## 2 Related Work and Comparisons between the Methods
### Further Discussion on Histogram Modification-Based Approaches  
The method of [PERSON] et al. [46] constitutes a rough framework and foundation for RDH based on histogram modification, and hence has been further developed in the follow-up research [5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. In these studies, a histogram is first generated from the prediction error of pixels, and then it is modified by expansion or shifting to achieve reversible embedding. Currently, such methods, modifying the _prediction error histogram_ (PEH), are collectively perceived as prediction error expansion (PEE). RDH based on histogram modification has the following two advantages:  
* Using histograms, especially PEHs, can effectively utilize image redundancy. * Modifying the histogram by expansion or shifting can control the maximum modification range of each pixel and the embedding distortion effectively. From the above points of view, the methods based on histogram modification, especially PEE based on PEH modification, have better embedding performance than other methods. Therefore, we focus on histogram generation and three-dimensional histogram modification. Note that the current RDH methods based on histogram modification mainly include the following aspects:  
* Generation method of histogram. Combined with PEE, the methods of this research direction mainly aim to generate a sharp and rapidly dropping PEH by using better image prediction methods, e.g., the methods of [12, 13, 19, 20, 23, 24]. * Modification method of histogram. Different from the early expansion methods [8, 9, 16, 24] using a peak in histogram, several authors [15, 25, 26, 27] proposed methods to expand the histogram by adaptively selecting with the frequency of pixels in the image histogram. These methods can significantly reduce the embedding distortion of PEE. * Selection of embedding location. This type of method firstly selects the image area that is more suitable for reversible embedding (usually smooth areas), and then uses the selected area as a new carrier for RDH. The effect of these methods are remarkable. Combining with PEE can effectively reduce the embedding distortion of PEE. Its idea was first proposed by [PERSON] et al. [18], and many subsequent works have also applied this method as an auxiliary means to further optimize the embedding performance. * High-dimensional histogram modification. Several authors [10, 21] proposed the methods based on high-dimensional histogram modification. They map high-dimensional redundant features of images to two-dimensional space, and then modify the two-dimensional histogram to achieve reversible embedding. In recent works [5, 14, 17], the methods based on three-dimensional or high-dimensional histogram modification are proposed. By mapping the redundant features of the image to a higher-dimensional space, the embedding capacity is increased and the image quality is maintained. This type of method can greatly improve the embedding performance of existing PEE algorithms. * Multi-histogram modification. In [11, 22], the reversible embedding methods based on using multi-histograms are proposed. Compared with the method of using a single histogram, the use of multiple histograms has greater flexibility and can further improve the performance of PEE algorithms. * PEH for color images. In [51], [PERSON] et al. applied 3D-PEH to color images. Their approach is to predict the pixel values of each RGB channel of a color image and establish the 3D prediction-error histogram. Their results yield low distortion for color images. Below we summarize two recent progress on the other perspectives on the histogram modification-based methods. * Histogram-shifting-imitated technique based on human visual system (HVS). [PERSON] et al. take human visual system into consideration [52] and improves previous work using histogram-shifting-imitated reversible data hiding method in [53].