In radio astronomy, visibility data, which are measurements of wave signals from radio telescopes, are transformed into images for observation of distant celestial objects. However, these resultant images usually contain both real sources and artifacts, due to signal sparsity and other factors. One way to obtain cleaner images is to reconstruct samples into dense forms before imaging. Unfortunately, existing reconstruction methods often miss some components of visibility in frequency domain, so blurred object edges and persistent artifacts remain in the images. Furthermore, the computation overhead is high on irregular visibility samples due to the data skew. To address these problems, we propose PolarRec, a transformer-encoder-conditioned reconstruction pipeline with visibility samples converted into the polar coordinate system. This coordinate system matches the way in which radio telescopes observe a celestial area as the Earth rotates. As a result, visibility samples distribute in the polar system more uniformly than in the Cartesian space. Therefore, we propose to use radial distance in the loss function, to help reconstruct complete visibility effectively. Also, we group visibility samples by their polar angles and propose a group-based encoding scheme to improve the efficiency. Our experiments demonstrate that PolarRec markedly improves imaging results by faithfully reconstructing all frequency components in the visibility domain while significantly reducing the computation cost in visibility data encoding. The code is available at https://github.com/RapidsAtHKUST/PolarRec.