Filtered back projection

One can then use the classical algorithm of tomographic reconstruction by summation of filtered back-projections ... 

A. Katsevich, Exact Filtered Back Projection (FPB) Algorithm for Spiral Computer Tomography, US Patent 6,574,277 (2003). 

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],... 

U van Stevendaal, J-P Schlomka, A Harding and M Grass (2003) A reconstruction algorithm for coherent scatter CT based on filtered back-projection. Med. Phys. 30, 2465-2474. 

One-dimensional images of toluic acid were obtained with a stationary sample. For two-dimensional images, the 4-bromobenzoic acid crystal was rotated about an axis orthogonal to the gradient direction in constant increments of either 3° or 6° over a range of 180° to collect sets of 60 or 30 one-dimensional projections, respectively. The two-dimensional images were calculated with the filtered back-projection reconstruction algorithm (36). 

Fig. 3 Transaxial slices reconstructed with filtered back projection (top row) and iterative reconstruction (bottom row). Negative reconstructed values have been set to zero for the filtered back projection reconstruction. The streak artifacts and count defects caused by focal activity deposition in the central airways are clearly evident on the reconstructions with filtered back projection. The iterative reconstruction provides artifact free images.

A simple method of recovering the truncated data is to linearly extrapolate the truncated CT projections to the extent of PET FOV and then setting the values to zero at the edge of the PET FOV. The attenuation correction by this method is adequate to remove the truncation artifacts in PET/CT images (Fig. 3.15). Using the iterative method rather than the filtered back projection technique also has shown to help to eliminate these artifacts. 

The images are shown in Fig. 6 al to a3. The device weighs 110 g and hence is portable. The images obtained are reconstructed by filtered back projection algorithm. The device is such that it can be connected to a computer in which the image processing can be performed. This can also thus be extremely crucial in telemedicine [3]. 

While the Fourier slice theorem implies that given a sufficient number of projections, an estimate of the two-dimensional transform of the object could be assembled and by inversion an estimate of the object obtained, this simple conceptual model of tomography is not implemented in practice. The approach that is usually adopted for straight ray tomography is that known as the filtered back-projection algorithm. This method has the advantage that the reconstruction can be started as soon as the first projection has been taken. Also, if numerical interpolation is necessary to compute the contribution of each projection to an image point, it is usually more accinate to conduct this in physical space rather than in frequency space. 

A formal description of the filtered back-projection process is best served by expressing the object function /(jc, y) defined in Eq. (26.64), in an alternative coordinate system. Here, die rectangular coordinate system in the frequency domain (u, v) is exchanged for the polar coordinate system iyv, 0). so that... 

The function Qff) is a filtered projection so that Eq. (26.70) represents a summation of back-projections from the filtered projections. This means that a particular Qfr) will contribute the same value at every point ix, y) in the reconstructed image plane that lies on the line defined by (r, ft) according to Eq. (26.59). This is equivalent to smearing the filtered back-projection along the line GH in the reconstructed image plane (Fig. 

The entire DSR system is conveniently divided into the DSR scanner section, the reconstruction section, and the data analysis section. The reconstruction system utilizes a filtered back-projection fan beam reconstruction system based on a floating point array processor (Robb et ai, 1980). Most important to the... 

Figure 2.5 A rat bone scan acquired in a microPET-FOCUS camera. The animal was scanned in three bed positions, 1.5 h postinjection of 0.88 mCi of Image reconstruction was performed using a FORE followed by 2D-filtered back projection.

The use of techniques of reconstruction such as filtered back-projection (1) makes it possible to rebuild the initial structure in the form of a 3D image made up of voxels (elementary volume of the 3D image, for estample 1024 x 1024 x 1024 voxels). This technique thus aims at finding the value of n for each voxel crossed by the X-rays. To obtain a faithful reconstruction, it is important to make sure in particular that the X-rays are not completely absorbed by the object and are absorbed (for all projections) only by the observed object. It is also necessary to observe the object at 180° to obtain a precise reconstruction although it is possible to find the structure, under certain conditions, when projections are truncated (local tomography) [1, 2],... 

---