We study the issue of quality assessment in tone mapping- based high-dynamic-range (HDR) image compression. In this, there are two stages at which a decision should be made regarding perceptual vis- ual quality: (a) for finding the optimal parameters of the dynamic range reduction function so that the visual quality is maximized, and (b) visual quality judgment of the decompressed image. We first investigate two objective optimization criteria, namely mean squared error and structural similarity index measure, toward optimization of a tone mapping model- based HDR image compression method. We then conduct a comprehen- sive subjective study to evaluate the visual quality of the compressed HDR images. Therefore, we consider both objective and subjective aspects for HDR image compression. To our knowledge, no systematic and compre- hensive studies exist in the current literature which shed light on the issue of quality assessment in HDR compression. So this study brings in new knowledge and perspective for the relatively less investigated topic of HDR compression from the view point of perceptual quality. We further evaluate the prediction performances of four objective methods on the 140 compressed HDR images that have been subjectively rated.