Reconstruction from projections

The first point has not been clearly recognized or appreciated in the early days of the method, so terms like radon transformation aad filtered backprojection were introduced [Herl, Manl], In practical realizations of image reconstruction from projections, however, numerical frers must, indeed, be used [Herl]. 

Use of the expression backprojection instead of reconstruction from projections is historical. Given a sufficient number of projections of an object acquired at different angles, the shape of the object can indeed be reconstructed with recognizable features, if the projections are just smnmed over the image plane in the directions over which 

If the magnetic-field gradient is applied in direction r in polar coordinates, which forms an angle p with the A -axis of a Cartesian reference frame, the projection P (r, p) is obtained according to (5.4.15) by integration of the spin density Mo(r, 5) along direction s, which is perpendicular to r (cf. Fig. 5.4.1), 

The backprojection approach uses straightforward addition of the projections acquired at different angles p (cf. Fig. 6.1.2). By taking as a continuous instead of a discrete variable for simplicity, the backprojection image Mq is obtained by integration over p, 

This means that each point of a given projection is added to all pixels in direction s perpendicular to r. The simple sum image of the circular disc obtained in this way demonstrates that all projections produce a desired signal increase in the centre at the position of the disc, but image is distorted in a way reminiscent of the rays of a star. These distortions can be eliminated by filtering the projections prior to summation. This approach has 

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Image reconstruction from projections is a basic process in computerized tomography (CT)... 

Finally, it is shown in terms of the presented example that the proposed adaptive reconstruction algorithm is valuable for image reconstruction from projections without any prior information even in the case of noisy data. The number of required projections can be determined by investigating the convergence properties of the reconstruction algorithm. 

G. T. Herman, Image Reconstruction from Projections, Academic Press, NY, 1980... 

Herman G.T. Image reconstruction from projections.. Computer Science and Applied Mathematics, New York Academic, 1980... 

Barbieri, M. 1974b. A criterion to evaluate three-dimensional reconstructions from projections of unknown structures. Journal of Theoretical Biology, 48,451-467. 

Gordon, R., and Herman, G.T. 1974. Three-dimensional reconstruction from projections a review of algorithms. International Review of Cytology, 38,111-151. 

Herman GT (1980) Image Reconstruction from Projection The Fundamentals of Computerized Tomography. New York, NY Academic Press. 

Fig. 6.1.4 Gradient paths for 3D reconstruction from projections. Only half a hemisphere is covered by the gradient paths, because signal for negative gradient values can be acquired by time inversion in echo techniques, (a) 3D space can be covered by a set of 2D projections, so that the 2D algorithm can be applied in two steps, (b) Optimization of the point density in 3D k space requires an integral approach to 3D reconstruction from projections. Adapted from [Lail] with permission from Institute of Physics.

The method of reconstruction from projections can be combined with spectroscopic resolution, so that an NMR spectrum can be assigned to each voxel. The spectroscopic information can be separated from the spatial information if a set of projections is acquired with different gradient strengths for each gradient orientation. Two approaches [Corl, Laul ] are considered below and illustrated for the case of a ID spin density with space coordinate r. 

The gradient is static in the sample ftame so that the rotating spin density leads to a time-independent signal. The phase tp is varied in small steps for imaging with reconstruction from projections, or it is switched by multiples of n/2 for Fourier imaging methods. 

Fig, 8.6.2 2D MARF images of compressed polycrystalline adamantane. Left MARF images obtained by reconstruction from projections. Right shapes of the objects and dimensions. Adapted from i Mar2] with permission from the author. 

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