## 2 Our Proposed Methods
### Concurrent Image Conversion and Adaptive Message Embedding  
In this process, we convert the input HDR image, \(I_{EXR}\), encoded by the OpenEXR format into the HDR image encoded by the RGBE format. Figure 2 shows the 48-bit/pixel OpenEXR and 32-bit/pixel RGBE formats. A pixel in an OpenEXR image is represented by a floating-point vector, \(X=(\mathbf{X}_{R},\ X_{G},\ X_{B})\). The floating-point value in the corresponding Red, Green, and Blue channels can be converted from 1-bit sign, 5-bit exponent, and 10-bit mantissa fields. In contrast, a pixel in an RGBE image is denoted by \(\mathbf{P}=(P_{R},\ P_{G},\ P_{B},\ P_{E})\), representing the respective Red, Green, Blue, and the Exponent channels with 8-bit storage. During the conversion, we concurrently and adaptively embed a secret message, thus constructing a new stego HDR RGBE image, \(I_{S}\). The process is completed by the following five steps:  
* **Step 1:** Image conversion. An \(I_{EXR}\) is converted into the HDR image encoded by the RGBE format. The expressions shown on the left expression in Equation (1) convert chromatic information into the four components in the RGBE format: \[ \begin{cases}P_{E}=\left\lceil\log_{2}[\max(X_{R},X_{G},X_{B})]+128\right\rceil,&X_{R}=\frac{P_{R}+0.5}{256}\times 2^{P_{E}-128}\\ P_{R}=\left\lfloor\frac{256\times X_{R}}{2^{P_{E}-128}}\right\rfloor,&X_{G}= \frac{P_{G}+0.5}{256}\times 2^{P_{E}-128},\\ P_{B}=\left\lfloor\frac{256\times X_{B}}{2^{P_{E}-128}}\right\rfloor,&X_{B}= \frac{P_{B}+0.5}{256}\times 2^{P_{E}-128},\end{cases} \] (1) where \(\mathbf{X}=(X_{R},\ X_{G},\ X_{B})\) represents the \(i\)-th pixel in scene-referred color in \(I_{EXR}\) and \(\mathbf{P}=(P_{R},\ P_{G},\ P_{B},\ P_{E})\) denotes the \(i\)-th pixel in the corresponding RGBE HDR image. Note that we omit the index \(i\) if the expression does not cause any ambiguity. In a special case when \(\max(X_{R},\ X_{G},\ X_{B})\) is less than \(10^{-38}\), the conversion is written out as \((0,\ 0,\ 0,\ 0)\). We can translate the integer representation in \(\mathbf{P}=(P_{R},\ P_{G},\ P_{B},\ P_{E})\) back to the scene-referred color, \(\mathbf{X}=(X_{R},\ X_{G},\ X_{B})\), using the right expressions in Equation (1), which is also known as reverse conversion. In a special case described above, we directly translate to \((0,\ 0,\ 0)\). Note that the four components, \(P_{R},\ P_{G},\ P_{B},\ P_{E}\), are integers within the range of [0, 255]. While the first three components, \(P_{R},\ P_{G},\ P_{B}\), represent the respective pixel values in the Red, Green, and Blue channels, the fourth component, \(P_{E}\), indicates the pixel value in the Exponent channel (hereafter E-channel). * **Step 2:** Optimal base determination. In this step, we determine an optimal base, \(\mathbf{B}=(b_{R},\ b_{G},\ b_{B})\), for the \(i\)-th pixel \(\mathbf{P}\), according to \(P_{E}\), i.e., this pixel's value in the E-channel. An optimal base is required so that we can embed a secret message in the \(P_{R},\ P_{G},\ P_{B}\). In this paper, we convey a secret message in the Red-, Green-, and Blue-channels and leave the E-channel intact because the E-channel is the exponent shared by the other three channels, so any changes in it, for example, plus or minus \(n\) values \((\pm n)\), will result in the magnitude of exponential changes, \((\text{R},\ \text{G},\ \text{B})\times 2^{\pm n}\) in three other components. Instead, we utilize the value of the E-channel in a pixel \(\mathbf{P}\) as a guide to determine its corresponding optimal base, thereby achieving adaptive message embedding. We detail this step as follows. Let \(P_{E}\) (\(0\leq P_{E}\leq 255\)) represent the value of the pixel \(\mathbf{P}\) in E-channel which is currently processed. Let \(E_{m}\) denote the median of the pixels in the E-channel derived from the HDR RGBE image which has just been converted from an OpenEXR HDR image. Let \(F\) represent the minimal notational number system where an \(F\)-ary secret message can be concealed in \(P_{R},\ P_{G},\ P_{B}\).