## 3 Data Embedding in PHL Spectrum  
The watermarking approach, presented in this paper, is based on inserting secret information in the PHL transform domain. The method assumes that the PHL spectrum is calculated only for the luminance channel of the given image, representing its grayscale version. To speed up the computations, the forward transform operation is performed on smaller subimages, i.e., blocks with the size: 8 \(\times\) 8 pixels, using Equation (10) and the matrices (6) and (7).  
As a result, after performing the above process to the input signal, we get its spectral coefficients in the PHL domain. Typically, a limited number of these coefficients carry most of the signal energy [30; 31].  
The PHL transform may be used for image compression purposes [32]. In this task, the spectral coefficients that are above a given threshold are kept while the remaining ones are set to zero. Following this approach, our method assumes embedding of the watermark by modification of the coefficients having relatively low values. To perform this operation, the PHL coefficients are split into channels. Each channel groups the spectral coefficients with the same indices from each block processed in the forward transform step. This way, we obtain 64 PHL transform channels. The study of a set of various images and their spectra indicates that the top-left channel cumulates most of the signal energy. It is well depicted in Figure 2 which shows the PHL spectrum coefficients after grouping into 64 channels.  
For the testing purpose and the presentation of the image manipulation detection method in the following section, the Optical Coherence Tomography (OCT) images, having the resolution of 1536 \(\times\) 496 pixels, were used [33]. The OCT is a non-invasive imaging examination that uses light waves to take cross-section pictures of the human retina. One sample image of this type is shown in Figure 3. The tests show that the blocks: 37-39, 45-47, and 53-55, marked in Figure 4, should be usually selected for the process of inserting secret information. This conclusion is based on the analysis of spectra of diverse images with varying content and characteristics. For the selection of the best channel for watermark embedding, the mean of all absolute values from each block is calculated. The channel with the lowest mean is chosen as the first candidate for the subsequent data embedding operation. To increase the capacity of the watermark, other blocks can be selected afterward, considering their mean values sorted in ascending order.  
Figure 2: PHL spectrum coefficients grouped into channels.  
The selected channel coefficients are replaced with the consecutive bits of the message that is to be hidden in the image. Subsequently, the channel coefficients need to be relocated back to their previous positions. The final step is the inverse PHL transform of the modified image spectrum that results in the image with an inserted watermark. The stages of the whole embedding process are presented in Figure 5.  
For the recovery of the embedded information, the same steps as previously need to be performed--the forward transform, the grouping of PHL coefficients, and finally extracting information from the selected channel or channels.  
The selection of nine blocks for watermark embedding can be performed adaptively, as described previously, or arbitrarily. In this way, the chosen order can be used as an additional key at the watermark extraction phase.  
Figure 4: Blocks selected for data embedding.  
Figure 3: Sample OCT image.