Back-projection

We present a novel method, called VIGRAL, to size and position the reflecting surface of a flaw. The method operates on standard B-scan recorded with traditional transducers, to extract Time-of-Flight (ToF) information which is then back-projected to reconstruct the reflecting surface of the flaw and characterize its radiation pattern. The VIGRAL method locates and sizes flaws to within k/2, and differentiates between flat and volumetric defects. 

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

To create a two-dimensional image, two gradients are applied along the X- and y-directions, and a series of one-dimensional images recorded in different directions in the xy-plane. A technique known as back-projection... 

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

Radermacher, M. (1992). Weighted Back-Projection Methods. Plenum, New York. 

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],... 

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). 

From N data points, N — k such equations can be generated, and as long as N — k is larger than the number of spectral lines, the problem is overdetermined and a set of coefficients af can be obtained by suitable mathematical procedures. With these coefficients the FID can be forward or back projected. For example, Fig. 3.9 illustrates the improvement in spectral quality as a result of replacement of a severely distorted data point. 

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