How do they make the descriptor rotation-invariant? And how do they make the descriptor illumination invariant?
How do they make the descriptor rotation-invariant?
This is explained by D. Lowe in the original SIFT paper (see Chapter 5. Orientation Assignment):
By assigning a consistent orientation to each keypoint based on local image properties, the keypoint descriptor can be represented relative to this orientation and therefore achieve invariance to image rotation.
How do they make the descriptor illumination invariant?
Here again this is detailed in this paper (see Chapter 6.1. Descriptor Representation):
[...] the feature vector is modified to reduce the effects of illumination change. First, the vector is normalized to unit length [...] we reduce the influence of large gradient magnitudes by thresholding the values [...] then renormalizing to unit length
According to the original paper, MOPS follows the same principles.