## 4 Experimental Results  
In this paper, the three gray-scale Lena, Peppers and Jet images, shown in Figure 3, are considered. The size of the three gray images is \(512\times 512\). The range of the block size \(s\) is from six to 38, which was already considered in [4; 5]. The function \(h_{w}\) to flip \(w\) LSBs is considered as \(h_{3}\), which is shown in (6). To compare the proposed data-hiding systems with the referenced systems, the error pattern of the data-hiding system is shown in Figure 4. For a Lena image and \(s=8\), error positions of [PERSON]'s system [4], [PERSON]'s system [5], the proposed system with fluctuation functions in (15) and (16) and the proposed system with the fluctuation function and RS(15,11) codes are shown in (a), (b), (c) and  
Figure 3: Test images used for the simulation. (**a**) Lena; (**b**) Peppers; (**c**) Jet [31]. (d) in Figure 4, respectively. The small black square in a large square in Figure 4 denotes \(s\times s\) error pixels in a Lena image. From Figures 4a to 4c, the recovery performance of the system with the proposed fluctuation functions in (15) and (16) is better than the ones of [PERSON]'s system [4] and [PERSON]'s system [5] for \(s=8\) and the Lena image. Moreover, there is no error pattern in (d) of Figure 4 when the proposed system with RS(15,11) codes is used. To investigate how RS codes can help recover the error in RDH systems, an example for the proposed system with RS codewords is shown in Figure 5. Figure 5a is similar to Figure 4c. One error in Figure 5a is located at \(y(9,26)\). It is assumed that the codewords of RS(15,11) start from \(y(0,0)\) and are assigned to the right sequentially. Then, the error at \(y(9,26)\) belong to the 11 th codeword, which starts at \(y(9,24)\) and ends at \(y(10,24)\). The gray rectangle in Figure 5a denotes the 11 th codeword, and the codewords are also shown in Figure 5b. The codeword is composed of 15 symbols, which corresponds to 60 bits. A blue rectangle in Figure 5b denotes a symbol of codewords. From Figure 5b, one symbol of the codeword is an error, and it is known that RS(15,11) can correct up to two error symbols. Therefore, the error pattern at \(y(9,26)\) can be recovered from the RS codes. Similarly, the other errors can be recovered by the help of RS codes. Figure 4: Error patterns of [PERSON]’s system [4], [PERSON]’s system [5], and the proposed scheme in the Lena image for \(s\)=8. (**a**) \(f_{Z}\) in (11) and (12); (**b**) \(f_{H}\) in (13) and (14); (**c**) \(f_{K}\) in (15) and (16); (**d**) \(f_{K}\) in (15) and (16) + RS(15,11) code. In Figure 6, the BER performances of the referenced systems and proposed system without RS codes are shown according to the size of \(s\) when the Lena, Peppers and Jet image in Figure 3 are considered. The 'Ref. [PERSON] [4]' and 'Ref. [PERSON] [5]' in these figures denote BERs for [PERSON]'s scheme in [4] and [PERSON]'s scheme in [5], respectively. The 'Pro. Fluc.' in these figures stands for BERs for the proposed fluctuation functions in (15) and (16). In Figure 6, as block size \(s\) increases, the BERs of two referenced system sand the proposed system without RS codes become better. BER performances of two referenced systems and the proposed system without RS codes in Figure 6a,b are better than BER performances in Figure 6c, since the Jet is a weakly correlated image. In Figure 6a,b, BER performances of the proposed fluctuation function are always better than the referenced systems. It is shown that the proposed fluctuation functions are effective for the RDH system without the help of RS codes. In Figure 6c, the BER of the proposed system without RS codes is the same or better than the BER of [PERSON]'s system. For a small length of \(s\), the BERs of three systems turn around since the reliability of the fluctuation function in the weakly correlated image is small.