Abstract
A new method for the three-dimensional reconstruction of a structure from projections of randomly oriented particles on a plane is proposed. Reconstruction is performed in two steps. First, we find mutual orientation of particles, i.e., Euler angles, describing the angle of one particle with respect to another. Almost all the paper is devoted to solving this problem. Then we perform the three-dimensional reconstruction of an object from its projections in already-known directions.
The stability of the method with respect to experimental errors is shown. Three-dimensional reconstruction of asymmetric biological objects might be one of its applications.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Radon, J.: Über die Bestimmung von Functionen durch ihre Integralwarte längs gewisser Mannigfaltigkeiten.Ber. Verh. Sächs. Acad. 69 (1917), 262–277.
Pikalov, V. V. and Preobrazhensky, N. G.: Computational tomography and physical experiment,Russian Phys. Surveys, 1983,141, No. 3 (1983), 469–498 (in Russian).
Wainstein, B. K.: Three-dimensional electron microscopy of biology micromolecules,Russian Phys. Surveys, 1973,109, No. 3 (1973) 455–497 (in Russian).
Kam, Z.: Three-dimensional reconstruction of aperiodic objects,J. Theoret. Biol. 82 (1980), 15–32.
Gelfand, I. M., Graev, M. J., and Vilenkin, N. Ya.:Integral Geometry and Related Problems of Representation Theory., Fizmatgiz, Moscow, 1962 (in Russian).
Gelfand, I. M., Graev, M. J., and Shapiro, Z. Ya.: Integral geometry onk-dimensional planes,Funct. Anal. Appl. No. 1 (1967), 15–31 (in Russian).
Gelfand, I. M., Gindikin, S. G., and Graey, M. J.:Integral Geometry in Affine and Projective Spaces. Results of Science and Technics (Modern Problems of Mathematics, v. 16), VINITI, Moscow, 1980 (in Russian).
Gelfand, I. M., Minlos, R. A., and Shapiro, Z. Ya.:Representations of the Rotation Group and the Loretz Group, Fizmatgiz, Moscow, 1958 (in Russian).
Wainstein, B. K. and Goncharov, A. B.: Recording of the spatial orientation of randomly arranged identical particles of unknown structure from their projections,Doklady USSR Acad. Sci. (1986) (in Russian).
Goncharov, A. B.: Methods of integral geometry and finding the mutual orientation of randomly arranged particles on the plane from their projections on the line,Doklady USSR Acad. Sci. (1986) (in Russian).
Goncharov, A. B.: Methods of integral geometry and recovering a function with compact support from its projections in unknown directions.Acta Applic. Math. 11 (1988), 213–222.
Goncharov, A. B., Wainstein, B. K., Ryskin, A. I., Vagin, A. A.: preprint.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Goncharov, A.B. Integral geometry and three-dimensional reconstruction of randomly oriented identical particles from their electron microphotos. Acta Applicandae Mathematicae 11, 199–211 (1988). https://doi.org/10.1007/BF00140118
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00140118