How do they make the descriptor rotation-invariant? And how do they make the descriptor illumination invariant?
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How do they make the descriptor rotation-invariant?
This is explained by D. Lowe in the original SIFT paper (see Chapter 5. Orientation Assignment):
How do they make the descriptor illumination invariant?
Here again this is detailed in this paper (see Chapter 6.1. Descriptor Representation):
According to the original paper, MOPS follows the same principles.