Projection reconstruction method

Vosegaard and Massiot [144] showed that it is possible to create a high-resolution 2D spectrum correlating the chemical shifts with the second-order quadrupolar lineshapes from several spectra recorded at different magnetic field strengths using a projection-reconstruction method called chemical shift-quadrupolar projection-reconstruction of one-dimensional spectra (CQ-PRODI). 

The projection-reconstruction approach is a technique unrelated to covariance processing which can provide data typically inaccessible to the natural product chemist. For example, 13C-15N correlation spectra were obtained for vitamin B12 at natural abundance.104 Compared with a conventional three-dimensional 13C-15N correlation experiment, the projection-reconstruction method provides a sensitivity enhancement of two orders of magnitude. The final 13C-15N spectrum was reconstructed from data obtained from ll l5N and H- C correlations acquired using a time-shared evolution pulse sequence that allowed all the information to be obtained in one experiment.104 Martin and co-workers also demonstrated the ability to generate 13C-15N correlation spectra using unsymmetrical indirect covariance NMR with vinblastine as an example.105 In the latter case, 13C-15N correlation spectra were obtained from - C HSQC data and H-1sN HMBC data that were acquired separately. Both methods provide access to correlations that would be inaccessible for most natural products at natural abundance. 

Radial sampling results when the incrementation of evolution times is coupled, and is the approach employed by GFT, RD, and back-project reconstruction methods. Radial sampling has also found application in MRl. When a fully-dimensional spectrum is computed from a set of radial samples (e.g., BPR, radial FT, MaxEnt), the radial sampling vectors are typically chosen to somewhat uniformly span the orientations from 0° to 90°. When the fuUy-dimensional spectmm is not reconstmcted,... 

Projection reconstruction methods Important image reconstruction procedures are ... 

Smith B.D. Image reconstruction from cone-beam projections necessary and sufficient conditions and reconstruction methods., IEEE Trans. Med. Imaging, V. 4, 1985, p. 14-28. 

Figure Bl.17.10. Principles of 3D reconstruction methods, (a) Principle of single axis tomography a particle is projected from different angles to record correspondmg images (left panel) this is most easily realized in the case of a helical complex (right panel), (b) Principle of data processing and data merging to obtain a complete 3D structure from a set of projections.

Some "fast methods" such as single-scan experiments,93 94 Hadamard95 and projection-reconstruction techniques96 take a few seconds to produce 2D spectra, but suffer from a lack of spectral resolution. As mentioned... 

The projected potential can be reconstructed. This reconstruction method is demonstrated in Fig. 10 on HREM images of K20-7Nb205. This method is especially powerful for crystals with high symmetries. 

Here the principles of constructing a 3D structure model from several HREM images of projections of inorganic crystals will be presented. Some of the principles may also be applied to non-periodic objects. A complex quasicrystal approximant v-AlCrFe is used as an example (Zou et al., 2003). Procedures for ab initio structure determination by 3D reconstruction are described in detail. The software CRISP, ELD. Triple and 3D-Map are used for 3D reconstruction. The 3D reconstruction method was demonstrated on the silicate mineral (Wenk et al. 1992). It was also applied to solve the 3D structures of a series mesoporous materials (Keneda etal. 2002). 

As we can see, the problem can be given a precise formulation, but what really counts is that it can also be given a solution. I have demonstrated that structures can indeed be reconstructed by using only 10% of the minimum number of projections (Barbieri, 1974a, 1974b, 1987), and an iterative algorithm which exploits memory matrices. More precisely, a reconstruction from incomplete projections is possible if two conditions are met (1) if the reconstruction method employs memory matrices where new information appears, and (2) if the reconstruction method employs codes, or conventions, which transfer information from the memory space to the real space. 

Because the reconstruction problem is ill-posed, the solution or reconstruction method must be tuned to the problem at hand as defined by the character of n x,y) and the number and distribution over s and 9 of the available projections p si,9j). 

The first NUS approaches have used experimentally weighted random sampling methods and variations of random sampling. However, other NUS methods, such as radial sampling with projection reconstruction, have been proposed (see introduction). Different algorithms have been used for reconstruction of NUS data, such as Maximum Entropy or Maximum Likelihood Methods. It is beyond the scope of this chapter to compare exhaustively the different approaches. Here we use just the FM reconstruction software to compare the performance of different random sampling schedules where the randomness is skewed by weighting functions. 

B. D. Smith, Image Reconstruction From Cone-Beam Projections—Necessary and Sufficient Conditions and Reconstruction Methods, IEEE Trans. Med. Imag., 4(1), March 1985. 

Figure 1. Schematic diagram of an ap and a lateral angiocardiographic projection. The videoline n is taken as an example to demonstrate the information to be gained from a biplane angiocardiogram by videometry and videodensitometry. The true area and shape of the real cross-section remain unknown unless e.g. a cast is taken and sliced mechanically, or by using tomographic reconstruction methods.

The projection reconstruction (PR) method originates from imaging techniques [43, 44] and is based on the projection theorem [45, 46]. The technique has recently attracted considerable interest in multi-dimensional NMR spectroscopy [47-49], especially in the field of protein NMR [2—4,17-19]. It is a particularly efficient time saving technique that works amazingly well with sparse spectra and medium size proteins. The technique has been applied in combination with the 2D/3D (HA)CA CO N NH experiment to a small protein, GBl [36]. The three-dimensional experiment was recorded in just 15 min. The three-dimensional spectrum was reconstructed from three planes - two orthogonal planes and a tilted plane recorded at an optimum projection angle of 69.5°. Figure 21 shows the comparison of the... 

To derive projection reconstruction using planar-integral data, let us first direct our attention to the FB method. Based on the FB method, recovery of the original volume image function fix, y, z) can be achieved through the inverse Fourier transform of Eq. (23),... 

NMR tomographic images can be formed by direct mapping, projection reconstruction, or Fourier imaging. To date, two kinds of potentially useful imaging methods—direct Fourier imaging and projection reconstruction—are most widely used. 

Two main types of imaging methods—direct Fourier imaging (Kumar-Welti-Emst method) and projection reconstruction—and the possibility of parameter imaging will be discussed in detail here. Hardware configurations and the related statistical aspects of image quality and imaging times will also be discussed briefly. 

The classical computer tomography (CT), including the medical one, has already been demonstrated its efficiency in many practical applications. At the same time, the request of the all-round survey of the object, which is usually unattainable, makes it important to find alternative approaches with less rigid restrictions to the number of projections and accessible views for observation. In the last time, it was understood that one effective way to withstand the extreme lack of data is to introduce a priori knowledge based upon classical inverse theory (including Maximum Entropy Method (MEM)) of the solution of ill-posed problems [1-6]. As shown in [6] for objects with binary structure, the necessary number of projections to get the quality of image restoration compared to that of CT using multistep reconstruction (MSR) method did not exceed seven and eould be reduced even further. 

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