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In a typical communication pipeline, images undergo a series of processing steps that can cause visual distortions before being viewed. Given a high quality reference image, a reference (R) image quality assessment (IQA) algorithm can be applied after compression or transmission. However, the assumption of a high quality reference image is often not fulfilled in practice, thus contributing to less accurate quality predictions when using stand-alone R IQA models. This is particularly common on social media, where hundreds of billions of user-generated photos and videos containing diverse, mixed distortions are uploaded, compressed, and shared annually on sites like Facebook, YouTube, and Snapchat. The qualities of the pictures that are uploaded to these sites vary over a very wide range. While this is an extremely common situation, the problem of assessing the qualities of compressed images against their pre-compressed, but often severely distorted (reference) pictures has been little studied. Towards ameliorating this problem, we propose a novel two-step image quality prediction concept that combines NR with R quality measurements. Applying a first stage of NR IQA to determine the possibly degraded quality of the source image yields information that can be used to quality-modulate the R prediction to improve its accuracy. We devise a simple and efficient weighted product model of R and NR stages, which combines a pre-compression NR measurement with a post-compression R measurement. This first-of-a-kind two-step approach produces more reliable objective prediction scores. We also constructed a new, first-of-a-kind dedicated database specialized for the design and testing of two-step IQA models. Using this new resource, we show that two-step approaches yield outstanding performance when applied to compressed images whose original, pre-compression quality covers a wide range of realistic distortion types and severities. The two-step concept is versatile as it can use a...