*5.1. PSNR Comparisons*

To demonstrate the effectiveness of our method, we conducted three experiments for comparing the image qualities of the proposed method with those of Imaizumi et al. [16] and Aryal et al. [17] under the same embedding capacity. The first two experiments assume that the location map needed by Imaizumi et al. and Aryal et al.'s methods is not embedded in the stego image, and it is sent through another channel. The third experiment assumes that the location map needed by Imaizumi et al. and Aryal et al.'s methods is embedded in the stego image; thus, the embedding capacity is reduced.

**Figure 1.** Twenty-five test images.

In the first experiment, Imaizumi et al.'s and the proposed methods with capacities of 15,000 and 30,000 bits were conducted. First, each image was divided into 128 × 128 blocks, each of which has size 2 × 2. Then, each block was embedded *k* bits, *k* = 1, 2. Part of results is depicted in Tables 3 and 4. These tables reveal that the image qualities of the proposed method were the best. This means that the proposed method actually has lower embedding distortion for each block such that each block can be used for data embedding. In the following, more details of explanation are described.


**Table 3.** Peak signal-to-noise ratios (PSNRs) of Imaizumi et al.'s [16] and the proposed methods.

**Table 4.** PSNRs of Imaizumi et al.'s [16], Aryal et al.'s [17], and the proposed methods with a capacity 45,000 bits and *k* = 3.


In Imaizumi et al.'s method, the palette is reordered; this increases the distance between two neighboring colors (i.e., two colors with index difference 1) when their indices are large. In data embedding, if one pixel's color *c* with index *i* is replaced by color *c*- , then the index of *c* is *i* − 1 or *i* + 1. However, when *i* is large, these two colors with the index difference of 1 may not be close; this increases the embedding distortion.

To provide more explanation, the image Lena was used with setting *k* = 1. Figure 2 illustrates the result of Tanaka et al.'s parity assignment. Figure 2a depicts all colors with parity 0, and Figure 2c illustrates all colors with parity 1. Figure 2b depicts the corresponding closest color with parity 1 of each color in Figure 2a. Figure 2d depicts the corresponding closest color with parity 0 of each color in Figure 2c. From these figures, we can determine that for each color with parity 0 (1), a similar color with parity 1 (0) can always be found.

Figure 3 illustrates the result of Imaizumi et al.'s palette reordering. Figure 3a displays the reordered color palette. Figure 3b denotes the best replacing one of each color in Figure 3a when embedding data. Figure 3c illustrates the enlarged part of the rectangle marked by red color in Figure 3a. Figure 3d depicts the enlarged part of the rectangle marked by red color in Figure 3b. These figures indicate that some colors may be replaced by an unsimilar color during data embedding.

(**c**) Colors with parity 1.

(**d**) Corresponding closest color with parity 0 for each color in (c).

**Figure 2.** Results of Tanaka et al.'s parity assignment with *k* = 1 for Lena.

(**a**) The reordering palette using Imaizumi et al.'s method.

(**b**) The corresponding replacing color for each color in (a).

(**d**) Enlarged part of the rectangle marked by red color in (b).

**Figure 3.** Results of Imaizumi et al.'s palette reordering for Lena.

In the second experiment, we compared the proposed method with the methods proposed by Imaizumi et al. [16] and Aryal et al. [17] with a capacity of 45,000 bits and *k* = 3. Note that for the image Couple, only 43,845 bits can be embedded using Aryal et al.'s method. Part of results is listed in Table 4, which reveals that the proposed method provides the best image quality under the same capacity. The reason is the same as that in experiment 1. Furthermore, our method modifies at most one pixel during data embedding; other methods [16,17] may modify more than one pixel during data embedding.

As mentioned previously, Imaizumi et al. and Aryal et al.'s methods need an extra location map to record which blocks are used for embedding, in experiment 3, the location map is embedded into a stego image. According to Table 2, the maximum valid embedding capacity of Imaizumi et al. and Aryal et al.'s methods are 12, 288 × *k* and 16384 × *k*, respectively. Table 5 shows part of image qualities of Imaizumi et al.'s, Aryal et al.'s, and our methods under the same embedding capacity. Note that to enforce embedding the location map in the first 4096 (5467) blocks for Imaizumi et al.'s (Aryal et al.'s) method, all indices 0 (255) of pixels in these blocks are changed to 1 (254) before embedding; this will avoid overflow/underflow after embedding. From these tables, we can see that image qualities of the proposed method are always superior to those of Imaizumi et al. and Aryal et al.'s methods under the same embedding capacity. The main reason is that embedding the location map will occupy several blocks and reduce the embedding capacity. Thus, under the same embedding capacity, those blocks used for embedding the location map in Imaizumi et al. and Aryal et al.'s methods will be skipped for data embedding in the proposed method. This will make the proposed method have higher image quality.



#### *5.2. Chi-Square Attack and RS Steganalysis*

In steganography, the main goal for a stego image is statistically undetectable [21] (p. 52). To test whether the embedded images are detectable, chi-square attack [23], RS steganalysis [24], and embedding efficiency [21] were conducted. Chi-square attack and RS steganalysis are discussed in this section, and embedding efficiency in Section 5.3.

Each of chi-square attack and RS steganalysis has two experiments, one embedded 15, 000 × *k* bits (*k* = 1, 2, 3) for Imaizumi et al.'s and the proposed methods. The other embedded 15, 000 × *k* bits (*k* = 3) for Aryal et al.'s and the proposed methods.

Chi-square attack employs Pearson's chi-square test to determine whether there is a statistically significant difference between the expected frequencies and the observed frequencies in one or more categories of an image, and it can detect whether a palette image is embedded by messages. The details of applying the chi-square attack to steganography methods are described as follows:

Step 1: Let the palette indices of a palette image be divided into *K* categories, each category contains a pair of indices (2*i*, 2*i* + 1), *i* = 0, ... , *K* − 1. Let *Ei* and *Oi* be the theoretically expected and observed frequencies of pixels with index 2*i* after embedding messages, respectively. Then

$$E\_i = \frac{\text{The number of pixels with color index } 2i \text{ or } 2i + 1}{2},\tag{17}$$

*Oi* = The number of pixels with color index 2*i*. (18)

Step 2: The χ<sup>2</sup> statistic is given as

$$\chi^2\_{K-1} = \sum\_{i=0}^{K-1} \frac{\left(O\_i - E\_i\right)^2}{E\_i} \text{ with } K - 1 \text{ degrees of freedom.} \tag{19}$$

Step 3: Let *pv* represent the embedding probability, *pv* can be calculated by the following equation:

$$p\_{\upsilon} = 1 - \frac{1}{2^{\frac{K-1}{2}} \Gamma\left(\frac{K-1}{2}\right)} \int\_{0}^{\chi\_{K-1}^{\frac{2}{2}}} e^{-\frac{x}{2}} x^{\frac{K-1}{2}-1} dx. \tag{20}$$

In chi-square attack, the results for three methods are *pv* = 0; this means that the three methods can resist chi-square attack. There are two reasons: (1) These three methods are block-based; the embedding capacities are limited and lower than 1 bpp; this will make chi-square attack fail [21]; (2) chi-square attack is used to detect LSB-based methods, but these three methods are not LSB-based methods.

Chi-square attack just uses sample counts and neglects spatial correlations among pixels in the stego image. Fridrich et al. [24] introduced RS steganalysis for detection of LSB embedding that utilizes sensitive dual statistics derived from spatial correlations in image.

In RS steganalysis, for a given image *I*, through a given local mask, two flipping functions, and a discrimination function, all pixels of the image can be classified into three groups: Regular, singular, and unchanged. Given a non-negative mask *m*, we can obtain the relative frequency of the regular group denoted as *R*+ and the relative frequency of the singular group denoted as *S*+. Then through the mask –*m*, we can obtain the relative frequency of the regular group denoted as *R*− and the relative frequency of the singular group denoted as *S*−. Let *pe* be the embedding rate in *f*, RS steganalysis can estimate *pe* by solving the following equation:

$$2(d\_1 + d\_0)z^2 + (d\_{-0} - d\_{-1} - d\_1 - 3d\_0)z + d\_0 - d\_{-0} = 0,\tag{21}$$

where

$$\begin{aligned} d\_0 &= R\_+ \left( p\_\varepsilon / 2 \right) - S\_+ \left( p\_\varepsilon / 2 \right), \; d\_1 = R\_+ \left( 1 - p\_\varepsilon / 2 \right) - S\_+ \left( 1 - p\_\varepsilon / 2 \right), \\\ d\_{-0} &= R\_- \left( p\_\varepsilon / 2 \right) - S\_- \left( p\_\varepsilon / 2 \right), \; d\_{-1} = R\_- \left( 1 - p\_\varepsilon / 2 \right) - S\_- \left( 1 - p\_\varepsilon / 2 \right), \end{aligned} \tag{22}$$

and *R*+(*p*/2) represents *R*<sup>+</sup> of a stego image with *p* pixels embedded (i.e., the LSBs of *p*/2 pixels flipped from a cover image). Thus, we can obtain *R*+(*pe*/2) easily through *f*; furthermore, by flipping the LSB of each pixel in *I* to get *I* - , we can also obtain *R*+(1 − *pe*/2) through *I* - . *S*+, *S*−, and *R*− can be obtained by the similar way. Then, *pe* is calculated from the root *z* of Equation (20), whose absolute value is smaller, by the following equation:

$$p\_c = \frac{z}{z - \frac{1}{2}}.\tag{23}$$

Two experiments are conducted, one for Imaizumi et al.'s [16] and the proposed methods uses block size 2 × 2 and three capacities (23%, 46%, and 69%); the other for Aryal et al.'s [17] and the proposed methods uses block size 1 × 3 and the same three capacities. In the experiments, the mask *m* is defined as [0110]. Tables 6 and 7 show part of the *pe*s of Imaizumi et al.'s, Aryal et al.'s, and the proposed methods. Note that both Imaizumi et al. and Aryal et al.'s methods embedded message by reordering the original palette, so there are two results for their methods by analyzing the original and the reordering palettes. From these tables, we can see that RS steganalysis cannot estimate *pe* accurately. The reason is that these three methods are not LSB-based methods. We can conclude that the three methods can resist RS steganalysis and are undetectable.



#### *5.3. Embedding E*ffi*ciency*

Another measurement for undetectability is embedding efficiency [21]. The embedding efficiency is defined as the number of embedded random message bits per embedding change [21]. According to this definition, embedding efficiency (*EF*) can be expressed as follows:

$$EF = \frac{\text{number of embedding bits}}{\text{number of embedding pixel changes}}.\tag{24}$$

Methods with higher embedding efficiency are more undetectable, because under the same capacity, higher embedding efficiency methods will change less pixels than lower embedding efficiency methods. For *k* = 3, according to Equation (24), the estimated embedding efficiency of the proposed method (*EFp*) can be calculated by Equation (25).

$$EF\_p = \frac{3}{0 \times p\_0 + 1 \times p\_1} \,\prime \tag{25}$$

where *pi* stands for the probability of *i* pixel changed in a block, here *p*<sup>0</sup> = 1/8 and *p*<sup>1</sup> = 7/8.

The estimated embedding efficiency of Imaizumi et al.'s method (*EFi*) can be calculated by Equation (26).

$$EF\_i = \frac{3}{0 \times p\_0 + 1 \times p\_1 + 2 \times p\_2 + 3 \times p\_3 + 4 \times p\_4},\tag{26}$$

where *pi* stands for the probability of *i* pixel changed in a block, here *p*<sup>0</sup> = 1/15, *p*<sup>1</sup> = *p*<sup>2</sup> = *p*<sup>3</sup> = 4/15, and *p*<sup>4</sup> = 2/15.

*Appl. Sci.* **2020**, *10*, 7820

The estimated embedding efficiency of Aryal et al.'s method (*EFa*) can be calculated by Equation (27).

$$EF\_a = \frac{3}{(p\_0) \times (1 \times p\_1 + 2 \times p\_2) + (1 - p\_0) \times ((1 + 1) \times p\_1 + (1 + 2) \times p\_2)}.\tag{27}$$

where *p*<sup>0</sup> stands for the probability of *B*<sup>0</sup> unchanged, *pi* stands for the probability of *i* pixels from {*B*1, *B*2} changed, here *p*<sup>0</sup> = 1/2, *p*<sup>1</sup> = 4/7, and *p*<sup>2</sup> = 2/7.

Table 8 shows the estimated embedding efficiencies of the proposed, Imaizumi et al.'s [16], and Aryal et al.'s [17] methods obtained by using Equations (25)–(27). Table 9 shows part of experimental embedding efficiency calculated by applying Equation (24) to each stego image, which is obtained by one of Imaizumi et al.'s, Aryal et al.'s and the proposed methods. From these tables, we can see that either estimated embedding efficiency or experimental one, the proposed method has higher embedding efficiency than those of Imaizumi et al. and Aryal et al.'s methods. This means that the proposed method is less detectable than Imaizumi et al. and Aryal et al.'s methods.


**Table 7.** The results of RS steganalysis for Aryal et al.'s [17] and the proposed methods with block size 1 × 3.

**Table 8.** Estimated embedding efficiencies of Imaizumi et al.'s [16], Aryal et al.'s [17], and the proposed methods with *k* = 3.



**Table 9.** Experimental embedding efficiencies of Imaizumi et al.'s [16], Aryal et al.'s [17], and the proposed methods with *k* = 3.

#### **6. Conclusions**

As mentioned previously, some BBMs produce large embedding distortions in some blocks; a location map is incorporated into these methods to record which blocks are used for data embedding. This will reduce the embedding capacity for secret data. To avoid this disadvantage, in this paper, we have proposed a BBM for palette images. The method modifies at most one pixel in a block. If modification is required, one optimal pixel with minimal embedding distortion is selected; this makes each block be used to embed secret data; that is, the embedding capacity of the proposed method is larger than that of the state-of-the-art BBMs. As to the undetectability, chi-square attack, RS steganalysis, and the embedding efficiency are used. Imaizumi et al.'s, Aryal et al.'s., and the proposed methods can resist chi-square attack and RS steganalysis. However, through the measure of embedding efficiency, both estimated and experimental efficiencies revealed that our method provided higher undetectability.

**Author Contributions:** Conceptualization, H.-Y.W. and L.-H.C.; methodology, H.-Y.W. and L.-H.C.; software, H.-Y.W.; validation, H.-Y.W. and L.-H.C.; writing—Original draft preparation, H.-Y.W.; writing—Review and editing, H.-Y.W. and L.-H.C.; visualization, H.-Y.W. and L.-H.C.; supervision, L.-H.C. and Y.-T.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


(BigMM), Laguna Hills, CA, USA, 13–16 September 2018; Institute of Electrical and Electronics Engineers (IEEE): New York, NY, USA, 2018; pp. 1–5.


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