## 2 Our Proposed Methods
### Concurrent Image Conversion and Adaptive Message Embedding
Inspired by the fact that human eyes are less sensitive to the pixel with low luminance (dark pixels), we derive \(F\) in Equation (2) as a function of \(P_{E}\) because the constant, \(ER_{tar}\), is given by the end user and the constant, \(E_{m}\), is derived from the HDR RGBE image,  
\[F=[2^{ER_{tar}}\times 2^{E_{m}-P_{E}}]. \tag{2}\]  
The derived \(F\) strongly relates to the difference between \(P_{E}\) and the median \(E_{m}\). If \(P_{E}=E_{m}\), \(F=2^{ER_{tar}}\). This means that, if a pixel's \(P_{E}\) is identical to the median, the derived \(F\) equals the two's power of the target embedding rate \((2^{ER_{tar}})\), as expected. However, if \(P_{E}>E_{m}\), the derived \(F\) will be smaller than \(2^{ER_{tar}}\), and the larger the \(P_{E}\), the larger the difference between \(F\) and \(2^{ER_{tar}}\); this means that fewer secret messages are conveyed in this pixel. In contrast, if \(P_{E}<E_{m}\), the derived \(F\) will be larger than \(2^{ER_{tar}}\), and the smaller the \(P_{E}\), the larger the difference between \(F\) and \(2^{ER_{tar}}\), implying that more secret messages will be conveyed in this pixel. This approach coincides with the concept that since human eyes are less sensitive to the pixel with low luminance (dark pixels), more secret message can be concealed in a pixel with smaller \(P_{E}\) (larger difference between \(E_{m}\) and \(P_{E}\)) without causing noticeable difference. In this paper, we set \(ER_{tar}=7.3\) bpp, and apparently, the end user can alter it without any restraint. Once \(F\) is determined, we can derive an optimal base, \(\mathbf{B}=(b_{R},\ b_{G},\ b_{B})\) provided that two conditions are held. First, \(b_{R}\times b_{G}\times b_{B}\geq F\); Second, \((b_{R},\ b_{G},b_{B})\) has minimal _expected_ mean squared error (_EMSE_), which can be determined by Equation (3):  
\[EMSE(\mathbf{B})=\frac{1}{3\times 12}\Big{\{}\Big{[}(b_{R})^{2}-(-2)^{(b_{R}+1)mod2 }\Big{]}+\Big{[}(b_{G})^{2}-(-2)^{(b_{G}+1)mod2}\Big{]}+\Big{[}(b_{B})^{2}-(-2 )^{(b_{B}+1)mod2}\Big{]}\Big{\}}. \tag{3}\]  
Figure 2: The bit breakdown for the 48-bit/pixel OpenEXR, 1-bit sign, 5-bit exponent, 10-bit mantissa, and 32-bit/pixel RGBE encodings, 8-bit in the Red, Green, Blue, and Exponent channel. We adopt the term, _expected_, implying that we assume the three components, \(b_{R}\), \(b_{G}\), \(b_{B}\), have an equal probability of appearing. We remark that EMSE represents the distortion in the R-, G-, B-channels rather than the entire HDR RGBE image. We also remark that, since \(\mathbf{B}\) depends on \(F\), \(\mathbf{B}\) is also a function of \(P_{E}\). Once \(\mathbf{B}=(b_{R},\ b_{G},\ b_{B})\) is discovered, we can compute \(M_{P_{E}}=b_{R}\times b_{G}\times b_{B}\), which may be equal to or larger than \(F\). Consequently, \(M_{P_{E}}\) represents the ultimate notational system that a secret message can be concealed in \(P_{R},\ P_{G},\ P_{B}\). We present an example to further illustrate how to determine an optimal base. If an end user demands a target embedding rate, \(ER_{tar}=7.3\) bpp, and the median of the HDR RGBE image is \(E_{m}=126\). We consider four representative pixels, \(\mathbf{P1}\), \(\mathbf{P2}\), \(\mathbf{P3}\), \(\mathbf{P4}\), as follows: \(\mathbf{P1}=(165,\ 85,\ 16,\ 126),\mathbf{P2}=(195,\ 89,\ 37,\ 126),\mathbf{P3}=(85,\ 128,\ 48,\ 125)\) and \(\mathbf{P4}=(195,\ 89,\ 37,\ 129)\). First, since \(P_{E}=126\) in \(\mathbf{P1}\) and this value equals the median, the corresponding \(F=[2^{7.3}\times 2^{126-126}]=158\). Thus, an optimal base is \(\mathbf{B}=(b_{R},\ b_{G},\ b_{B})=(5,\ 5,\ 7)\) because \(5\times 5\times 7=175>F\) and \(EMSE(\mathbf{B})=\frac{1}{3\times 12}[24+24+48]=2.667\) is minimal. Since \(M_{P_{E}}=b_{R}\times b_{G}\times b_{B}=175\), \(\mathbf{P1}\) can conceal a \(175\)-ary secret message.