Filtered backprojection filters

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. 

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]. 

A linear filter performs a convolution of the input function with the Fourier transform of the filter transfer function. According to the convolution theorem (cf. Section 4.2.3) application of a filter in one domain corresponds to multiplication of the Fourier transform of the function to be filtered with the filter-transfer function. To filter a backprojection image, eqn (6.1.3) is Fourier transformed,... 

Multiplication of the FID p k, p) by Ikl prior to Fourier transformation for use by the backprojection method (6.1.3) can be interpreted in terms of filtering the projection P r, o) by a filter the transfer function of which is given by f k) = k. For this reason, calculation of the image by proper transformation of the raw data from cylindrical to... 

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. 

Several factors affect the filtered backprojection. Adequate sampling of projections (both linear and angular projections related to r and of the sinogram) is needed for accurate backprojection. Data noise, positron range, noncolinearity, scattering, and random events are not taken into consideration in the method. Also in this model, detectors are assumed to be point... 

Figure 4.5. Different windows that are used in combination with a ramp filter to suppress the higher frequency noise in backprojection method.

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. 

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). 

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. 

Dasch G J (1992) One-dimensional tomography a comparison of Abel, onion-peeling, and filtered backprojection methods. Applied Optics 31 1146-1153. 

---