Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Klyachin A. A., Klyachin V. A. Uniqueness theorems for recovering the inverse image under degenerate transformations. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2022, vol. 22, iss. 1, pp. 15-27. DOI: 10.18500/1816-9791-2022-22-1-15-27, EDN: FIPUFI

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.03.2022
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Language: 
Russian
Heading: 
Mathematics
Article type: 
Article
UDC: 
514.17
DOI: 
10.18500/1816-9791-2022-22-1-15-27
EDN: 
FIPUFI

Uniqueness theorems for recovering the inverse image under degenerate transformations

Autors: 
Klyachin Alexey A., Volgograd State University
Klyachin Vladimir Aleksandrovich, Volgograd State University
Abstract: 

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.

Key words: 
set projection
rotation transform
spatial reconstruction
spatial orientation
Acknowledgments: 
This work was supported by the Ministry of Education and Science of Russia (project “Development of Virtual 3D Reconstruction of Historical Objects Technique”, scientific theme code 2019–0920, project number in the research management system FZUU-0633-2020-0004).
References: 
  1. Mousavian A., Anguelov D., Flynn J., & Kosecka J. 3D Bounding Box Estimation Using Deep Learning and Geometry. 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017, pp. 5632–5640. https://doi.org/10.1109/cvpr.2017.597
  2. Gordeev A. Y., Klyachin V. A. Determination of the Spatial Position of Cars on the Road Using Data from a Camera or DVR. In: E. G. Popkova, B. S. Sergi, eds. “Smart Technologies” for Society, State and Economy. ISC 2020. Lecture Notes in Networks and Systems, vol. 155. Cham, Springer, 2021, pp. 172–180. https://doi.org/10.1007/978-3-030-59126-7_20
  3. Gordeev A. Y., Klyachin V. A., Kurbanov E. R., Driaba A. Y. Autonomous Mobile Robot with AI Based on Jetson Nano. In: K. Arai, S. Kapoor, R. Bhatia, eds. Proceedings of the Future Technologies Conference (FTC), 2020, vol. 1. FTC 2020. Advances in Intelligent Systems and Computing. Vol. 1288. Cham, Springer, 2021, pp. 190–204. https://doi.org/10.1007/978-3-030-63128-4_15
  4. Ren M., Pokrovsky A., Yang B., Urtasun R. SBNet: Sparse Blocks Network for Fast Inference. 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2018, pp. 8711–8720. https://doi.org/10.1109/CVPR.2018.00908
  5. Hu H., Cai Q., Wang D., Lin J., Sun M., Krhenbhl P., Darrell T., Yu F. Joint Monocular 3D Vehicle Detection and Tracking. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. Seoul, Korea, 27 October – 2 November 2019, pp. 5390–5399.
  6. Huang S., Qi S., Zhu Y., Xiao Y., Xu Y., & Zhu S. C. Holistic 3d scene parsing and reconstruction from a single rgb image. In: Proceedings of the European Conference on Computer Vision (ECCV), 2018, pp. 187–203.
  7. Jackson A. S., Bulat A., Argyriou V., Tzimiropoulos G. Large pose 3D face reconstruction from a single image via direct volumetric CNN regression. 2017 IEEE International Conference on Computer Vision (ICCV). IEEE, 2017, pp. 1031–1039.
  8. Ferkova Z., Urbanova P., Cerny D., Zuzi M., Matula P. Age and gender-based human face reconstruction from single frontal image. Multimedia Tools and Applications, 2020, vol. 79, pp. 3217–3242. https://doi.org/10.1007/s11042-018-6869-5
  9. Klyachin V. A., Grigorieva E. G. Algorithm for automatic determination of camera orientation parameters in space based on the characteristic elements of the photograph. Tendencii razvitiya nauki i obrazovaniya, 2018, no. 45, pt. 6, pp. 10–20 (in Russian). https://doi.org/10.18411/lj-12-2018-125
  10. Klyachin V. A., Grigorieva E. G. A 3D reconstruction algorithm of a surface of revolution from its projection. Journal of Applied and Industrial Mathematics, 2020, vol. 14, pp. 85–91. https://doi.org/10.1134/S1990478920010093
  11. Kamyab S., Ghodsi A., Zohreh Azimifar S. Deep structure for end-to-end inverse rendering. 2017. ArXiv, abs/1708.08998.
  12. Kamyab S., Zohreh Azimifar S. End-to-end 3D shape inverse rendering of different classes of objects from a single input image. 2017. ArXiv, abs/1711.05858.
  13. Penczek P. A. Fundamentals of three-dimensional reconstruction from projections. Methods in Enzymology, 2010, no. 482, pp. 1–33. https://doi.org/10.1016/S0076-6879(10)82001-4
Received: 
26.12.2020
Accepted: 
07.10.2021
Published: 
31.03.2022
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