## 2 Our Proposed Methods
### Concurrent Image Conversion and Adaptive Message Embedding
We remark that although other bases, such as \((5,\ 6,\ 6)\) or \((5,\ 6,\ 7)\), do satisfy the first condition, \(5\times 6\times 6\geq F\) and \(5\times 6\times 7\geq F\), unfortunately, they hold larger expected mean squared errors, where \(EMSE(\mathbf{B1})=2.778\) and \(EMSE(\mathbf{B2})=3.056\). Consequently, \(\mathbf{B}=(5,\ 5,\ 7)\) does actually represent an optimal base for \(\mathbf{P1}\). Since \(\mathbf{P2}\) and \(\mathbf{P1}\) have the same \(P_{E}\), they share the same optimal base able to conceal a \(175\)-ary secret message. With regard to the \(\mathbf{P3}\) pixel, we derive \(F=[2^{7.3}\times 2^{126-125}]=316\) and the optimal base is \(\mathbf{B}=(7,\ 7,\ 7)\), thereby producing \(EMSE(\mathbf{B})=\)\(4.0\). Since \(M_{P_{E}}=7\times 7\times 7=343\), the \(\mathbf{P3}\) pixel can carry a \(343\)-ary secret message, conforming to our design goal for embedding a more secret message when \(P_{E}\) is smaller than \(E_{m}\). Finally, an optimal base for the \(\mathbf{P4}\) pixel is \(\mathbf{B}=(3,\ 3,\ 3)\) because \(F=[2^{7.3}\times 2^{126-129}]=20\) and \(M_{P_{E}}=27\). As expected, a limited \(27\)-ary secret message will be concealed in this pixel, as its \(P_{E}=129\) is much larger than the median, \(E_{m}=126\). * **Step 3:** Real embedding rate determination. Deriving the real embedding rate, \(ER_{rea}\), to comply with the end user's request, we verify that the real embedding rate is greater than or equals the target embedding rate (\(ER_{rea}\geq ER_{tar}\)). In this step, we derive \(ER_{rea}\) using Equation (4): \[ER_{rea}=\frac{1}{H\times V}\sum_{P_{E}=0}^{255}N_{P_{E}}\times log_{2}\big{(}M _{P_{E}}\big{)},\] (4) where \(N_{P_{E}}\) represents the number of pixels in the entire HDR RGBE image holding \(P_{E}\) value in E-channel; for example, \(N_{132}\) denotes the number of pixels holding \(P_{E}=132\). Since \(P_{E}\) is within the range of [0, 255], it is certain that \(N_{P_{E}}=N_{0}+N_{1}\ldots+N_{255}\) equals the image resolution of \(H\times V\). In case \(ER_{rea}<ER_{tar}\), we adjust the median \(E_{m}\) until we adhere to the end user's request. Finally, we will construct an E-channel mapping table, as shown in Table 1a, which maps every \(P_{E}\) value to \(N_{P_{E}}\), \(F\), \(M_{P_{E}}\), \(\mathbf{B}=(b_{R},\ b_{G},\ b_{B})\), and \(EMSE(\mathbf{B})\). In addition, we can derive the real embedding rate, \(ER_{rea}\), and the expected mean squared error for three channels, \(EMSE(RGB)\). Table 1a presents the E-channel mapping table constructed after Step 2 for the HDR image \"memorial\". The end user requests the embedding rate of \(ER_{tar}=7.3\) bpp and the median of E-channel in this image is \(E_{m}=126\). We list \(N_{P_{E}}\) according to \(P_{E}\) in ascending order, and the associate optimal bases. Referring to Equation (4), the real embedding rate, \(ER_{rea}=7.578\) bpp, is larger than the user's request. Also shown in the right part of table is the E-channel mapping table built when the requested embedding rate is \(ER_{tar}=9.0\) bpp. As can be seen from the table, when a user requests different \(ER_{tar}\), the same \(P_{E}\) corresponds to different \(N_{P_{E}}\), \(F\), \(M_{P_{E}},\mathbf{B}=(b_{R},\ b_{G},\ b_{B})\), and \(EMSE(\mathbf{B})\). Nevertheless, both cases show that our scheme offers a more embedding rate than that request by the end user. Table 1b shows another E-channel mapping table constructed for the HDR image \"display1000\". This image has the median of E-channel, \(E_{m}=125\). We can observe from the table that, when a pixel has a smaller \(P_{E}\), a larger optimal base is determined, thereby enabling it to conceal a larger \(M_{P_{E}}\)-ary secret message. Again, the real embedding rate, \(ER_{rea}=7.618\) bpp, is larger than the request, \(ER_{tar}=7.3\) bpp. These two tables demonstrate that our scheme can adaptively conceal a secret message according to the features of pixels in an HDR RGBE image.