## 2 Methodology
### Watermark Embedding
#### 2.3.1 Frequency Domain Transformation  
The insertion of the watermark image is performed in the frequency domain, and this involves converting the host image to the frequency domain using DTCWT. This transformation technique has gained attention due to its advantageous properties over DWT. The most notable properties are approximate shift invariance, directionally selective filters, and perfect reconstruction [29]. DTCWT utilizes two real DWT trees to produce the real and imaginary parts of the transformed image. After applying DTCWT, the image is decomposed into eight sub-bands, including Low-Low (LL), Low-High (LH), High  
Figure 3: The logo used for embedding in this research. The actual logo size is 100 \(\times\) 112, and the logo is black and white.  
Figure 2: The overall design of the proposed method.  
Low (HL), and High-High (HH), for both the real and imaginary parts of a one-level decomposition. Figure 4 illustrates a multi-level decomposition for the LL sub-band. The coefficients from the LL sub-band are selected for embedding the watermark image.  
The dual-tree complex wavelet transform differs from DWT in that it uses two orthogonal parallel transforms of the real wavelet, with the first DWT tree producing the real part and the second tree producing the imaginary part. Each tree contains a unique set of orthogonal filters. This results in DTCWT being able to obtain coefficients of six directions, namely \(\pm 15^{\circ}\), \(\pm 45^{\circ}\), and \(\pm 75^{\circ}\), which solves the problem of poor directionality. At each level of decomposition, DTCWT produces two complex low-frequency sub-bands and six high-frequency complex detailed sub-bands, which are oriented by these angles. This property provides superior perceptual characteristics compared to DWT and enables the development of a watermarking approach with high imperceptibility. In mathematical terms, the low-frequency coefficients can be expressed as shown in Equation (1) [29].  
\[\mathrm{x}\left(\lambda,\mathrm{L},\mathrm{u},\mathrm{v}\right)=\mathrm{Re} \left(\mathrm{x}\left(\lambda,\mathrm{L},\mathrm{u},\mathrm{v}\right)\right)+ \mathrm{jIm}\left(\mathrm{x}\left(\lambda,\mathrm{L},\mathrm{u},\mathrm{v} \right)\right) \tag{1}\]  
where \(\lambda\) is the scaling factor (wavelength), \(\mathrm{L}\) is the decomposition level, \(\mathrm{Re}\) is the real part of the tree, \(\mathrm{Im}\) is the imaginary part of the tree, \(\mathrm{u}\) and \(\mathrm{v}\) indicate the location of the coefficient in each sub-band, and \(\mathrm{j}\) represents the square root of \(-1\).  
In this equation, the image is factorized using both real and imaginary parts to preserve information during processing.  
The high-frequency coefficients can be written as in Equation (2).  
\[\mathrm{x}\left(\overset{\prime}{,}\theta,\mathrm{u},\mathrm{v}\right)=\mathrm{ Re}\left(\mathrm{x}\left(\overset{\prime}{,}\theta,\mathrm{u},\mathrm{v}\right)\right)+ \mathrm{jIm}\left(\mathrm{x}\left(\overset{\prime}{,}\theta,\mathrm{u},\mathrm{ v}\right)\right) \tag{2}\] \[\mathrm{\theta}\in\left\{-75,\,-45,\,-15,\,+15,\,+45,\,+75\right\}\]  
In the embedding process, the frequency component at locations \(\mathrm{u}\) and \(\mathrm{v}\) in the decomposition is represented by the complex coefficient after applying the DTCWT. Therefore, by modifying the least significant bits of these coefficients, the watermark data can be embedded without significantly altering the image's appearance.  
Figure 4: DTCWT sub-bands using multi-level decomposition.