  360$^\circ$ videos have received widespread attention due to its realistic
and immersive experiences for users. To date, how to accurately model the user
perceptions on 360$^\circ$ display is still a challenging issue. In this paper,
we exploit the visual characteristics of 360$^\circ$ projection and display and
extend the popular just noticeable difference (JND) model to spherical JND
(SJND). First, we propose a quantitative 2D-JND model by jointly considering
spatial contrast sensitivity, luminance adaptation and texture masking effect.
In particular, our model introduces an entropy-based region classification and
utilizes different parameters for different types of regions for better
modeling performance. Second, we extend our 2D-JND model to SJND by jointly
exploiting latitude projection and field of view during 360$^\circ$ display.
With this operation, SJND reflects both the characteristics of human vision
system and the 360$^\circ$ display. Third, our SJND model is more consistent
with user perceptions during subjective test and also shows more tolerance in
distortions with fewer bit rates during 360$^\circ$ video compression. To
further examine the effectiveness of our SJND model, we embed it in Versatile
Video Coding (VVC) compression. Compared with the state-of-the-arts, our
SJND-VVC framework significantly reduced the bit rate with negligible loss in
visual quality.
