Since underwater imaging is affected by the complex water environment, it often leads to severe distortion of the underwater image. To improve the quality of underwater images, underwater image enhancement and restoration methods have been proposed. However, many underwater image enhancement and restoration methods produce over-enhancement or under-enhancement, which affects their application. To better design underwater image enhancement and restoration methods, it is necessary to research the underwater image quality evaluation (UIQE) for underwater image enhancement and restoration methods. Therefore, a subjective evaluation dataset for an underwater image enhancement and restoration method is constructed, and on this basis, an objective quality evaluation method of underwater images, based on the relative symmetry of underwater dark channel prior (UDCP) and the underwater bright channel prior (UBCP) is proposed. Specifically, considering underwater image enhancement in different scenarios, a UIQE dataset is constructed, which contains 405 underwater images, generated from 45 different underwater real images, using 9 representative underwater image enhancement methods. Then, a subjective quality evaluation of the UIQE database is studied. To quantitatively measure the quality of the enhanced and restored underwater images with different characteristics, an objective UIQE index (UIQEI) is used, by extracting and fusing four groups of features, including: (1) the joint statistics of normalized gradient magnitude (GM) and Laplacian of Gaussian (LOG) features, based on the underwater dark channel map; (2) the joint statistics of normalized gradient magnitude (GM) and Laplacian of Gaussian (LOG) features, based on the underwater bright channel map; (3) the saturation and colorfulness features; (4) the fog density feature; (5) the global contrast feature; these features capture key aspects of underwater images. Finally, the experimental results are analyzed, qualitatively and quantitatively, to illustrate the effectiveness of the proposed UIQEI method.