## 1 Introduction  
High dynamic range (HDR) images [1, 2] have attracted much interest in recent years because they offer a more dynamic range, allowing displays to show images in a clearer way, especially in the details of the highlights and shadows. Highlights can be shown brighter and truly dark, and deep shadows can be displayed. Thus, the original image can be shown more completely and faithfully, closer to what the human eye can see and perceive. Dynamic range is the ratio between the brightest and darkest values that a display can show. Using the floating-point number to store crucial information, HDR images can faithfully represent a large range of luminance and colors, thereby offering great potential to become the leading image standard. There are three main HDR image encoding formats: RGBE [3], LogLuv [4], and OpenEXR [5]. Any two formats can be converted in either a forward or backward manner. Tone-mapping [1, 2, 6], an essential step for HDR images, consists of adjusting the tonal values of an image with a high dynamic range so that it can be viewed on digital displays. Tone-mapping operator, TMO, scales down the dynamic range while attempting to preserve the appearance of the original image. It is thus applied to HDR images to reveal their full details and give them a dynamic twist and realistic look, providing the best image presentation for visualizing an HDR image. Data hiding [7], also known as steganography, is a technique developed to embed secret messages into digital media. The camouflaged media looks like the original harmless one, thus arousing no suspicion among malicious eavesdroppers. The media before message embedding is called a cover, and the media after the message concealment is referred to as a stego. In the era of internet transmission and cloud computing, data hiding provides a method for communicating secretly. While time-varying media, such as video, can be used as a carrier [8], static images are the most popular because they are widely transmitted over the Internet. Data hiding schemes using conventional low dynamic range (LDR) images, such as binary, grayscale, color, or palette images, have been presented in the extant literature [7; 9; 10]. Despite HDR images having the potential to become a leading standard, limited numbers of HDR data hiding works have been proposed in the literature. Similar to LDR image data hiding, the number of bits that an HDR image can carry is called the embedding capacity (EC). Taking the image resolutions into consideration, the embedding rate (ER) represents the number of bits concealed in a pixel, also denoted as bits per pixel (bpp). One of the goals pursued by a data hiding algorithm is providing an embedding rate that is large enough for practical applications. [PERSON] and [PERSON] [11] proposed an adaptive data hiding and authentication algorithm for HDR RGBE images. They provided an embedding rate in the range of 5.13 to 9.69 bpp. The peak signal-to-noise ratios (PSNR) of the tone-mapped stego images are only slightly greater than the 30 dB. [PERSON] et al. [12] adopted HDR images with a 48-bit TIFF format and improved the embedding rate of [PERSON] and [PERSON]'s scheme. Their algorithm offered an average embedding rate of 26 bpp, with the PSNR of the tone-mapped stego images between 30.47 and 37.00 dB. [PERSON] et al. [13], [PERSON] et al. [14], and [PERSON] et al. [15] took advantage of modifying a homogeneity index to achieve distortion-free data hiding in HDR RGBE images. However, the penalty of this approach limited the embedding rate to only 0.127-0.145 bpp. [PERSON] et al. [16] presented a novel data hiding algorithm (DHOB) and an aggressive bit encoding and decomposition (ABED) scheme for OpenEXR images. Depending on the parameters, the embedding rate was between 2.433 and 20.002 bpp. [PERSON] et al. [17] and [PERSON] et al. [18] employed prediction error expansion to embed a secret message in every individual Red, Green, and Blue color channel. Their algorithms provided embedding rates ranging from 1.202 to 2.85 bpp. [PERSON] et al.