Image and video quality assessment (QA) is a critical issue in image and video processing applications. General full-reference (FR) QA criteria such as peak signal-to-noise ratio (PSNR) and mean squared error (MSE) do not accord well with human subjective assessment. Some QA indices that consider human visual sensitivity, such as mean structural similarity (MSSIM) with structural sensitivity, visual information fidelity (VIF) with statistical sensitivity, etc., were proposed in view of the differences between reference and distortion frames on a pixel or local level. However, they ignore the role of human visual attention (HVA). Recently, some new strategies with HVA have been proposed, but the methods extracting the visual attention are too complex for real-time realization. We take advantage of the phase spectrum of quaternion Fourier transform (PQFT), a very fast algorithm we previously proposed, to extract saliency maps of color images or videos. Then we propose saliency-based methods for both image QA (IQA) and video QA (VQA) by adding weights related to saliency features to these original IQA or VQA criteria. Experimental results show that our saliency-based strategy can approach more closely to human subjective assessment compared with these original IQA or VQA methods and does not take more time because of the fast PQFT algorithm.