The underlying idea of the tomographic method is
The underlying idea of the tomographic method is in reconstructing f(x, y), i.e. the density distribution function (or distribution of any other quantity) in the plane . The initial data are sets of Radon images Rφ(s) that are projections of the functions f(x, y) on the set of the corresponding straight lines s in the xy plane, forming the angles with the axis x (i is the number of a projection). A set of functions that are reciprocal projections of Rφ(x, y) is then calculated from the set of Radon images. This projection Rφ(x, y) is a function of two variables (x, y) repeating the corresponding Radon image Rφ(s) in any section by a plane parallel to the straight line s. The summary projection
is found from a set of reciprocal projections.
The function S(x, y) is the function f(x, y) reconstructed from Radon images. Eq. (2) holds only for an infinite number of Radon images, which is impossible to realize in practice. This problem is solved by a procedure of discrete-sampling Eq. (2), which results in reciprocal-projection integration around the angle φ (an angle with the x axis) becoming summation:
where N is a finite number of Radon images.
For the finite number of projections the function S(x, y) repeats the initial function f(x, y) only with certain accuracy. Similar reasoning is also valid for the general case when f(x1, x2, …, ) is a function of density distribution (or distribution of any other quantity) in the n-dimensional space. In this case the n-dimensional space in projected on the (n−1)-dimensional one.
In our problem, the three-dimensional permittivity distribution produced by the studied object is projected in the space on a set of planes (with a two-dimensional distribution of the optical path alk pathway recorded for each of them) corresponding to each receiver–transmitter pair. Then the total projection S(x, y, z) and the reciprocal-projection Rφ (x, y, z) become functions of three variables, and the Radon image Rφ (s) is a projection of the three-dimensional distribution on the set of planes s forming the angles with the xy plane (i is the number of a projection). Then the sought-for spatial permittivity distribution is described by the function f(x, y, z).
To facilitate the calculations, the studied zone was, as in Ref. , evenly divided into unit cells of known dimensions, since the procedure of calculating the precise geometrical intersection of the straight lines containing the receiver–transmitter points from all possible directions would be extremely tedious. The number of straight lines intersecting each unit cell was calculated in advance. The sum of optical path elongations measured for each straight line intersecting the unit cell was entered in each unit cell.
Measurements, results, and their discussion Two samples were used as objects in the conducted experiments: one made of beeswax, and one of polyvinylchloride (PVC). Their pre-measured permittivities were 2.6 ± 0.2 and 3.8 ± 0.2, respectively. Sample dimensions were 17 × 14 × 8 cm and 17 × 15 × 8 cm, respectively. Each sample was placed in the studied zone of the experimental setup in various positions relative to the receiving and transmitting elements. The complex amplitude of the electromagnetic field that had passed through a sample was measured at all receive antennas and at all frequencies. The measured field amplitudes were subjected to an inverse Fourier transform (1); then the elongation of the optical path was found from the shift in the transforms principal maximum for each receiver–transmitter direction. The elongation of the optical path calculated for each receiver–transmitter pair was entered in every tomographic projection. Then the density distribution of optical elongation in the studied zone was calculated from the set of tomographic projections using the three-dimensional case (3). Finally, the permittivity distribution was calculated from the given one. Fig. 2 shows tomograms (the xz-plane cross-sections of density distributions of optical path elongation in the studied zone) for beeswax and PVC obtained using the linked cluster algorithm. This algorithm found the spatial areas in the cross-section of the tomogram that matched the position of the sample. The dimensions of the studied zone were 140 × 50 × 60 cm, and the coordinates of the zone\'s spatial boundaries (in cm) were from −70 to +70, from −25 to +25, from −40 to +20 in the x-, y-, z- directions, respectively.