Backprojection reconstruction

Reconstruction method used Choice of filters with a selected cutoff frequency in the filtered backprojection reconstruction method may introduce additional degradation of the spatial resolution of the scanner. For example, a filter with a too high cutoff value introduces noise and thus degrades spatial resolution. An error (KT) due to the reconstruction technique is usually a factor of 1.2-1.5 depending on the method (Huesman, 1977). 

In order to ensure perpendicular beam incidence on the cylindrical specimen, the circular B-scan profiles were acquired by high frequency (narrow beam) transducers in a synthetic circular aperture array. From these profiles two-dimensional reflection tomograms were reconstructed using a filtered backprojection technique. Straight line propagation was assumed. Several artificial discontinuity types in a cylindrical Plexiglas (Perspex) specimen were compared with similar artificial discontinuities in a cylindrical A/Si-alloy [2]. Furthermore, examples of real discontinuities (an inclusion and a feed head) in the cylindrical AlSi-alloy are presented. 

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

Typical tomographic 2D-reconstruction, like the filtered backprojection teelinique in Fan-Beam geometry, are based on the Radon transform and the Fourier slice theorem [6]. 

The Filtered Backprojection (FBP) method may be used to process by reconstructing the original image from its projections in two steps Filtering and Backprojection. 

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... 

Different approaches can be taken to obtain radial images. Radial field gradients can be applied by the use of dedicated hardware [Hakl, Leel, Lee2]. Alternatively, a 2D image can be reconstructed from one projection by the backprojection technique, and a radial cross-section can be taken through it. The most direct way to access the radial image from a projection consists in computing the inverse Hankel transformation (cf. Section 4.4.2) of the FID measured in Cartesian k space (cf. Fig. 4.4.1) [Majl]. But in practice, the equivalent route via Fourier transformation of the FID and subsequent inverse Abel transformation (cf. Section 4.4.3) is preferred because established phase and baseline correction routines can be used in the calculation of the projection as an intermediate result. 

For each image pixel (x,y) at each projection angle o, r is calculated by Eq. (4.1). The measured counts in the projection sinogram corresponding to the calculated r are added to the (x,y) pixel in the reconstruction matrix. This is repeated for all projection angles. Thus, the backprojected image pixel... 

Figure 4.1. Principle of backprojection in image reconstruction, (a) An object with two hot spots (solid spheres) is viewed at three projection angles (at 120° angle) and the acquired data are backprojected for image reconstruction (b) When many views are obtained, the reconstructed image represents the activity distribution with hot spots, but the activity is blurred around the spots (c) Blurring effect described by 1 /r function, where r is the distance away from the central point. (Reprinted with the permission of The Cleveland Clinic Center for Medical Art Photography 2009. All Rights Reserved)...

Finally, the inverse Fourier transformation is performed to obtain filtered projection data in the spatial domain, which are then backprojected in the same manner as in the simple backprojection. With the use of faster computers, the Fourier technique of filtered backprojection has gained wide acceptance in reconstruction of images in nuclear medicine. 

Figure 4.10. Comparison of filtered backprojection (FBP) and iterative (OSEM) methods with attenuation correction, (a) Lungs, (b) Normal Liver, (c) Liver with tumor, (d) Breast. FBP images with attenuation corrections are noisier than OSEM images with attenuation correction. (Reprinted by permission of Society of Nuclear Medicine from Riddell C et al (2001) Noise reduction in oncology FDG PET images by iterative reconstructions a quantitative assessment. J Nucl Med 42 1316)...

The filtered backprojection can be applied to 3D image reconstruction with some manipulations. The 3D data sinograms are considered to consist of a set of 2D parallel projections, and the FBP is applied to these projections by the Fourier method. The iteration methods also can be generally applied to the 3D data. However, the complexity, large volume, and incomplete sampling of the data due to the finite axial length of the scanner are some of the factors that limit the use of the FBP and iterative methods directly in 3D reconstruction. To circumvent these difficulties, a modified method of handling 3D data is commonly used, which is described below. 

The final step is tomographic reconstruction, presently done using filtered backprojection using the IDL Riemann function. Reconstruction times are a few hours for a 512x512x512 3-D data set using a single 450 MHz Pentium PC running Windows NT. Visualization is currently performed in IDL on the local PC at the beamline. 

The first one is a coded mask telescope of 1.5m length. The mask would have a size of 70 X 70cm and elements of 1X Imm. A detector of 35 X 35cm would be needed, consisting of 36 XMM CCD wafers, to provide a field of view of 24°. In order to use it for continuous scanning, a new reconstruction method has been developed (direct backprojection of each photon) which very effectively smoothes the fixed pattern noise. The scientific requirements are met very well, in particular at the high energy part of the band, for X-... 

Projection reconstruction 2-D Projection reconstruction Filtered backprojection (FB)... 

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