# A Robust and Secure Watermarking Approach Based on Hermite Transform and SVD-DCT
## 3 Materials and Methods
### Discrete Cosine Transform  
The DCT serves as the foundation for numerous image compression techniques and lossy digital image compression systems: Joint Photographic Experts Group (JPEG) for still images and Moving Picture Experts Group (MPEG) for video images [48]. It is a mathematical tool to perform frequency analysis without complex numbers and to approximate a typical signal using fewer coefficients (low-, high-, and middle-frequency components), i.e., it can pack the most information in the fewest coefficients and pack energy in the low-frequency regions [49; 50]. One of the most-usual applications is in signal and image processing for lossy compression because of its property to compact strong energy, creating predictions according to its local uniqueness. Besides, as mentioned in [51], this transform has entropy retention, decorrelation, and energy retention-energy concentration, among which energy concentration is of great significance to digital image encryption. Therefore, the most-important DCT advantages, such as a high compression ratio and low error rates [52], are taken into account in different applications, such as digital image encryption, because the energy concentration is a very important element. When we applied the DCT to an image (matrix), we obtained a DCT coefficient matrix that contained the DC coefficient and the AC coefficient. The energy was concentrated in the DC element. As an example, the _Lena_ image and its transformation applying the DCT are shown in Figure 8.  
Figure 8: Discrete cosine transform. (**a**) Original _Lena_ image; (**b**) DCT of _Lena_ image.  
Another application is in steganography systems and watermark systems, which embed the information of a signal in the transform domain. These systems are more robust if they operate in the transform domain (Discrete Fourier Transform (DFT), Discrete Cosine Transform (DCT), Discrete Wavelet Transform (DWT), Contourlet Transform (CT), etc.). Specifically, in the DCT domain, the algorithms are more robust against common processing operations (JPEG and MPEG compression) compared with spatial domain techniques, and also, the DCT offers the possibility of directly realizing the embedding operator in the compressed domain (i.e., inside a JPEG or MPEG encoder) to minimize the computation time [53].  
The 2D DCT of a grayscale image \(I(x,y)\) is as follows (Equation (5)):  
\[I(u,v)=\sum_{y=1}^{Y}\sum_{x=1}^{X}w(u,v)I(x,y)cos\bigg{(}\frac{(2x-1)(u-1)\pi} {2M}\bigg{)}cos\bigg{(}\frac{(2y-1)(v-1)\pi}{2N}\bigg{)}, \tag{5}\]  
where \(X\) represents the number of columns and \(Y\) the number of rows of the image \(I(x,y)\), \(x,y\) its spatial coordinates, \(u,v\) the corresponding frequency coordinates, and  
\[w(u,v)=\begin{cases}\frac{1}{\sqrt{X}Y}&\text{when }u=1,v=1\\ \sqrt{\frac{2}{XY}}&\text{when }u=1,v\geq 2\\ \sqrt{\frac{4}{XY}}&\text{otherwise},\end{cases}\]  
where \(w(u,v)\) is a weight factor, \(y,v\in[1,\dots,Y]\) and \(x,u\in[1,\dots,X]\)[54].  
The 2D Inverse Discrete Transform (IDCT) is given as follows (Equation (6)):  
\[I(x,y)=\sum_{v=1}^{Y}\sum_{u=1}^{X}w(u,v)I(u,v)cos\bigg{(}\frac{(2x-1)(u-1)\pi }{2X}\bigg{)}cos\bigg{(}\frac{(2y-1)(v-1)\pi}{2Y}\bigg{)}. \tag{6}\]