(x,y,\sigma_{I},\sigma_{D})=\sigma_{D}^{2}G(\sigma_{I})\star\left[\begin{array} []{cc}L_{x}^{2}(x,y,\sigma_{D})&L_{x}L_{y}(x,y,\sigma_{D})\\ L_{x}L_{y}(x,y,\sigma_{D})&L_{y}^{2}(x,y,\sigma_{D})\end{array}\right] \tag{3}\]  
where \(\sigma_{I}\) and \(\sigma_{D}\) are the integration scale and local scale, respectively, and \(L\) is the derivative computed in an associated direction by a Gaussian. Given \(\sigma_{D}\), the uniform Gaussian multiscale space representation \(L\) is defined by:  
\[L(x,y,\sigma_{D})=G(x,y,\sigma_{D})\star I^{\prime} \tag{4}\]  
where \(G\) is the associated uniform Gaussian kernel with a standard deviation \(\sigma_{D}\) and a mean of zero.  
Given \(\sigma_{I}\) and \(\sigma_{D}\), the scale-adapted Harris corner strength _cornerness_ used to quantitatively describe the stability under variations in imaging conditions can be computed. The original _cornerness_ measure function needs an empirical parameter, which may float for different images. Therefore, in this paper, we will adopt another _cornerness_ measure function, that is, the Alison Noble measure [51]:  
\[\text{\emph{cornerness}}(x,y,\sigma_{I},\sigma_{D})=\det(M(x,y,\sigma_{I}, \sigma_{D}))/(\text{trace}\left(M(x,y,\sigma_{I},\sigma_{D})\right)+\text{ \emph{eps}}) \tag{5}\]  
where \(\det(\cdot)\) and \(\text{trace}(\cdot)\) denote computation of the determinant of the matrix and the trace of the matrix, respectively. _eps_ is the smallest integer to ensure that the denominator is nonzero. The feature points obtained by this method are more robust under variations in imaging conditions [51].  
At each level of the scale space, the candidate points are extracted as follows:  
\[\left\{\begin{array}{ll}\text{\emph{cornerness}}(x,y,\sigma_{I},\sigma_{D}) >\text{\emph{cornerness}}(\hat{x},\hat{y},\sigma_{I},\sigma_{D})&\forall(\hat {x},\hat{y})\in\text{A}\\ \text{\emph{cornerness}}(x,y,\sigma_{I},\sigma_{D})>t_{n}\end{array}\right. \tag{6}\]  
where A represents the points within the \(3\sigma_{I}\) radius neighborhood, and \(t_{n}\) is the threshold, which is \(0.1\cdot\max(\text{\emph{cornerness}}_{I})\).  
The automatic scale selection of the feature points is performed. To select the characteristic scale of the local structure, a scale-normalized derivative LoG operator is defined as:  
\[\text{LoG}(x,y,\sigma_{I})=\sigma_{I}^{2}\big{|}L_{xx}(x,y,\sigma_{I})+L_{yy}( x,y,\sigma_{I})\big{|} \tag{7}\]  
where \(L_{xx}\) and \(L_{yy}\) are second partial derivatives with respect to \(x\) and \(y\), respectively.  
For each candidate point, we apply an iterative method to determine the location and scale of the feature points. Given the initial point \(p\) with the scale \(\sigma_{I}\), the iteration steps are presented as follows.  
1. Find the local extremum over the scale of LoG for the point \(p_{k}\); otherwise, reject the point. The investigated range of scales is limited to \(\sigma_{I}^{(k+1)}=t\cdot\sigma_{I}^{(k)}\) with \(t\in[0.7,\dots,1.4]\).
2. Detect the spatial point \(p_{k+1}\) of a maximum of the SHCS closest to \(p_{k}\) for the selected \(\sigma_{I}^{(k+1)}\).
3. Go to step 1 if \(\sigma_{I}^{(k+1)}\
eq\sigma_{I}^{(k)}\) or \(p_{k+1}\
eq p_{k}\).