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We examine the relationships between the differential invariants of objectsand of their images under a surjective map. We analyze both the case when theunderlying transformation group is projectable and hence induces an action onthe image, and the case when only a proper subgroup of the entire group actsprojectably. In the former case, we establish a constructible isomorphismbetween the algebra of differential invariants of the images and the algebra offiber-wise constant (gauge) differential invariants of the objects. In thelatter case, we describe residual effects of the full transformation group onthe image invariants. Our motivation comes from the problem of reconstructionof an object from multiple-view images, with central and parallel projectionsof curves from three-dimensional space to the two-dimensional plane serving asour main examples.

Invariants of objects and their images under surjective maps