Virtual Reality (VR), which brings immersive experiences to viewers, has been gaining popularity in recent years. A key feature in VR systems is the use of omnidirectional content, which provides 360-degree views of scenes. In this work, we study the human quality perception of omnidirectional images, focusing on different zones surrounding the foveation point. For that purpose, an extensive subjective experiment is carried out to assess the perceptual quality of omnidirectional images with non-uniform quality. Through experimental results, the impacts of different zones are analyzed. Moreover, nineteen objective quality metrics, including foveal quality metrics, are evaluated using our database. It is quantitatively shown that the zones corresponding to the fovea and parafovea of human eyes are extremely important for quality perception, while the impacts of the other zones corresponding to the perifovea and periphery are small. Besides, the investigated metrics are found to be not effective enough to reflect the quality perceived by viewers.