Reconstruction algorithm

Simulations of the adaptive reconstruction have been performed for a single slice of a porosity in ferritic weld as shown in Fig. 2a [11]. The image matrix has the dimensions 230x120 pixels. The number of beams in each projection is M=131. The total number of projections K was chosen to be 50. For the projections the usual CT setup was used restricted to angels between 0° and 180° with the uniform step size of about 3.7°. The diagonal form of the quadratic criteria F(a,a) and f(a,a) were used for the reconstruction algorithms (5) and (6). 

The comparison of curve 1 and 2 in Fig. 3 yields, that the convergence with respect to the number of projections k is not strongly influenced by noise because of the properties of the reconstruction algorithm. Nevertheless, the noise increases the asymptotic value of o(n)/0 ... 

Finally, it is shown in terms of the presented example that the proposed adaptive reconstruction algorithm is valuable for image reconstruction from projections without any prior information even in the case of noisy data. The number of required projections can be determined by investigating the convergence properties of the reconstruction algorithm. 

The reconstruction algorithm proposed in this work is based on a special choice of basis flinctions to expand the unknown refractive index profile. The following set of functions is used here ... 

In this section, two illustrative numerical results, obtained by means of the described reconstruction algorithm, are presented. Input data are calculated in the frequency range of 26 to 38 GHz using matrix formulas [8], describing the reflection of a normally incident plane wave from the multilayered half-space. 

Nielsen, S.A. Borum, K.K. and Gundtoft, H.E (1995). Verifying an ultrasonic reconstruction algorithm for non-destructive tomography. Proc. of 1st World Congress on Ultrasonics, Berlin, Vol. 1, 446-450. 

Grangeat P. Description of a 3-D reconstruction algorithm for diverging X-ray beam., Radiol. Soc. North. America Conf Proc., Nov.1985. 

Likewise, efficient interface reconstruction algorithms and mixed cell thermodynamics routines have been developed to make three-dimensional Eulerian calculations much more affordable. In general, however, computer speed and memory limitations still prevent the analyst from doing routine three-dimensional calculations with the resolution required to be assured of numerically converged solutions. As an example. Fig. 9.29 shows the setup for a test involving the oblique impact of a copper ball on a hardened steel target... 

Image Space Reconstruction Algorithm. ISRA (Daube-Witherspoon and MuehUehner, 1986) is a multiplicative and iterative method which yields the constrained maximum likelihood in the case of Gaussian noise. The ISRA solution is obtained using the recursion ... 

Daube-Witherspoon, M.E., Muehllehner, G., 1986, An iterative image space reconstruction algorithm suitable for volume etc., IEEE Trans. Med. Imaging, 5, 61... 

At present, most PET scanners can acquire in both a two-dimensional as well as a three-dimensional mode, whereas SPECT cameras measure in a three-dimensional mode. The physical property of the dual-positron gamma-rays emission lends itself to mathematical reconstruction algorithms to produce three-dimensional images in which the calculations are much closer to exact theoretical ones than those of SPECT. This is, in part, due to the two-photon as opposed to single-photon approach. PET can now achieve resolutions, for example in animal-dedicated scanners, in the order of 1 or 2 mm. The resolution is inherently limited theoretically only by the mean free path or distance in which the positron travels before it annihilates with an electron, e.g. those in biological water 2-8 mm. SPECT, although achieving millimeter resolution with the appropriate instrumentation, cannot quite achieve these levels. 

Tarantola, G., Zito, F. and Gerundini, P. PET instrumentation and reconstruction algorithms in whole-body applications. /. Nucl. Med. 44 756-769, 2003. 

Fig. 8. Reconstruction of Young s modulus map in a simulated object. A 3D breast phantom was first designed in silico from MR anatomical images. Then a given 3D Young s modulus distribution was supposed with a 1 cm diameter stiff inclusion of 200 kPa (A). The forward problem was the computing of the 3D-displacement field using the partial differential equation [Eq. (5)]. The efficiency of the 3D reconstruction (inverse problem) of the mechanical properties from the 3D strain data corrupted with 15% added noise can be assessed in (B). The stiff inclusion is detected by the reconstruction algorithm, but its calculated Young s modulus is about 130 kPa instead of 200 kPa. From Ref. 44, reprinted by permission of Wiley-Liss, Inc., a subsidiary of John Wiley Sons, Inc.

E. E. Van Houten, M. I. Miga, J. B. Weaver, F. E. Kennedy and K. D. Paulsen, Three-dimensional subzone-based reconstruction algorithm for MR elastography, Magn. Reson. 

The reconstruction algorithms to be used for conventional sources are substantially different from the codes described in the previous section, due to the different acquisition geometry related to the conical shape of the beam (Burch and Lawrence, 1992 Feldkamp et al., 1984). 

For the second case, a hollow ball was dipped inside a jar filled with water and scanned. The diameters of the ball and jar were 6.98 and 18.95 cm, respectively, as shown in Fig. 5a. A scan area of 27 cm in diameter was reproduced using the reconstruction algorithm. The dimensions of the objects as reproduced by the scan were 6.97 and 19.20 cm, respectively, as shown in Fig. 5b, which give a maximum spatial error of about 2.5 mm. This is good enough to resolve relatively small maldistribution, if it exists, inside the 30.48-cm-diameter column used in this study. The figure shows that the overall error in the estimated total holdup is within 12.8%. 

To achieve that, an object reconstruction algorithm detects connected components (blobs) in the binary images of each material class. In a first step, blobs that do not meet pre-defined size restrictions are considered as classification errors and filtered out (e.g. the black line due to a faulty camera pixel in object D). The binary image of each material class (material and overlay) is then morphologically dilated... 

Yorkey, T. J., Comparing reconstruction algorithms for electrical impedance tomography," PhD Thesis, University of Wisconsin (1986). 

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],... 

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. 

---