2015/07/01 by Irina A. Kogan, Peter J. Olver
Computer Science · Mathematics · Medicine · #Algebra over a field #Artificial intelligence #Computer science #Crystallography #Differential (mechanical device) #Group (periodic table) #Image (mathematics) #Isomorphism (crystallography) #Mathematics #Medical Image Segmentation Techniques #Medical Imaging Techniques and Applications #Physics #Pure mathematics #Space (punctuation) #Surjective function #Topological and Geometric Data Analysis #Transformation (genetics) #acm:14H50 #acm:14L24 #acm:53A55 #acm:68T45 #cs.CV #math.DG #msc:14H50 #msc:14L24 #msc:53A55 #msc:68T45
paper · pdf · doi:10.1134/s1995080215030063
published as Lobachevskii J. Math. 36 (2015), 260--285 · This paper includes corrections and additions to the published version
openalex publication_date 2015/07/01 · arxiv created 2015/09/22 · arxiv updated 2015/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We examine the relationships between the differential invariants of objects and of their images under a surjective maps. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the image, and the case when only a proper subgroup of the entire group acts projectably. In the former case, we establish a constructible isomorphism between the algebra of differential invariants of the images and the algebra of fiber-wise constant (gauge) differential invariants of the objects. In the latter case, we describe residual effects of the full transformation group on the image invariants. Our motivation comes from the problem of reconstruction of an object from multiple-view images, with central and parallel projections of curves from three-dimensional space to the two-dimensional plane serving as our main examples.