Cone beam reconstruction

Finch D. Cone-beam reconstruction with sources on a curve., SIAM J. Appl. Math., V. 45(4), 1985,p.665-673. 

Peyrin F. The generalized backprojection theorem for cone-beam reconstruction., IEEE Trans. Nucl. Sci., V. NS-32, 1985, p.1512-1519. 

Smith B.D., Chen J. Implementation, investigation and improvement of a novel cone-beam reconstruction method., IEEE Trans. Med. Imaging, V. 11, 1992, p. 260-266. 

Zhang Y, Chan HP, Sahiner B, et al (2006) A comparative study of limited-angle cone-beam reconstruction methods for breast tomosynthesis. Med Phys 33 3781-3795... 

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. 

To apply these relations in practice the cone beam reconstruction algorithm would involve the following arithmetic operations. 

The scanning schemes that have been adapted to the above principle may involve the spiral/ helical motion that requires lateral displacement with rotation. Extensions of the Feldkamp algorithm, for quite general three-dimensional scaiming loci, have been developed by Wang et al. For further discussions on the approximate and accurate cone-beam reconstruction the reader is referred to Ref. 17. 

The specimen motion required for cone-beam reconstruction is a lateral axes (x, z) translation and a vertical axis P rotation. The scanning cycle needs to be under computer control in order to synchronize mechanical movement with data acquisition. Further, the control must provide a level of accuracy that, at the very least, matches the measured resolution of the x-ray source. In practice, an encoded accuracy in lateral translation of 10,000 counts per mm and a rotational accuracy of 2000 counts per degree of revolution can be achieved with commercial components. 

Hein I, Taguchi K, Silver M D, Kazarna M, Mori I (2003) Feld-kamp-based cone-beam reconstruction for gantry-tilted helical multislice CT. Med Phys 30 3233-3242 Hsieh J (2001) Investigation of the slice sensitivity profile for step-and-shoot mode multi-slice computed tomography. Med Phys 28 491-500... 

The basic principles of micro-CT or cone-beam CT have been described by a number of papers. The topics range from mathematical theories on cone-beam reconstruction, or detector and X-ray source design, to practical applications. Jorgensen et al. (20), Ritman (21,22), and Riiegsegger et al. (23) give a good overview of the technical basis for micro-CT, and micro-CT continues to advance with improving hardware and software. 

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. 

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. ... 

The practical implementation of cone-beam systems requires a choice of scanning geometry and of a reconstruction method. The good answers to both these questions simultaneously can hardly be obtained. 

Grangeat P. Mathematical framework of cone beam three-dimensional reconstruction via the first derivative of the Radon transform.. Math. Methods in Tomography, V.1947 of Springer Lecturre Notes in Math-cs, Springer-Verlag, Berlin, 1991, p.66-97. 

In traditional Fan-Beam CT the radiation emitted from the X-ray tube is collimated to a planar fan, and so most of the intensity is wasted in the collimator blades (Fig. 2a). Cone-Beam CT, where the X-rays not only diverge in the horizontal, but also in the vertical direction, allows to use nearly the whole emitted beam-profile and so makes best use of the available LINAC photon flux (Fig. 2b). So fast scanning of the samples three-dimensional structure is possible. For Cone-Beam 3D-reconstruction special algorithms, taking in consideration the vertical beam divergence of the rays, were developed. 

The main disadvantage of Feldkamp s approaeh is the fact, that it is mathematically correct only in tire midplane of the beam. With larger Cone-Beam angles the error grows and over 30 degrees severe artefacts can be observed In the reconstruction. 

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. 

Fourier Methods in 3D-Reconstruction from Cone-Beam Data... 

Reconstruction in Spiral Cone-Beam CT at small Cone-Angles... 

Therefore it is reasonable to prepare already the data acquisition for a three dimensional evaluation in cone-beam-technique by means of two-dimensional detectors. The system is already prepared to integrate a second detector- system for this purpose. An array of up to four flat panel detectors is foreseen. The detector- elements are based on amorphous silicon. Because of the high photon energy and the high dose rates special attention was necessary to protect the read-out electronics. Details of the detector arrangement and the software for reconstruction, visualisation and comparison between the CT results and CAD data are part of a separate paper during this conference [2]. 

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],... 

To contribute to a voxel (r, s, z) for z 0 in the cone beam geometry, the fan beams must be tilted out of the r, s plane to intersect the particular voxel (r, s, z) from various x-ray source orientations. As a result, the location of the reconstruction point in the tilted system is now determined by a new coordinate system (r, s) (Fig. 26.25). Consequently, the fan beam geometry in these new coordinates will change. Specifically, the new source distance is defined by... 

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. 

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

Shi H, Scarfe W, Farman A (2006) Maxillary sinus 3D segmentation of reconstruction from cone beam CT data sheets. Inti J CARS 83-89... 

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