Radon transforms, alignment, and 3d-reconstruction from random projections

1996 ◽  
Vol 54 ◽  
pp. 588-589
Author(s):  
Michael Radermacher
Keyword(s):  
Radon Transform ◽  
Fourier Transforms ◽  
Three Dimensional ◽  
Radon Transforms ◽  
One Dimensional ◽  
Dimensional Reconstruction ◽  
Section Theorem ◽  
Common Lines ◽  
The Common ◽  
Polar Grid

Since their inception three-dimensional reconstruction techniques have been based on the theory of Radon transforms. Only much later have Radon transforms been recognized as powerful tools for image processing and pattern recognition. Techniques like the “common lines ” technique for finding the orientation of projections of highly symmetrical particles, which had been developed using Fourier transforms, can easily be translated into a technique that uses Radon transforms. In contrast to Fourier transforms Radon transforms have the advantage of being real valued which simplifies many interpolation steps. The central section theorem known for Fourier transforms also applies to Radon transforms. The two- or three-dimensional Fourier transform on a polar grid can be obtained from the two- or three-dimensional Radon transform by a one-dimensional (radial) Fourier transforms and vice versa. Fourier transforms obtained by a one-dimensional transformation of the Radon transform will be referred to a Fourier/Radon transforms.

scholarly journals Fast and Robust Orientation of Cryo-Electron Microscopy Images

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Guoliang Xu ◽  
Xia Wang ◽  
Ming Li ◽  
Zhucui Jing
Keyword(s):  
Sparse Matrix ◽  
Three Dimensional ◽  
Image Plane ◽  
Real Space ◽  
Computation Method ◽  
Common Line ◽  
The Matrix ◽  
Common Lines ◽  
The Common ◽  
Dimensional Object

AbstractWe present an efficient and reliable algorithm for determining the orientations of noisy images obtained fromprojections of a three-dimensional object. Based on the linear relationship among the common line vectors in one image plane, we construct a sparse matrix, and show that the coordinates of the common line vectors are the eigenvectors of the matrix with respect to the eigenvalue 1. The projection directions and in-plane rotation angles can be determined fromthese coordinates. A robust computation method of common lines in the real space using aweighted cross-correlation function is proposed to increase the robustness of the algorithm against the noise. A small number of good leading images, which have the maximal dissimilarity, are used to increase the reliability of orientations and improve the efficiency for determining the orientations of all the images. Numerical experiments show that the proposed algorithm is effective and efficient.

Three-Dimensional reconstructions of ’light’ and ’intermediate’ capsids of equine herpes virus

1989 ◽  
Vol 47 ◽  
pp. 822-823
Author(s):  
T. S. Baker ◽  
W. W. Newcomb ◽  
F. P. Booy ◽  
J. C. Brown ◽  
A. C. Steven
Keyword(s):  
Herpes Virus ◽  
Carbon Film ◽  
Three Dimensional ◽  
Viral Envelope ◽  
Step Size ◽  
Total Particle ◽  
Dimensional Reconstruction ◽  
Common Lines ◽  
Herpès Virus ◽  
Morphological Species

Equine herpes virus type 1 (EHV-1) belongs to an extensive family of large, genetically complex, and medically important animal viruses. The virion consists of an icosahedral nucleocapsid (T=16) separated from the viral envelope by a proteinaceous tegument layer. Assembly occurs in the nucleus of infected cells where capsids assemble, are packaged with DNA, then bud through the nuclear membrane. Two morphological species of EHV-1 capsids have been distinguished: “lights” which are abortive particles incapable of packaging DNA and “intermediates” which are precursors in the assembly of mature virions. Purified “intermediates” contain an additional proteinP22 (46 kDa), which is not present in “lights” and accounts for ˜10% of the total particle mass. In order to characterize the capsid structures of the two particle types and explore differences between them, we have applied three-dimensional reconstruction techniques to electron micrographs of frozen-hydrated specimens.The Kentucky A strain of EHV-1 was propagated in L-929 cells, and the two types of capsids were obtained as separate fractions from Renografin-76 density gradients. Neither fraction contained any significant amount of DNA. Cryo-electron microscopy of capsids on carbon film substrates was performed as described earlier. Micrographs were recorded at a nominal magnification of x36,000 at 2-3μm underfocus and digitized with a 50/μn step size (˜1.38nm sampling at the specimen). Three-dimensional reconstructions of the two particle types were computed to 4.5nm resolution using modified “common-lines” procedures. Images of 37 “light” and 39 “intermediate” particles were separately combined to compute three-dimensional density distributions.

Exact solutions of the three-dimensional scalar wave equation and Maxwell's equations from the approximate solutions in the wave zone through the use of the Radon transform

1989 ◽  
Vol 422 (1863) ◽  
pp. 351-365 ◽  
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Radon Transform ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Three Dimensional ◽  
Asymptotic Solutions ◽  
Scalar Wave ◽  
Wave Zone ◽  
One Dimensional

In our earlier paper we have shown that the solutions of both the three-dimensional scalar wave equation, which is also the three-dimensional acoustic equation, and Maxwell’s equations have forms in the wave zone, which, except for a factor 1/ r , represent one-dimensional wave motions along straight lines through the origin. We also showed that it is possible to reconstruct the exact solutions from the asymptotic forms. Thus we could prescribe the solutions in the wave zone and obtain the exact solutions that would lead to them. In the present paper we show how the exact solutions can be obtained from the asymptotic solutions and conversely, through the use of a refined Radon transform, which we introduced in a previous paper. We have thus obtained a way of obtaining the exact three-dimensional solutions from the essentially one-dimensional solutions of the asymp­totic form entirely in terms of transforms. This is an alternative way to obtaining exact solutions in terms of initial values through the use of Riemann functions. The exact solutions that we obtain through the use of the Radon transform are causal and therefore physical solutions. That is, these solutions for time t > 0 could have been obtained from the initial value problem by prescribing the solution and its time-derivative, in the acoustic case, and the electric and magnetic fields, in the case of Maxwell’s equations, at time t = 0. The role of time in the relation between the exact solutions and in the asymptotic solutions is made very explicit in the present paper.

Three-parameter Radon transform based on shifted hyperbolas

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. V39-V48 ◽  
Author(s):  
Ali Gholami ◽  
Toktam Zand
Keyword(s):  
Radon Transform ◽  
Seismic Data ◽  
High Performance ◽  
Fourier Transforms ◽  
Real Data ◽  
Data Sets ◽  
Radon Transforms ◽  
Slice Theorem ◽  
Ground Roll ◽  
Zero Offset

The focusing power of the conventional hyperbolic Radon transform decreases for long-offset seismic data due to the nonhyperbolic behavior of moveout curves at far offsets. Furthermore, conventional Radon transforms are ineffective for processing data sets containing events of different shapes. The shifted hyperbola is a flexible three-parameter (zero-offset traveltime, slowness, and focusing-depth) function, which is capable of generating linear and hyperbolic shapes and improves the accuracy of the seismic traveltime approximation at far offsets. Radon transform based on shifted hyperbolas thus improves the focus of seismic events in the transform domain. We have developed a new method for effective decomposition of seismic data by using such three-parameter Radon transform. A very fast algorithm is constructed for high-resolution calculations of the new Radon transform using the recently proposed generalized Fourier slice theorem (GFST). The GFST establishes an analytic expression between the [Formula: see text] coefficients of the data and the [Formula: see text] coefficients of its Radon transform, with which a very fast switching between the model and data spaces is possible by means of interpolation procedures and fast Fourier transforms. High performance of the new algorithm is demonstrated on synthetic and real data sets for trace interpolation and linear (ground roll) noise attenuation.

Three-Dimensional Electron Microscopy of Individual Biological Objects

1976 ◽  
Vol 31 (6) ◽  
pp. 645-655 ◽  
Author(s):  
W. Hoppe ◽  
H. J. Schramm ◽  
M. Sturm ◽  
N. Hunsmann ◽  
J. Gaßmann
Keyword(s):  
Electron Microscopy ◽  
Three Dimensional ◽  
Image Point ◽  
Two Dimensional ◽  
Dimensional Electron ◽  
Biological Objects ◽  
Dimensional Reconstruction ◽  
The Common ◽  
Measuring Scheme

In this paper methods and results of three-dimensional electron microscopy of individual molecules will be presented. Part I describes the general experimental and theoretical methods (microgoniometer, measuring scheme, two-dimensional and three-dimensional reconstruction, determination of the common origin of the projections). Special attention will be given to the image point shapes under different reconstruction conditions

scholarly journals Green's function for anisotropic dispersive poroelastic media based on the Radon transform and eigenvector diagonalization

2019 ◽  
Vol 475 (2221) ◽  
pp. 20180610 ◽  
Author(s):  
Qiwei Zhan ◽  
Mingwei Zhuang ◽  
Yuan Fang ◽  
Jian-Guo Liu ◽  
Qing Huo Liu
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Radon Transform ◽  
Hyperbolic Equations ◽  
Three Dimensional ◽  
Constitutive Relationship ◽  
Dimensional System ◽  
Characteristic Analysis ◽  
One Dimensional ◽  
Poroelastic Media

A compact Green's function for general dispersive anisotropic poroelastic media in a full-frequency regime is presented for the first time. First, starting in a frequency domain, the anisotropic dispersion is exactly incorporated into the constitutive relationship, thus avoiding fractional derivatives in a time domain. Then, based on the Radon transform, the original three-dimensional differential equation is effectively reduced to a one-dimensional system in space. Furthermore, inspired by the strategy adopted in the characteristic analysis of hyperbolic equations, the eigenvector diagonalization method is applied to decouple the one-dimensional vector problem into several independent scalar equations. Consequently, the fundamental solutions are easily obtained. A further derivation shows that Green's function can be decomposed into circumferential and spherical integrals, corresponding to static and transient responses, respectively. The procedures shown in this study are also compatible with other pertinent multi-physics coupling problems, such as piezoelectric, magneto-electro-elastic and thermo-elastic materials. Finally, the verifications and validations with existing analytical solutions and numerical solvers corroborate the correctness of the proposed Green's function.

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