One of the most critical missions of sonar is to capture deep-sea pictures to depict sea floor and various objects, and provide an immense understanding of biology and geology in deep sea. Due to the poor condition of underwater acoustic channel, the captured sonar images very possibly suffer from several typical types of distortions before finally reaching to users. Unfortunately, very limited efforts have been devoted to collecting meaningful sonar image databases and benchmark reliable objective quality predictors. In this paper, we first generate a sonar image quality database (SIQD), including 840 images. All distorted images were collected without artificially introducing any distortions beyond those occurring during compression and transmission. The subjective quality assessment was conducted for gathering mean opinion score (MOS) to represent the image quality and existence of target (EOT) which describes whether the image is useful. Based on the built SIQD database, state-of-the-art general image quality metrics were found to poorly correlate with "ground-truth" MOS. As a consequence, this paper further develops a novel full-reference local entropy backed sonar image quality predictor (LESQP). The experimental results demonstrate the superiority of our LESQP metric over the available quality measures.