1.3.6.2. Image reconstruction using a regular sampling found from analysis of the parameters of vortices in random field

It has been shown [12] that the field at the Fourier domain (at far field) can be represented by so-called product of the wave functions:

, (1.43)

where is the phase change in the vicinity of the -th vortex.

Replacing the system of anisotropic vortices by the system of isotropic ones, one can rewrite Eq. (1.43) in the form:

, (1.44)

where is the true amplitude, , , the origin of the coordinates coincides with the center of the th vortex, and is the topological charge of the th isotropic vortex.

It follows from Eq. (1.44) and from accounting that the resulting phase of a field is the sum of elementary phases associated with the each vortex, that the sum of the initial phases of isotropic vortices,

, (1.45)

forms only the constant phase shift [12], which is the same for the whole field. In this sense, the initial phase of an isotropic vortex is not relevant and can be neglected in consideration of the field formation. Evidently, this conclusion leads to the new insight in replacing of the anisotropic vortices by isotropic ones. The field in the vicinity of any individual speckle is formed by “equal” vortices, which differ from each other by the sign of the topological charge alone. In its turn, it means that the field reconstruction using the shifted sampling must also be different. Indeed, one can conclude that even for considerable magnitude of the sampling shift, when any reading passes from one speckle to another, this point is under “influence” of the equivalent vortex.

This conclusion is confirmed by the results of computer simulation represented in Figure 1.20. This figure illustrates the field reconstruction using the shifted samplings, when anisotropic vortices at far zone are replaced by isotropic ones. Similarly to the previous case, the shift of sampling coordinates is performed in directions of both the - axis and the -axis. One can see that in contrast to the results shown in Figure 1.20, the dark area at the center of the image remains even for the shift 2 , cf. Figure 1.20(f).

Figure 1.20. Reconstruction of the field using the shifted sampling, when anisotropic vortices are replaced by isotropic ones. (a) – original image; (b) – Illustrating the principle of formation of the shifted sampling; – grey points correspond to the positions of amplitude zeroes, black points indicate the positions of the readings of a sampling, – shift of the readings in respect to the vortex coordinates; (c)-(f) correspond to various magnitudes of : (c) – 0.1 ; (d) – 0.5 ; (e) – ; (f) – 2 .

In conclusion of this section, we represent the results of reconstruction of the initial field using the Shannon sampling formed using the characteristics of the system of isotropic vortices. There are the principles of the sampling formation:

1. spacing of the readings is of the order of magnitude 0.5 ;

2. amplitude of the field at the reading point is the same as its true magnitude found as the square root from intensity;

3. phase of the field at point of any reading is determined as

, (1.46)

where are the coordinates of the nearest to the point isotropic vortex, and are the topological charge and the initial phase of this vortex, respectively.

In fact, a phase at the arbitrary point of the field is determined as the phase of the nearest isotropic vortex, while the influence of a vortex on the field structure is extended over the distance comparable with a half of the correlation length of a field, as it has been shown above. This circumstance is taken into account, while the spacing of readings equals 0.5 .

N

Figure 1.21. Field reconstruction using the Shannon sampling formed using the characteristics of the system of isotropic vortices. (a) – original image; (b) – the results of reconstruction; (c) – illustrating the principle of sampling formation. – grey points correspond to the positions of isotropic vortices, and black points correspond to the positions of readings. Arrows indicate the coordinates of readings, whose phase is “determined” by the corresponding vortex.

aturally, this statement is true only in statistical sense. In practice, the distance between amplitude zeroes varies in wide limits. That is why the influence of some vortex of a random field is extended at various distances. The estimations show that the phase of a field reconstructed following Eq. (1.46) coincides with the true phase in 75-80% of events. The Rayleigh criterium can be used as the “criterium of identity” of a phase. It is assumed that the phase calculated following Eq. (1.46) is equal to the true phase of a field, is the difference of these two phases does not exceed .

In our consideration, an average deviation of the computed phase from the true one did not exceed = , and in some points the computed phase was equal to the true one exactly. An average deviation, , in the points where the difference between two mentioned phases exceeded , was not larger than (20 – 25% of the total number of points).

The results of the field reconstruction using such Shannon sampling are represented in Figure 1.21(b). It is seen from figure that the field reconstructed using the regular sampling (with computed following Eq. (1.46)) approaches the initial image of the test-object. Relatively dark area at the center of the image is explained as the result of combined influence of the system of isotropic vortices.

Thus, the obtained results lead to the following conclusions.

1. “Phase influence” of some vortex is extended on the distance comparable with the correlation length of the field.

2. The reconstructed field is modulated by the “vortex” amplitude function, if the reconstructing sampling of the point sources coincides in coordinates with the shifted net of amplitude zeroes.

3. The net of anisotropic vortices of a random field can be replaced by the system of isotropic vortices. For that, the influence of such system of isotropic vortices and of individual vortices per se on the field structure is extended much more far than the influence of the associated system of anisotropic vortices, viz. at distances much exceeding the correlation length of the field.

4. A phase at the arbitrary point of the field can be determined reliably (with error non exceeding ) by analyzing the characteristics of the system of isotropic vortices.