The Wide-field Imager for Solar PRobe (WISPR) obtained the first high-resolution images of coronal rays at heights below 15 R$_odot$ when the Parker Solar Probe (PSP) was located inside 0.25 au during the first encounter. We exploit these remarkable images to reveal the structure of coronal rays at scales that are not easily discernible in images taken from near 1 au. To analyze and interpret WISPR observations, which evolve rapidly both radially and longitudinally, we construct a latitude versus time map using the full WISPR dataset from the first encounter. From the exploitation of this map and also from sequential WISPR images, we show the presence of multiple substructures inside streamers and pseudostreamers. WISPR unveils the fine-scale structure of the densest part of streamer rays that we identify as the solar origin of the heliospheric plasma sheet typically measured in situ in the solar wind. We exploit 3D magnetohydrodynamic models, and we construct synthetic white-light images to study the origin of the coronal structures observed by WISPR. Overall, including the effect of the spacecraft relative motion toward the individual coronal structures, we can interpret several observed features by WISPR. Moreover, we relate some coronal rays to folds in the heliospheric current sheet that are unresolved from 1 au. Other rays appear to form as a result of the inherently inhomogeneous distribution of open magnetic flux tubes.
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