Algebraic reconstruction

Greenleaf, J.F. Johnson S.A. Samayoa, W.F. and Duck, F.A. (1975). Algebraic reconstruction of spatial distributions of acoustic velocities in tissue from their time-of-flight profiles. In Acoustical Holography, Vol. 6, Ed. N. Booth, Plenum Press, 71-90. 

Gordon R., Bender R., Herman G.T. Algebraic reconstruction techniques (ART) for three-dimensional electron micrographs and X-ray photography., J. Theor. Biol., V. 29, 1970, p. 471-481. 

Gordon, R., Bender, R. and Herman G.T. 1970. Algebraic Reconstruction Techniques (ART) for three-dimensional electron microscopy and X-ray photography Journal of Theoretical Biology, 29,471 481. 

It is useful to divide Wald s work into three main historical periods The first is devoted to thermodynamics, particularly entropy (approximately 1889-1893), the second to an algebraic reconstruction of stoichiometry on the basis of Gibbs phase rule (approximately 1893-1907), and the third to a polydimensional deduction of chemical variety on the basis of the notion of phase (approximately 1907-1930). 

Algebraic reconstruction technique (ART) Maximum likelihood reconstruction (MLR) Direct Fourier reconstruction (DFR)... 

Algebraic Reconstruction Teohnique and Maximum Likelihood Reoonstruction... 

The algebraic methods of reconstruction give result at incomplete and complete set of initial projection data. But the iterative imhlementation of these methods requires large computing resources. Algebraic method can be used in cases, when the required accuracy is not great. 

Algebraically, the reconstruction of the values of X has been defined by the matrix product of the scores S with the transpose of the loadings L (eq. (31.22)). Geometrically, one reconstructs the value Xy by perpendicular projection of the point represented by upon the axis represented by s, as shown in Fig. 31.3c ... 

Because protein ROA spectra contain bands characteristic of loops and turns in addition to bands characteristic of secondary structure, they should provide information on the overall three-dimensional solution structure. We are developing a pattern recognition program, based on principal component analysis (PCA), to identify protein folds from ROA spectral band patterns (Blanch etal., 2002b). The method is similar to one developed for the determination of the structure of proteins from VCD (Pancoska etal., 1991) and UVCD (Venyaminov and Yang, 1996) spectra, but is expected to provide enhanced discrimination between different structural types since protein ROA spectra contain many more structure-sensitive bands than do either VCD or UVCD. From the ROA spectral data, the PCA program calculates a set of subspectra that serve as basis functions, the algebraic combination of which with appropriate expansion coefficients can be used to reconstruct any member of the... 

T. E. Oliphant, A. Manduca, R. L. Ehman and J. F. Greenleaf, Complex-valued stiffness reconstruction for magnetic resonance elastography by algebraic inversion of the differential equation, Magn. Reson. Med., 2001, 45, 299-310. 

Once the weighting factors are calculated, the reconstruction values are obtained by equations 3.3 with simple additions and multiplications. This classic algebraic method, known as matrix inversion, is rigorous and straightforward, but in practice it is employed... 

Matrix inversion is not widely used in practice, but from a theoretical point of view is extremely useful, because it allows us to calculate the minimum number of projections that are required for a complete reconstruction. If we have p projections of a structure, and each projection contains r rays, a reconstruction procedure amounts to solving a system of p-r equations in n2 unknowns, and algebra tells us that a solution exists only if the number of linearly independent equations is equal to the number of the unknowns. 

It is important to notice that, in real-life applications, the actual number of projections must always be greater (often much greater) than the theoretical minimum, because of the need to compensate the inevitable loss of information which is produced by various types of noise. It is also important to notice that the theoretical minimum obtained with non-algebraic methods (Crowther et al., 1970) is never inferior to the algebraic minimum. Equation 3.5, in other words, is the lowest possible estimate of the minimum number of projections that are required for a complete reconstruction of any given structure. 

In this paper, an inverse problem for galvanic corrosion in two-dimensional Laplace s equation was studied. The considered problem deals with experimental measurements on electric potential, where due to lack of data, numerical integration is impossible. The problem is reduced to the determination of unknown complex coefficients of approximating functions, which are related to the known potential and unknown current density. By employing continuity of those functions along subdomain interfaces and using condition equations for known data leads to over-determined system of linear algebraic equations which are subjected to experimental errors. Reconstruction of current density is unique. The reconstruction contains one free additive parameter which does not affect current density. The method is useful in situations where limited data on electric potential are provided. 

The first step can be addressed from a purely group theoretical viewpoint [i.e., using formulas such as (4.12)]. In lieu of the simple conversion laws for vibrational quantum numbers [Eqs. (4.31) and (4.53)], it is easier to directly reconstruct local vibrational basis states rather than purely algebraic ones. The routine for generating a convenient vibrational basis will thus be given by a simple code for the calculation of integer partitions of the total vibrational quantum number... 

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