RU2533502C1 - Method of forming sub-diffraction resolution image - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: rectangular grid x0y is input into the plane of an observed object, in the nodes of said grid of which a two-dimensional array u_i,j of intensity values of the formed image is created, said values being defined by minimising variation of the intensity of the image L(u).

EFFECT: high reliability of the image in the presence of diffraction restrictions, noise and other interference.

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Description

The invention relates to methods and devices for radio-vision, radar and radio astronomy (hereinafter collectively, radio-vision devices) of millimeter, terahertz and far infrared ranges of electromagnetic radiation and can be used to construct radio images of various objects that are sources of primary and / or secondary radiation of radio waves of these ranges.

Radio-vision devices are used to obtain images of various objects of artificial and natural origin, which are sources of electromagnetic radiation. Such a device includes an optical system that performs spatial transformation of the radiation field of the sources. A typical option for constructing an optical system is a system having two focal planes. An ideal optical system of this type converts the field of radiation sources located in one focal plane into an electromagnetic field (image field) in another focal plane. In this case, the transformed field is identical to the initial field with an accuracy of linear displacement and a scaling factor specifying the compression or extension of the initial field. In an imperfect optical system, there are optical image distortions (aberrations) and interference noise and other nature. The proposed method of image formation is intended to rid the converted field of distortion and interference in an imperfect optical system and the formation (construction) of the corresponding image, as close as possible to the ideal.

The plane in which the radiation sources are located is called the plane of the image sources. The reconstructed image is formed in this plane. Points in the plane of the image sources are described in the x0y coordinate system.

The second focal plane of the optical system in which the transformed field is created is simply called the focal plane of the optical system. It can be the only focal plane in the case when the source plane is removed at a sufficiently large distance from the optical system, which can be considered infinite. This situation is typical for radio astronomy devices. Points in the focal plane of the optical system are described in the coordinate system ξ0η.

As an optical system, mirror antennas can be used. Known optical system in the form of a two-mirror Cassegrain antenna (Sazonov D.M. Antennas and microwave devices. M .: Higher school. 1988). It can be focused on both the finite and the infinite distance to the radiation source.

Known optical systems based on dielectric lenses (Zelkin E.G., Petrova R.A. Lens antennas. M .: Soviet radio, 1974). To build optical systems, more complex lenses are also used, for example, the Rotman lens and the Luneberg lens (Kornblit S., Microwave optics, M .: Svyaz, 1980).

Stationary and scanning optical systems are known.

The claimed invention relates to scanning optical systems. They have the ability to move the field in the focal plane. This movement is achieved in many ways. The simplest of these is the movement of the entire optical system. In radio astronomy, in the study of distant objects, a particular type of displacement of the optical system is used — its rotation. The advantage of scanning by mechanical movement of the optical system is the absence of additional distortions - aberrations associated with the scanning process.

Known scanning optical system based on a Cassegrain two-mirror antenna, in which scanning is achieved by moving an auxiliary mirror - a subreflector (May T., Zieger G., Anders S., Zakosarenko VM, Meyer H.-G., Thorwirth G., Kreysa E ., Passive stand-off Terahertz imaging with 1 Hertz frame rate // Proc. SPIE, Vol.6949, 6949C (2008); DOI: 10.1117 / 12.777952). Moving a subreflector having a mass substantially less than the mass of the entire optical system is much simpler. Therefore, the scanning speed in this case is much higher than when moving the entire optical system. However, the movement of the subreflector leads to the appearance of additional aberrations of the optical system.

It was noted above that an ideal optical system converts a field in the plane of image sources into a field in the focal plane, performing two operations with it: moving and scaling. As a rule, optical systems are constructed so that they compress the field. Mathematically, the transformation of the source field with intensity U (x, y) into a field in the focal plane U _f (ξ, η) is described for an ideal optical system as follows:

$$U_f\left(\xi ,\eta \right)=A{\displaystyle \underset{S}{\int }U\left(x,y\right)\delta \left(\alpha \left(\xi -\xi \text{'}\text{'}\right)-x\right)\delta \left(\alpha \left(\eta -\eta \text{'}\text{'}\right)-y\right)dxdy,\left(\mathrm{one}\right)}$$

where A is the amplitude factor, α is the field scaling factor, ξ ', η' are parameters defining the field shift, δ (x) is the delta function, S is the portion of the source plane in which the image sources are located.

The function K _OS (x, y, ξ, η), of the form

$$K_{OS}\left(x,y,\xi ,\eta \right)=\delta \left(\alpha \left(\xi -\xi \text{'}\text{'}\right)-x\right)\delta \left(\alpha \left(\eta -\eta \text{'}\text{'}\right)-y\right),\left(2\right)$$

under the integral in (1) is called the hardware function of the optical system. For a specific real optical system, it is described in a more complex way, which allows one to take into account the aberrations existing in it.

Aberrations have a different physical nature and affect the appearance of the hardware function of the optical system in different ways. Geometrical-optical aberrations are known: astigmatism, coma, etc. They are connected with the fact that the optical system focuses the field imperfectly. Aberrations of this type are not fundamental. They can be reduced by improving the optical system. As ways to improve, you can specify an increase in perfect focus points, that is, a transition to polyfocal systems. An ideal non-aberrational optical system, for example, is the Luneberg lens mentioned above, which is a sphere with a dielectric constant with variable radius.

There are also aberrations associated with the wave nature of the electromagnetic field. They are of a fundamental nature, since it is impossible to eliminate them by improving the design of the optical system. The effect of such aberrations on the hardware function of the optical system K _OS (x, y, ξ, η) is expressed in the fact that even in the absence of geometric-optical aberrations, it cannot be represented as a product of delta functions. A typical hardware function of an optical system without geometrical optical aberrations is as follows:

$$\begin{array}{l}{K}_{OS}\left(x,y,\xi ,\eta \right)=\frac{J_{\mathrm{one}}\left(br\left(x,y,\xi ,\eta \right)\right)}{r\left(x,y,\xi ,\eta \right)},\left(3\right)\\ r\left(x,y,\xi ,\eta \right)=\sqrt{{\left(\left(\xi -\xi \text{'}\text{'}\right)-x/\alpha \right)}^2+{\left(\left(\eta -\eta \text{'}\text{'}\right)-y/\alpha \right)}^2,}\end{array}$$

where b is the parameter determined by the optical system, J ₁ (x) is the first-order Bessel function. The parameter b = πD / λF is inversely proportional to the wavelength λ.

It can be seen from expressions (2) and (3) that any point source of the image, independent or included in the composition of the image sources, located in the plane of the image sources, turns into a “spot” in the focal plane, the width of which is determined by the parameter b and the properties of the optical system. Distortions of this type were called diffraction distortions, and the spot named above was called the Airy spot (Sivukhin D.V., General Physics Course, Volume IV, Optics, Moscow, Nauka, 1985). The diameter of the Airy spot h is determined by the so-called Rayleigh criterion, which sets the limit of concentration (focusing) of the electromagnetic field using optical systems:

$$h=2.44\frac{\lambda }{D}\cdot F,\left(\mathrm{four}\right)$$

where λ is the radiation wavelength, D is the diameter of the primary mirror or lens of the optical system, F is the focal length of the optical system. For a single-mirror or single-lens system, the equivalent focal length is equal to the focal length of a mirror or lens. For more complex multi-element systems, it can be determined by known relations.

The diameter of the Airy spot h is an important parameter of the optical system, which determines its own resolution in the focal plane. It shows the minimum distance between the field of point sources in the focal plane that this optical system can register. It is important to note that due to the fact that the optical system performs a scaling transformation of the field in the plane of the radiation sources to the image field in the focal plane, it is impossible to directly use the parameter h to determine the minimum distance of distinguishable point sources in the plane of the sources ρ, since the indicated distance is associated with it in the following way:

h = kρ,

where k is the field compression coefficient, which is usually less than unity. For k> 1, it is more correct to speak of field extension in the focal plane.

The parameter ρ determines the intrinsic resolution of the optical system in the plane of the sources.

Another important element of a radio video system is the matrix receiver. The matrix receiver is a grid of receiving elements that convert the energy of the electromagnetic field of the image incident on them into electrical signals. The receiving elements are located in the focal plane of the optical system. Typical is the placement of receiving elements periodically in nodes of a rectangular or hexagonal grid.

The matrix receiver contains M elements, each of which is described by its number m, 1≤m≤M. The electrical signal at the output of the mth receiving element p _m is connected to the field in the focal plane as follows:

$$p_m={\displaystyle \underset{S}{\int }U\left(x,y\right)K_m\left(x,y\right)dxdy,\left(5\right)}$$

Where

$$K_m\left(x,y\right)=A{\displaystyle \underset{S_m}{\int }F\left(\xi ,\eta \right)K_{OS}\left(x,y,\xi ,\eta \right)d\xi d\eta ,}$$

F (ξ, η) is the known weight function, which is determined by the structure of the receiving element, S _m is the region in the focal plane in which the function F (ξ, η) is nonzero, and A is the amplitude factor.

The functions K _m (x, y,) are called the optical

systems together with a matrix receiver. The hardware function of a scanning optical system generally changes during the scanning process. To describe them, we use functions with the index K _{OS, n} (x, y, ξ, η), which are the hardware functions of the optical system at each scanning step. Accordingly, for a scanning optical system, together with the matrix receiver, the electrical signals and hardware functions also have two indices: p _{n, m} and K _{n, m} (x, y,).

The hardware function of the complex “optical system plus matrix receiver” is the most important characteristic of a radio-vision system. It can be calculated, and in the practical case it can be obtained as a result of its direct measurement using an independent point radiation source with the characteristic U (x, y) = Сδ (xx ₀ ) δ (у-у ₀ ), where С is the intensity of the emitted this source of signal, with the stepwise movement of this point source in the plane of the image sources and the formation of an array of values of the corresponding electrical signals at the outputs of the matrix receiver. From the above relations it is seen that in this case the signals p _{n, m} coincide with the desired hardware functions. To distinguish them from electrical signals obtained when measuring radiation from an image, we denote them by q _{n, m} . Further this procedure is considered in more detail.

There are various methods for obtaining images of radiation sources using a scanning optical system. They are based on the conversion of the specified radiation field at each scan step into electrical signals at the outputs of the matrix receiver, which is located in the focal plane of the optical system, and the formation of an array of values of the indicated electrical signals p _{n, m} , 1≤n≤N, 1≤m≤M , N is the total number of steps of the radiation field moving during the scanning cycle, M is the total number of elements of the matrix receiver.

The simplest image acquisition method is known (Heinz E., May T., Zieger G., Born D., Anders S., Thorwirth G., Zakosarenko V.M., Schubert M., Krause T., Starkloff M., Kriiger A., Schulz M., Bauer F., Meyer H.-G., Passive Submillimeter-wave Stand-off Video Camera for Security Applications, J. Infrared Milli Terahz Waves, DOI 10.1007 / s10762-010-9716-y, 28 Sept. 2010). This method of constructing an image is based on the use of the mapping functions x (ξ, η), y (ξ, η). The display functions are determined during the calibration of the radio system. For each position of the receiving element in the focal plane, characterized by the coordinates ξ, η, the maximum of the hardware function can be found, which is located on the source plane at the point with coordinates x, y. Display functions relate the coordinates of points in the source plane to the coordinates of points in the focal plane.

In the framework of the above method, an array of values is formed

image intensities u _{n, m} as follows:

$$u_{n,m}=p_{n,m},\left(6\right)$$

where p _{n, m} is the generated array of electrical signals that are received from the outputs of the receiving elements located at the points ξ _{n, m} , η _{n, m} . Using the display functions, the coordinates of the points x _{n, m} , y _{n, m} on the plane of the sources that correspond to the array of image intensities u _{n, m are found} . An array of discrete values of the image intensity carries enough information to construct by approximation methods a continuous function U (x, y) that describes the image intensity in a certain region on the source plane.

The disadvantage of this method of constructing an image is its low resolution. The direct use of an array of electrical signals p _{n, m} to form an array of image intensities u _{n, m} does not allow us to eliminate the influence of diffraction distortions associated with the final size of the Airy spot. As a result, the image is “blurry”. It is not able to transmit rapid changes in the intensity distribution of radiation sources located in the plane of the sources.

The closest way to build the image to the claimed is described in the patent of the Russian Federation (Vystavkin A.N., Pestryakov A.V., Banks S.E., Chebotarev V.M., Patent of the Russian Federation for invention No. 2398253 "Image forming device with subdiffraction resolution" , priority of March 18, 2009). In this known method, the image of radiation sources in the ranges of millimeter, terahertz and far infrared waves located in the plane of the sources is formed using an optical system configured to stepwise move the radiation field, which is created by radiation sources in the focal plane of the optical system, by converting the specified field radiation at each step of moving into electrical signals at the outputs of the matrix receiver, which is located in the focal plane of the optical system, and the subsequent formation of an array of values of the indicated electrical signals p _{n, m} . The radiation field in the focal plane of the optical system in a known method of constructing an image moves in a circle, that is, performs a circular movement. In this case, the center of this movement is offset from the center of the matrix receiver. In the source plane, a rectangular grid with nodes at the coordinates x _i and y _j , 1≤i≤N _x , 1≤j≤N _y , is introduced, where N _x is the number of cells of the rectangular grid along the 0x axis, and N _y is the number of cells rectangular grid along the 0y axis. Next, a two-dimensional array of u _{i, j} values of the image intensity is formed at the nodes of the rectangular grid, a two-dimensional array u _{i, j} is determined from the condition for minimizing the functional L (u):

$$\begin{array}{l}L\left(u\right)={\displaystyle \sum _{n=\mathrm{one}}^N{\displaystyle \sum _{m=\mathrm{one}}^M{|\; |\; |p_{n,m}-{\displaystyle \underset{S}{\int }U\left(x,y\right)K_{n,m}\left(x,y\right)dxdy}|\; |\; |}^2+\left(7\right)}}\\ +\alpha {{\displaystyle \underset{S}{\int }|\; |\; |U\left(x,y\right)|\; |\; |}}^{2}dxdy,\end{array}$$

Where

U (x, y) = u _{i, j}

x _i ≤x≤x _{i + 1} ; y _i ≤y≤y _{i + 1}

and S is the region in which the image is built, α is a small parameter.

This method of constructing an image allows you to get images with a resolution that exceeds the limit set by the Rayleigh criterion.

The disadvantage of this method of constructing an image is its instability with respect to noise and interference of other types. Modeling the practical application of this image-building method (Bankov S.E., Chebotarev V.M., Cherepenin VA, Korjenevsky A.V., Pestryakov A.V., Vystavkin AN, Image production with sub-diffraction resolution in radio vision devices of the terahertz range using receiving arrays and image scanning procedure, Vestnik Novosibirsk State University, Series: Physics, 2010, Volume 5, issue 4) showed that image formation using it is possible with a signal-to-noise ratio at the outputs of the matrix receiver of at least 50. The signal at the output of the matrix receiver when determining the signal-to-noise ratio is understood as the signal that is obtained when setting up the hardware function uu optical system, together with a matrix receiver to maximum. Noise refers to the intrinsic noise of the matrix receiver.

A similar picture holds for other cases of applying the named method using the functional minimization procedure (7).

The proposed solution is aimed at obtaining a technical result, expressed in increasing the resolution of the image in the presence of optical, specifically diffraction distortion, as well as noise and other noise. The result obtained is expressed in the construction of images of radiation sources in the ranges of millimeter, terahertz and far infrared waves with a clarity exceeding the Rayleigh criterion for an optical system with given dimensions in the presence of interference of a different nature. Achieving the specified technical result allows to reduce the cost of radio-vision systems by reducing the size of the optical system at a fixed level of image resolution or to increase the resolution at given dimensions of the optical system.

The proposed method of forming an image created by radiation sources that emit in the ranges of millimeter, terahertz and far infrared waves and are located in the plane of the sources, using an optical system configured to stepwise move the radiation field, which is created by radiation sources in the focal plane of the optical system, by converting the specified radiation field at each movement step into electrical signals at the outputs of the matrix receiver, which is located in the focal plane of the optical system, and the subsequent formation of an array of values of the indicated electrical signals p _{n, m} , 1≤n≤N, 1≤m≤M, where p _{n, m} is the magnitude of the electrical signal at the nth step of the radiation field moving by m -th output of the matrix receiver, N is the total number of steps to move the radiation field, M is the total number of elements of the matrix receiver, solves the problem of image formation with increased resolution in the presence of diffraction distortion, as well as noise and other interference in the optical system and the environment.

This problem is solved due to the fact that in the plane of the sources a rectangular grid is introduced with nodes at points with coordinates x _i and y _i , 1≤i≤N _x , 1≤j≤N _y , where N _x is the number of cells of the rectangular grid along the axis 0x, and N _y is the number of cells of the rectangular grid along the 0y axis, an independent point source is sequentially placed in the nodes of the rectangular grid, the radiation field of the point source is stepwise moved, and an array of values of electrical signals is generated at the outputs of the matrix receiver q _{n, m, I, j} , at each step of the radiation field of the target source for each position of the point source in the nodes of the rectangular grid, a two-dimensional array of image intensities u _{i, j} of the image intensities in the nodes of the rectangular grid is formed, and the two-dimensional array u _{i, j} is formed from the condition of minimizing the variation in image intensity L (u):

$$L\left(u\right)={\displaystyle \sum _{i=\mathrm{one}}^{N_x}{\displaystyle \sum _{j-\mathrm{one}}^{N_y-\mathrm{one}}|\; |\; |u_{i,j+\mathrm{one}}-u_{i,j}|\; |\; |}+{\displaystyle \sum _{i=\mathrm{one}}^{N_x-\mathrm{one}}{\displaystyle \sum _{j-\mathrm{one}}^{N_y}|\; |\; |u_{i+\mathrm{one,}j}-u_{i,j}|\; |\; |}\left(8\right)}}$$

under restrictions

$$\begin{array}{l}\underset{\begin{array}{l}\mathrm{one}\le n\le N\\ \mathrm{one}\le m\le M\end{array}}{{\displaystyle \max }}|\; |\; |p_{n,m}-{\displaystyle \sum _{i=\mathrm{one}}^{N_x-\mathrm{one}}{\displaystyle \sum _{j=\mathrm{one}}^{N_y-\mathrm{one}}{u}_{i,j}{q}_{n,m,i,j}\Delta x_i\Delta y_j}}|\; |\; |\le \delta \},\left(9\right)\\ u_{i,j}\ge 0\end{array}$$

where Δx _j = x _{j + 1} -X _j , Δy _j = y _{j + 1} -y _j , δ is the specified error in the formation of an array of image intensity values in the nodes of a rectangular grid, the dimension of the array of quantities of electrical signals at the outputs of the matrix NM detector is selected with a larger number nodes of a rectangular grid N _x N _y . This ensures data redundancy over the number of variables in the problem, which is necessary to obtain its solution.

Figure 1 shows one of the possible embodiments of a device that implements the proposed method of image formation. The device contains an optical system 1 and a matrix receiver 2. The optical system 1 is made in the form of a two-mirror telescope according to the Cassegrain scheme. It contains the main mirror 3 and the subreflector 4. The subreflector 4 is arranged to move, and specifically, to rotate and change the angle of inclination relative to the axis of the main mirror 3. The possibility of rotation and change of the angle of inclination is provided by a set of two engines 5. The matrix receiver is located in the focal planes of the optical system 1.

It should be noted that close variants of the device that implements the proposed method of image formation are possible, for example, with the installation of a rotating K-mirror at the output of the optical system, or another moving mirror or a system of moving mirrors that step-by-step moves the radiation field in the optical system. The described method of imaging with subdiffraction resolution can be implemented with their help. A further description of the method of imaging with subdiffraction resolution is carried out on the example of an optical system with a Cassegrain two-mirror antenna.

The signals p _{n, m} and q _{n, m, I, j} from the outputs 6 of the matrix receiver 2 are received in the device for storing arrays of values of electrical signals 7, which can be made in the form of a memory register. The generated arrays of values of electrical signals can be transmitted from the output of the device for storing arrays of values of electrical signals 7 to an external device for their further use in the implementation of the imaging method.

Figure 2 shows, as one of the options, the element of the matrix receiver 2, which is made in the form of an immersion lens 8, which is combined with the substrate. The substrate serves as a metal screen in which a double-slot antenna 9 is made. On the opposite side of the substrate is a microstrip conductor. A receiving element 10 is included in the gap of said conductor, which can be made, for example, in the form of a superconducting bolometer. Superconducting bolometers have a low level of intrinsic noise and are used in terahertz radiation receivers. The low-frequency signal from the output of the superconducting bolometer enters the microstrip line, which is used as output 6 of the matrix receiver 2.

Figure 3 shows the placement of the optical system 1 relative to the plane of the sources 11. The radiation sources have a different nature. These can be real sources of radiation in the form of emitters, to which power is supplied from the generator. Also, radiation sources can serve as reflecting objects that scatter radiation from external sources located outside the plane of sources 11. It is also possible the appearance of virtual sources, for example, in the form of radiation passing through the plane of sources 11.

Regardless of the type of source, the task of a device that implements an image forming method is to construct an image of sources located in the plane of sources 11.

The plane of the sources 11 can be at a finite distance from the optical system 1. In this case, the optical system 1 must be focused on the plane of the sources 11. In radio astronomy, the plane of the sources 11 is located at a distance that is many times larger than the dimensions of the optical system 1. Such a distance can be considered infinitely large . In this case, the optical system should be focused on infinity. Focusing at different distances is achieved by choosing the shape of the main mirror 3. At a finite distance, it has an elliptical shape, and at infinite - parabolic.

To form (build) the image, as indicated above, in the plane of the sources 11, a rectangular grid 12 is introduced, as shown in Fig. 4. The nodes of the rectangular grid 12 are characterized by indices i, j, which vary within:

$$\begin{array}{l}\mathrm{one}\le i\le N_x,\left(10\right)\\ \mathrm{one}\le j\le N_y,\end{array}$$

where N _x and N _y are the numbers of nodes of a rectangular grid 12 along the axes 0x and 0y, respectively. The total number of nodes of the rectangular grid 12 is N _x N _y . Rectangular mesh 12 can be performed with a constant step, as shown in Fig.4. It can also have a variable pitch Δx _j = x _{j + 1} -x _i along the 0x axis or Δy _j = y _{j + 1} -y _j along the 0y axis.

The dimension of the array of quantities of electrical signals at the outputs of the matrix receiver NM is selected by a larger number of nodes of a rectangular grid N _x N _y , as mentioned above. This inequality determines the parameters of the displacement of the radiation field, which is created by radiation sources in the focal plane of the optical system 1.

Consider in detail the formation of the image in accordance with the proposed method. The image is formed in several stages. At the first stage, the optical system 1 and the matrix receiver 2 receive signals from radiation sources, which we call the studied radiation sources. The image of the investigated sources should be formed in accordance with the proposed method.

The radiation of the studied sources is converted by the optical system 1 in the radiation field in the focal plane of the optical system 1. The optical system 1 is configured to stepwise move the radiation field in the focal plane. The cycle of moving the radiation field in the focal plane or the scanning cycle includes N steps. Each scan step is described by the number n, 1≤n≤N. The matrix receiver 2 at each step of the movement of the radiation field in the focal plane of the optical system 1 converts the radiation field in the focal plane of the optical system into electrical signals p _{n, m} , 1≤m≤M, at the outputs 6 of the matrix receiver 2. Here m is the number of output 6 matrix receiver 2, and M is the total number of outputs 6 of matrix receiver 2. The set of electrical signals p _{n, m} forms an array of values of electrical signals p, which from outputs 6 goes to a memory device for the array of values of electrical signals 7.

смещен относительно центра

матричного приемника. На фиг.5 представлено четыре кадра, отображающих круговое перемещение для числа шагов N=4. В реальной процедуре сканирования их намного больше с тем, чтобы обеспечить условие NM>>N_xN_y, о котором говорилось выше и которое необходимо для получения решения задачи.The scanning process is illustrated in Fig. 5, which shows the stepwise movement of the radiation field, i.e. image in the focal plane of the optical system 1 shown in Fig.1. The optical system 1 in this example is focused on infinity and takes the radiation field, which is created by the spiral galaxy M101. In Fig. 5, seven circles of the receiving elements of the matrix receiver 2 are conventionally shown in white circles. The radiation field in the focal plane makes a circular motion relative to the stationary matrix receiver. In this case, the center of the image

offset from the center

matrix receiver. Figure 5 presents four frames displaying circular movement for the number of steps N = 4. In the real scanning procedure, there are much more of them in order to ensure the condition NM >> N _x N _y , which was mentioned above and which is necessary to obtain a solution to the problem.

At the second stage, an independent point source of high intensity is emitted into the plane of sources 11, emitting in the same range of radio waves as the radiation sources under study. By high intensity is meant such a level of radiated power that provides at each output 6 of the matrix receiver 2 in a position corresponding to the maximum of the hardware function, the signal-to-noise ratio is much larger than unity. In this case, the effect of noise is insignificant. A point source has geometrical dimensions much shorter than the wavelength in free space. As such a source, for example, a radiating electric dipole can be used.

A point source is sequentially placed in all nodes of a rectangular grid 12 with indices i, j. For each position of the point source, a stepwise circular movement of the radiation field in the focal plane of the optical system 1 is carried out according to the same strategy as the image movement in Fig. 5. Simultaneously, the outputs 6 of the matrix receiver 2 are removed electrical signals q _{n, m, I, j} , which are received in the device for storing arrays of electrical signals 7. Thus, an array of values of electrical signals q is formed. Arrays

the values of the electrical signals p and q are used in the next stages of image formation.

The image to be formed is described by the function U (x, y), which is defined in the plane of sources 11. This function is proportional to the intensity (power) of the source at a given point with x, y coordinates on the plane of sources 11. In accordance with the proposed image formation method, the latter It is an array of image intensity values u _{i, j,} the nodes of a rectangular grid 12. Knowing the image intensity at discrete points, we can, if necessary, to find its value at an arbitrary point on the plane 11 of sources, using the approximation process. The well-known linear approximation method allows you to record the image intensity in the form of a bilinear function:

$$\begin{array}{l}U\left(x,y\right)=\frac{\mathrm{one}}{\Delta x_i\Delta y_j}(\left(x-x_{i+\mathrm{one}}\right)\left(y-y_{j+\mathrm{one}}\right)u_{i,j}-\left(x-x_{i+\mathrm{one}}\right)\left(y-y_j\right)u_{i,j+\mathrm{one}}-\\ -\left(x-x_i\right)\left(y-y_{j+\mathrm{one}}\right)u_{i+\mathrm{one,}j}-\left(x-x_i\right)\left(y-y_j\right)u_{i+\mathrm{one,}j+\mathrm{one}}),\left(\mathrm{eleven}\right)\end{array}$$

x _i ≤x≤x _{i + 1} , y _j ≤y≤y _{j + 1} .

At the third stage, an array of values of the image intensity u _{i, j} is determined at the nodal points of the rectangular grid 12. It is determined, as already mentioned, by minimizing the variation in the image intensity L (u), which is determined through the sum of the moduli of the differences in the values of the image intensity in neighboring points of a rectangular grid 12 by relation (8). Minimizing the variation in the image intensity L (u) in entries (8) and (9) reduces to a linear programming problem, which is solved by the method of successive improvement of the solution, also known as the simplex method (Yudin DB, Holstein EG, Problems and linear programming methods, Sovetskoe Radio, Chapter 3, “Method of sequential plan improvement”, Moscow, 1961), which, in turn, reduces to sorting the vertices of a convex polyhedron using a modified method of sequential plan improvement, known as the inverse method trica, the implementation of which uses a priori information about the solution, which makes it possible to avoid loss of accuracy in the process of sequentially improving the solution due to the overdetermination of the original problem.

Minimizing the variation in image intensity ensures the stability of the generated image with respect to noise and noise of a stochastic nature, since it eliminates the possibility of sharp random jumps in the image that occur when using the known method of image formation, by which the image is formed by minimizing the functional (7).

Condition (9) ensures that the generated image is closer to the true distribution of the studied radiation sources in the plane of the sources 11, described by the values of u _{i, j} , at the nodal points of the rectangular grid 12. The approximation is achieved due to the fact that the minimum variation in the image intensity is sought in the region that is specified inequality (9), which includes elements of an array of values of electrical signals q _{n, m, i, j} obtained in the second stage. Since at the second stage an independent point source is used as the source, the values of electric signals q _{n, m, i, j} coincide with the values of the hardware function of the optical system 1, which includes the matrix receiver 2, at the points that are determined by the numbers m of outputs 6 of the matrix receiver 2 at the n-th scanning step when placing a point source in nodes with indices i, j.

Thus, the expression on the left side of inequality (9) corresponds to the maximum set deviation δ between the signals of two types at the outputs 6 of the matrix receiver 2 at all scanning steps. The first type of signals is real signals from the array P _{n, m} , which was formed at the first stage. They are created by the studied radiation source (object of study). The second type of signals are signals that are generated using the described method with minimizing variations in the image intensity L (u) in records (8) and (9) in the plane of sources 11 with radiation intensity at the nodal points of rectangular grid 12 with indices i, j, equal intensities of the components of the formed (desired) image, that is, the elements of the array u _{i, j} . Inequality (9) shows that the specified deviation should not exceed the specified error δ, which is a guarantee that the generated (desired) image is close to the intensity distribution of the studied radiation sources (to the real image).

The requirement that the elements of the array of values of the image intensity values u _{i, j} be positive excludes physically unrealizable elements from the specified array, since the image intensity is, by definition, a positive value.

An additional embodiment of the image forming method is possible, characterized in that the sizes of the cells of the rectangular grid Δx _i , Δy _j are selected in accordance with the condition:

$$\Delta x_i,\Delta y_j\le \rho /K,\left(12\right)$$

K> 1,

where ρ is the intrinsic resolution of the optical system in the plane of the image sources, K is the coefficient of image resolution improvement.

Condition (12) is a condition for improving image resolution compared to the intrinsic resolution of optical system 1, which is determined by diffraction effects. The coefficient of improvement in image resolution K shows how many times the minimum resolvable distance between two point radiation sources located in the plane of sources 11, which can be distinguished in the generated image, decreases compared to the parameter ρ.

An additional embodiment of the image forming method is possible, characterized in that the step of moving the radiation field s in the focal plane of the optical system is selected from the condition:

$$s\le h/K,\left(13\right)$$

where h is the intrinsic resolution of the optical system in the focal plane, K> 1 is the same coefficient of image resolution improvement. This option makes it possible to strengthen the inequality NM> N _x M _y , mentioned above.

An additional embodiment of the image forming method is possible, characterized in that the element of the matrix receiver 2 is made with an area exceeding h ² . Matrix receivers of millimeter, terahertz and infrared ranges are very expensive devices. Traditionally, the size of the element of the matrix receiver 2 is chosen close to the diameter of the Airy spot h. It is believed that this choice of size provides acceptable energy efficiency without compromising image resolution. In this case, to increase the area in which the image is formed, it is necessary to increase the number of elements of the matrix receiver 2, since an increase in the size of its element will lead to a deterioration in image resolution.

In the proposed embodiment of the image forming method, it is possible to increase the area in which the image is formed by increasing the area of the element of the matrix receiver 2. Moreover, the image resolution does not deteriorate due to the fact that increasing the area of the element of the matrix receiver 2 is equivalent to increasing the size of the Airy spot, i.e. the deterioration of the resolution of the optical system 1 in the focal plane. However, this deterioration is not fundamental for the proposed method of image formation, since its resolution can be improved due to the above General procedure. Thus, a reduction in the number of elements of the matrix receiver 2 can be achieved and the cost of the entire radio-vision system can be reduced.

The effectiveness of the proposed method of image formation is confirmed by the following example of numerical simulation. The problem of forming an image created by one-dimensional radiation sources that are distributed in the plane of sources 11 along the Ox axis was solved. 6, the solid curve shows the intensity distribution of the studied sources in the plane of sources 11. It is described by a step function that models two studied separately located sources having an intensity of 0.86 over an interval of 0.12 length and located at a distance of 0.25 from each other. It can be seen that the distribution of the studied radiation sources in this example has sharp jumps in intensity. Note that in the example of numerical modeling, dimensionless coordinates (relative units) were used, since the absolute values of the source sizes for modeling the proposed method of image formation do not matter. An important parameter is the ratio of their sizes to the width of the hardware function.

The radiation of the sources passes through the optical system 1. In this example, the hardware function of the optical system 1 is described by the following expression:

$$K\left(x,\xi \right)=\frac{\mathrm{one}}{\mathrm{one}+\frac{{\left(x-\xi \right)}^2}{0.03}}.\left(\mathrm{fourteen}\right)$$

From formula (14) it can be seen that the width of the hardware function at a level of 0.5 from the maximum is finite and equal to 0.35. The hardware function has the same width in the plane of sources 11 and in the focal plane, that is, its compression ratio is equal to unity. Thus, its own resolving powers ρ and h in the indicated planes are identical and equal to 0.35.

The simulated matrix receiver 2 consists of three virtual receivers located with a period of 0.3. The element size of the matrix receiver is also 0.3. The field in the focal plane of the optical system moved with a step s = 0.015. Matrix receiver 2 provided at its outputs 6 a signal-to-noise ratio of 30.

The weight function F (ξ) describing the element of the matrix receiver 2 was assumed to be equal to unity in the region occupied by the element of the matrix receiver 2 and equal to zero outside it.

In accordance with the claimed method of image formation, at the first stage, an array of values of electrical signals p was generated, which were created by the radiation source under study (points in Fig. 6). The points describe an array of signal values p _n , which were obtained from all outputs 6 of the matrix receiver 2 at all steps of moving the radiation field of the sources in the focal plane of the optical system 1.

The dimension of this array N was equal to 201. From the distribution of signal values (points in Fig. 6) it can be seen that image formation is one of the known methods, according to which the elements of the array p are taken as the values of the image intensity (see relation (6) ) cannot reflect sharp jumps in the distribution of the radiation intensity of the studied sources. 6 also shows the noise component in the recorded signals. Thus, the use of this known method of image formation does not eliminate distortions due to diffraction effects and noise interference.

At the second stage, a one-dimensional grid with nodes at the points x _i , i = 1 ... N _x was introduced in the plane of the sources. The grid had a uniform pitch Δx = 0.03. Parameter N _x = 100. After that, an array of values of electric signals q was formed, which were created at the outputs 6 of the matrix receiver 2 by an independent point source. 7 shows the distribution of the values of electrical signals in the array q by index i for fixed values of index n.

At the third stage, an array of intensity values was formed

images u _i . The formation of an array of image intensity values was carried out by minimizing the variation in image intensity L (u):

$$L\left(u\right)={\displaystyle \sum _{i=\mathrm{one}}^{N_x-\mathrm{one}}|\; |\; |u_{i+\mathrm{one}}-u_i|\; |\; |}\left(\mathrm{fifteen}\right)$$

with restrictions:

$$\begin{array}{l}\underset{\begin{array}{l}\mathrm{one}\le n\le N\\ \mathrm{one}\le m\le M\end{array}}{{\displaystyle \max }}|\; |\; |p_n-{\displaystyle \sum _{i=\mathrm{one}}^{N_x-\mathrm{one}}{u}_i{q}_{n,i}\Delta x}|\; |\; |\le \delta ,\\ u_i\ge 0\end{array}\}\left(16\right)$$

where the parameter δ was chosen equal to 0.001. Relations (15) and (16) are a one-dimensional analogue of relations (8) and (9). Minimization of the variation in the image intensity L (u) was carried out on a computer using the modified simplex method.

Figure 8 shows the intensity distributions of the image formed in accordance with different methods. Curve 8.1 corresponds to the initial distribution of the radiation intensity of the studied radiation sources; it is constructed in a slightly different vertical scale and corresponds to the solid curve in Fig. 6, on which an additional frame denotes the region of Fig. 8. Curve 8.2 shows the image obtained from the radiation sources in Fig. 8.1, passed through the optical system, as described above, and formed using the proposed method of image formation. Curves 8.3 and 8.4 are obtained using the known method of image formation, in which the functional (7) is minimized, also reduced to a one-dimensional form; curves 8.3 and 8.4 correspond to different values of the parameter α.

From Fig. 8 it can be seen that the image formed in accordance with the proposed method (curve 8.2) most closely conveys the characteristic features of the intensity distribution of the two studied sources. In particular, it has the same sharp jumps that are present in the intensity distribution of the studied sources. It can also be noted that in the generated image, two separate sources separated in space are clearly distinguished, which indicates a high degree of resolution obtained using the proposed image forming method.

Images formed in accordance with known methods (curves 8.3 and 8.4 in Fig. 8) differ from the intensity distribution of the studied sources much more strongly than the image formed by the proposed method. In them, the images of the two studied sources merge, which indicates a relatively low degree of resolution of the formed image. Thus, the above example shows that the use of the proposed method of image formation ensures the achievement of the stated goal (technical result), which consists in increasing the resolution of the image in the presence of diffraction, noise and other types of interference.

In conclusion of the description, we note that the information contained in it indicates the fulfillment of the following set of conditions when using the claimed invention:

- a method of forming an image created by the studied radiation sources that emit in the ranges of millimeter, terahertz and far infrared waves, as applied to the device that implements it, are intended for use in industry, namely in radio-vision systems;

- for the inventive method of forming an image in the form as described in the claims, the possibility of its implementation using the methods described in the application is confirmed;

- the claimed method of image formation allows you to implement the following technical result: to preserve the resolution of the images of the studied radiation sources in the ranges of millimeter, terahertz and far infrared waves while reducing the size of the optical system and thereby reduce the cost of radio-vision systems of these ranges or increase the resolution of images of the studied radiation sources with the same size of the optical system.

Claims (5)

1. The method of forming an image created by radiation sources that emit in the ranges of millimeter, terahertz and far infrared waves and are located in the plane of the image sources, using an optical system configured to stepwise move the radiation field that is created by radiation sources in the focal plane of the optical system , by converting at each step of the movement of the specified radiation field from sources into electrical signals at the outputs of the matrix receiver, which located in the first focal plane of the optical system, and the subsequent formation of an array of values of said electrical signals p _{n, m, 1≤n≤N,} 1≤m≤M, where p _{n, m} - the electric signal to the n-th step of moving the radiation field at the mth output of the matrix receiver, N is the total number of steps for moving the radiation field, M is the total number of elements of the matrix receiver, characterized in
that to increase the resolution of the image in the presence of diffraction restrictions, noise and other noise in the source plane, a rectangular grid is introduced with nodes at the coordinates x _i and y _j , 1≤i≤N _x , 1≤j≤N _y , where N _x - the number of nodes of a rectangular grid along the 0x axis, and N _y is the number of nodes of a rectangular grid along the 0y axis, an independent point source of radiation is placed sequentially in the nodes of a rectangular grid, this point source is moved step by step, and an array of values of electric signals is generated at the matrix outputs of the second receiver q _{n, m, i, j} , at each step of moving the radiation field of the point source for each position of the point source at the nodes of the rectangular grid, a two-dimensional array of u _{i, j} values of the image intensity at the nodes of the rectangular grid is independently formed, and this two-dimensional array is formed from the condition of minimizing the variation in image intensity L (u):

under restrictions

where Δx _i = x _{i + 1} -x _i , Δy _j = y _{j + 1} -y _j are the sizes of the cells of the rectangular grid, δ is the specified error in the formation of the array of values of the image intensity at the nodes of the rectangular grid, the dimension of the array of quantities of electrical signals at the outputs of the matrix the NM receiver selects a larger number of nodes of a rectangular grid N _x N _y . 2. The method according to paragraph 1, characterized in that the cell sizes of the rectangular grid Δx _j , Δy _j are selected in accordance with the condition:
Δx _i , Δy _j ≤ρ / K,
K> 1,
where ρ is the intrinsic resolution of the optical system in the plane of the image sources, K is the coefficient of image resolution improvement. 3. The method according to paragraph 1, characterized in that the step of moving the radiation field s in the focal plane of the optical system is selected from the condition:
s≤h / K,
where h is the intrinsic resolution of the optical system in the focal plane. 4. The method according to paragraph 1, characterized in that the minimum variation in image intensity is sought by successively improving the solution in accordance with the modified simplex linear programming method. 5. The method and device according to claim 1, characterized in that the element of the matrix receiver is made with an area exceeding h ² , where h is the intrinsic resolution of the optical system in the focal plane.

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