Uniqueness theorems for recovering the inverse image under~degenerate transformations

A. A. Klyachin ; V. A. Klyachin

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 22 (2022) no. 1, pp. 15-27.

Résumé

When solving problems of three-dimensional reconstruction of objects from images, the problem of determining the conditions under which such a reconstruction will have one or another degree of uniqueness is relevant. It is these conditions that make it possible to apply, in particular, deep machine learning methods using convolutional neural networks to determine the spatial orientation of objects or their constituent parts. From a mathematical point of view, the problem is reduced to determining the conditions for restoring the preimage for transforming the projection. In this article, we prove a number of uniqueness theorems for this kind of restoration. In particular, it has been proved that the parameters of a rotation transformation close to identical can be uniquely determined from the projection of the result of such rotation of an object with a given structure. In addition, the article found the conditions under which the spatial orientation of an object can be calculated from its projection.

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