Cone-beam projection

For the maximum use of advantages of a cone beam at collecting of data, it is necessary to study and develop effective algorithms of reconstmction for cone-beam projection data. The theory of reconstruction in cone beams is referred to as cone-beam tomography. It had developments in papers of many researchers. ... 

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. 

Another efficient and practical method for exact 3D-reconstruction is the Grangeat algorithm [11]. First the derivative of the three-dimensional Radon transfomi is computed from the Cone-Beam projections. Afterwards the 3D-Object is reconstructed from the derivative of the Radon transform. At present time this method is not available for spiral orbits, instead two perpendicular circular trajectories are suitable to meet the above sufficiency condition. 

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

In Dynamic Spatial Reconstructor at the expense of use 2D matrix of detectors there was the opportunity to use a divergent cone beam of source emission. This system had a number of lacks. In particular the number of projections is rigidly limited by the number of x-ray sources. The dispersion of source emission results in errors of data collected.. However the system confirmed basic advantages of application of conic beams and 2D matrices of detectors for collecting information about 3D object. 

The systems of such type have been developed of all last 10 years. We shall bring some characteristics of one of the last development within the framework of European BRITE project, carried out in LETT This 3D cone-beam tomograph is referred to as EVA Bench or Equipment for Voludensimetry Analysis. It is oriented on NDT of industrial products from ceramics and other composites. One of the main task of this tomograph is achievement of high resolution at study of whole internal volume of researched object. For test sample of the size 10mm spatial resolution in 50mm was obtained [14]. 

It is beyond the scope of this chapter to discuss filtered back-projection (FBP) reconstruction procedures or their 3-D extensions to cone-beam CT for which reference should be made to other works [39],... 

Here A is the average wavelength of the used radiation. The reconstruction algorithm is based on FBP. Filtering is carried out in the same manner as in 3-D (cone-beam) transmission CT, and back-projection is performed along curved trajectories. Details of the FBP algorithm can be found elsewhere [40],... 

The Cone Beam Reconstruction. With a cone beam of x-rays, a projection is formed by the illumination of a fixed area of detector cells (Fig. 26.24). A common detector structure in this respect is the equally spaced collinear cell array. The projection data for this geometry is represented by the function R (po, qo). where j3 is the source angle, the horizontal position, and q the vertical position, on the detector plane. 

The system is ideal for examining the rearrangement of microstractural composition of soft-solid materials with variation in temperature. An example of a material that is practically impossible to image in the natural state by conventional optical microscopy is shown in (Fig. 26.58a). The volumes depicted are child volumes, containing a region of interest (ROl), extracted fipom the reconstruction of a frozen four-phase soft-solid structure. The reconstructed volume is derived from 720 filtered and back-project frames using the Feldkamp cone-beam algorithm, with a resolution of 512 X 512 pixels per frame. 

Anoop KP, Rajgopal K (2007) Estimation of missing data using windowed linear prediction in laterally truncated projections in cone-beam CT. Conf Proc IEEE Eng Med Biol Soc 2007 2903-2906... 

Early spiral-CT scanners (34) provided stacks of cross-sectional CT sections. However, the slices were separated in time. The DSR acquired all the slices simultaneously but cone-beam reconstructions required a prohibitive amount of computation time. Tomography slices in medical CT have, until recently, been generated from one-dimensional projections via fan-beam methods. 

Figure 6.9 Radiographic projections or shadow images of a mouse acquired with a cone-beam micro-CT at two different angles.

Colijn A P, Beekman F J (2004) Accelerated simulation of cone beam x-ray scatter projections IEEE Trans. Med.Imaging 23 584-590. 

Malusek A, Sandborg M P et al. (2003) Simulation of scatter in cone beam CT effects on projection image quality SPIE Medical Imaging 2003 Physics of Medical Imaging (San Diego, CA, USA) SPIE vol 5030 pp 740-751. 

A transmission Laue pattern is made of a cubic crystal in the orientation of Prob. 3-1. By means of a stereographic projection similar to Fig. 3-8, show that the beams diffracted by the planes (2T0), (2T3), and (211), all of which belong to the zone [l20], lie on the surface of a cone whose axis is the zone axis. What is the angle between the zone axis and the transmitted beam ... 

Generally the detectors are set up perpendicular to the primary beam, with the intersection of the primary beam at the detector centre. This setting has some advantages the entire Bragg cones are detected and the deviation of the cone projection from an ideal circle is usually small. Sometimes a detector can be placed off-centre and non-orthogonally to the primary beam. This can enlarge the detectable q-space in a very cost effective manner. The downsides are the strongly elliptical conical projections and the loss of the entire azimuthal information of a diffraction cone. 

According to the Laue equation a S = h, the projection of S onto a is qual to hfa. The locus of all the vectors S which satisfy this equation is a series/of planes perpendicular to a and separated by 1/a. These planes make up the reciprocal space of a one-dimensional crystal. Their intersections with the Ewald sphere of radius 1/A (Fig. 3.20) define the directions s of the diffracted beams. This results in a series of coaxial cones around the a axis. The angle between the incident wave Sq and a is ocq. The half-opening angles a of the cones are obtained from the Laue equation,... 

A Debye-Scherrer camera consists of a metal cylinder provided with a photographic film. The primary beam is perpendicular to its axis. The distance between two symmetrical lines, produced by the intersection of a cone with the cylinder, is 46R, 6 being the Bragg angle (in radians) and R the radius of the camera. The interval is derived from Bragg s law. The powder method gives us only the norms of the reciprocal vectors. The set of norms corresponds to the projection of the reciprocal lattice onto a straight line. 

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