IMAGING DEVICE AND IMAGING METHOD

Abstract

An imaging device includes: a plurality of transmitters that each transmit a wave to a measurement area; a plurality of receivers that each receive a scattered wave of the wave from the measurement area; and an information processing circuit that images an object in the measurement area using measurement data of the scattered wave. The information processing circuit: derives a scattering field function using the measurement data and a velocity vector of the object; derives an imaging function that is defined using an amount output from the scattering field function in response to inputting an imaging target position into the scattering field function; and images the object in the measurement area using the imaging function. The information processing circuit derives the scattering field function from the measurement data using a coordinate system in which the position of the object is fixed.

Claims

1. An imaging device comprising: a plurality of transmitters that each transmit a wave to a measurement area; a plurality of receivers that each receive a scattered wave of the wave from the measurement area; and an information processing circuit that images an object in the measurement area using measurement data of the scattered wave, wherein in imaging the object, the information processing circuit: derives, using the measurement data and a velocity vector of the object, a scattering field function that receives a transmission position of the wave and a reception position of the scattered wave as input and outputs an amount of the scattered wave at the reception position; derives an imaging function that receives an imaging target position as input and outputs an image intensity at the imaging target position, and is defined using an amount output from the scattering field function in response to inputting the imaging target position into the scattering field function as the transmission position and the reception position; and images the object in the measurement area using the imaging function, and in deriving the scattering field function, the information processing circuit derives the scattering field function from the measurement data using a coordinate system that is determined using the velocity vector and in which a position of the object is fixed in at least two directions.

2. The imaging device according to claim 1, wherein the plurality of transmitters and the plurality of receivers are arranged along a straight line parallel to a y-axis, the scattering field function is expressed as: $\left[\mathrm{Math}.1\right]$ $\left(x,y_1,y_2,z,k\right)=\frac{1}{{\left(2\right)}^3}\underset{-}{\overset{}{}}\underset{-}{\overset{}{}}\underset{-}{\overset{}{}}{e}^{-i\left(k_{x}x+k_{y1}{y}_1+k_{y2}{y}_2\right)}\stackrel{~}{}\left({\mathrm{uk}}_x+{\mathrm{wk}}_{y_1}+{\mathrm{wk}}_{y_2},k_{y_1},k_{y_2},k\right).Math.e^{i\left\{\sqrt{{\left(\sqrt{k^2-k_{y_1}^2}+\sqrt{k^2-k_{y_2}^2}\right)}^2-k_x^2}\right\}z}{\mathrm{udk}}_x{\mathrm{dk}}_{y_1}{\mathrm{dk}}_{y_2}$ where x and z of the scattering field function respectively represent an x-coordinate and a z-coordinate of the transmission position and the reception position, y.sub.1 of the scattering field function represents a y-coordinate of the transmission position, y.sub.2 of the scattering field function represents a y-coordinate of the reception position, k represents a wavenumber of the wave, and k.sub.x, k.sub.y1, and k.sub.y2 respectively represent variables corresponding to wavenumbers with respect to x, y.sub.1, and y.sub.2 of the scattering field function, $\left[\mathrm{Math}.2\right]$ $\stackrel{}{}$ represents the measurement data that has been Fourier transformed, and u and w are defined as: $\left[\mathrm{Math}.3\right]$ $u=-v_x{.Math.\sqrt{v_x^2+v_y^2}.Math.}^{-1}$ $w=-v_y{.Math.\sqrt{v_x^2+v_y^2}.Math.}^{-1}$ where v.sub.x and v.sub.y respectively represent an x-component and a y-component of the velocity vector.

3. The imaging device according to claim 2, wherein the imaging function is expressed as: $\left[\mathrm{Math}.4\right]$ $\left(x,y,z\right)=\frac{1}{{\left(2\right)}^3}\underset{0}{\overset{}{}}\underset{-}{\overset{}{}}\underset{-}{\overset{}{}}\underset{-}{\overset{}{}}{e}^{-i\left(k_{x}x+k_{y1}y+k_{y2}y\right)}\stackrel{~}{}\left({\mathrm{uk}}_x+{\mathrm{wk}}_{y_1}+{\mathrm{wk}}_{y_2},k_{y_1},k_{y_2},k\right).Math.e^{{\mathrm{ik}}_{z}z}\left(\frac{\mathrm{dk}}{{\mathrm{dk}}_z}\right){\mathrm{udk}}_x{\mathrm{dk}}_{y_1}{\mathrm{dk}}_{y_2}{\mathrm{dk}}_z$ where x, y, and z of the imaging function respectively represent an x-coordinate, a y-coordinate, and a z-coordinate of the imaging target position, and k.sub.z and dk/dk.sub.z are defined as: $\left[\mathrm{Math}.5\right]$ $k_z=\sqrt{{\left(\sqrt{k^2-k_{y_1}^2}+\sqrt{k^2-k_{y_2}^2}\right)}^2-k_x^2}$ $\frac{\mathrm{dk}}{{\mathrm{dk}}_z}=\frac{k_z\sqrt{k^2-k_{y_1}^2}\sqrt{k^2-k_{y_2}^2}}{k\left(k_x^2+k_z^2\right)}$

4. The imaging device according to claim 1, wherein the plurality of transmitters are arranged along a first straight line parallel to a y-axis, the plurality of receivers are arranged along a second straight line parallel to the y-axis, the scattering field function is expressed as: $\left[\mathrm{Math}.6\right]$ $\left(x_1,y_1,x_2,y_2,z,k\right)=\frac{1}{{\left(2\right)}^3}\underset{-}{\overset{}{}}\underset{-}{\overset{}{}}\underset{-}{\overset{}{}}{e}^{-i\left(k_{x_1}{x}_1+k_{y1}{y}_1+k_{y2}{y}_2\right)}.Math.e^{s_3\left(x_2-d\right)}{e}^{s_{4}z}u\stackrel{~}{}\left(k_{x}u+k_{y_1}w+k_{y_2}w,k_{y_1},k_{y_2},k\right){\mathrm{dk}}_x{\mathrm{dk}}_{y_1}{\mathrm{dk}}_{y_2}$ where z of the scattering field function represents a z-coordinate of the transmission position and the reception position, x.sub.1 and y.sub.1 of the scattering field function respectively represent an x-coordinate and a y-coordinate of the transmission position, x.sub.2 and y.sub.2 of the scattering field function respectively represent an x-coordinate and a y-coordinate of the reception position, k represents a wavenumber of the wave, k.sub.x, k.sub.y1, and k.sub.y2 respectively represent variables corresponding to wavenumbers with respect to x, y.sub.1, and y.sub.2 of the scattering field function, and d represents a distance between the first straight line and the second straight line, $\left[\mathrm{Math}.7\right]$ $\stackrel{}{}$ represents the measurement data that has been Fourier transformed, s.sub.3 and s.sub.4 are defined as: $\left[\mathrm{Math}.8\right]$ $s_3=\frac{-{\mathrm{ik}}_{x_1}\sqrt{k^2-k_{y_2}^2}}{\sqrt{k^2-k_{y_1}^2}}$ $s_4=i\sqrt{{\left(\sqrt{k^2-k_{y_1}^2}+\sqrt{k^2-k_{y_2}^2}\right)}^2-{\left(k_{x_1}+{\mathrm{is}}_3\right)}^2}$ and u and w are defined as: $\left[\mathrm{Math}.9\right]$ $u=-v_x{.Math.\sqrt{v_x^2+v_y^2}.Math.}^{-1}$ $w=-v_y{.Math.\sqrt{v_x^2+v_y^2}.Math.}^{-1}$ where v.sub.x and v.sub.y respectively represent an x-component and a y-component of the velocity vector.

5. The imaging device according to claim 4, wherein the imaging function is expressed as: $\left[\mathrm{Math}.10\right]$ $\left(x,y,z\right)=\frac{1}{{\left(2\right)}^3}\underset{0}{\overset{}{}}\mathrm{dk}\underset{-}{\overset{}{}}\underset{-}{\overset{}{}}\underset{-}{\overset{}{}}{e}^{-i\left(k_{x}x+k_{y1}y+k_{y2}y\right)}.Math.e^{-s_{3}d}{e}^{s_{4}z}u\stackrel{~}{}\left(k_{x}u+k_{y_1}w+k_{y_2}w,k_{y_1},k_{y_2},k\right){\mathrm{dk}}_x{\mathrm{dk}}_{y_1}{\mathrm{dk}}_{y_2}$ where x, y, and z of the imaging function respectively represent an x-coordinate, a y-coordinate, and a z-coordinate of the imaging target position.

6. The imaging device according to claim 5, wherein the information processing circuit images the object using the imaging function in which an x-coordinate is shifted by x=L+, and the imaging function in which the x-coordinate is shifted is expressed as: $\left[\mathrm{Math}.11\right]$ $\left(,y,z\right)=\frac{1}{{\left(2\right)}^3}\underset{0}{\overset{}{}}\mathrm{dk}\underset{-}{\overset{}{}}\underset{-}{\overset{}{}}\underset{-}{\overset{}{}}{e}^{-i\left(k_x+k_{y1}y+k_{y2}y\right)}.Math.e^{-s_{3}d}{e}^{s_{4}z}u\left\{e^{-{\mathrm{ik}}_{x}L}\stackrel{~}{}\left(k_{x}u+k_{y_1}w+k_{y_2}w,k_{y_1},k_{y_2},k\right)\right\}{\mathrm{dk}}_x{\mathrm{dk}}_{y_1}{\mathrm{dk}}_{y_2}$ where L represents an amount of shift in the x-coordinate, and represents the x-coordinate shifted by L.

7. The imaging device according to claim 1, wherein the information processing circuit: derives a provisional scattering field function using the measurement data without using the velocity vector; derives a provisional imaging function using the provisional scattering field function; and derives the velocity vector using the provisional imaging function.

8. The imaging device according to claim 7, wherein in deriving the velocity vector, the information processing circuit: derives the velocity vector using an arithmetic expression represented as: $\left[\mathrm{Math}.12\right]$ $\left(\begin{array}{c}{v}_x\\ v_y\\ v_z\end{array}\right)=-{\left(\begin{array}{ccc}2a+2& e& g\\ e& 2b+2& f\\ g& f& 2c+2\end{array}\right)}^{-1}\left(\begin{array}{c}h\\ m\\ n\end{array}\right)$ where a, b, c, e, f, g, h, m, and n in the arithmetic expression are defined as: $\left[\mathrm{Math}.13\right]$ $a=\frac{1}{{\left(2\right)}^3}{k}_x^2{.Math.{}_k.Math.}^2{\mathrm{dk}}^3$ $b=\frac{1}{{\left(2\right)}^3}{k}_y^2{.Math.{}_k.Math.}^2{\mathrm{dk}}^3$ $c=\frac{1}{{\left(2\right)}^3}{k}_z^2{.Math.{}_k.Math.}^2{\mathrm{dk}}^3$ $e=\frac{1}{{\left(2\right)}^3}{k}_x{k}_z{.Math.{}_k.Math.}^2{\mathrm{dk}}^3+\mathrm{cc}.$ $f=\frac{1}{{\left(2\right)}^3}{k}_y{k}_z{.Math.{}_k.Math.}^2{\mathrm{dk}}^3+\mathrm{cc}.$ $g=\frac{1}{{\left(2\right)}^3}{k}_z{k}_x{.Math.{}_k.Math.}^2{\mathrm{dk}}^3+\mathrm{cc}.$ $h=\frac{-i}{{\left(2\right)}^3}{k}_x{}_k.Math.{}_{}{\stackrel{}{}}_k{\mathrm{dk}}^3+\mathrm{cc}.$ $m=\frac{-i}{{\left(2\right)}^3}{k}_y{}_k.Math.{}_{}{\stackrel{}{}}_k{\mathrm{dk}}^3+\mathrm{cc}.$ $n=\frac{-i}{{\left(2\right)}^3}{k}_z{}_k.Math.{}_{}{\stackrel{\_}{}}_k{\mathrm{dk}}^3+\mathrm{cc}.$ v.sub.x, v.sub.y, and v.sub.z respectively represent an x-component, a y-component, and a z-component of the velocity vector, F represents a positive value, .sub.k represents the provisional imaging function that includes as one of a plurality of input variables and is Fourier transformed with respect to x, y, and z, and k.sub.x, k.sub.y, and k.sub.z in the arithmetic expression respectively represent wavenumbers with respect to x, y, and z of the provisional imaging function, $\left[\mathrm{Math}.14\right]$ $k$ represents a wavenumber vector of k.sub.x, k.sub.y, and k.sub.z in the arithmetic expression, represents a time unit corresponding to a number of data collections, and cc. represents a complex conjugate of a term immediately preceding cc., and $\left[\mathrm{Math}.15\right]$ ${\stackrel{\_}{}}_k$ represents the complex conjugate of .sub.k.

9. An imaging method comprising: transmitting, by each of a plurality of transmitters, a wave to a measurement area; receiving, by each of a plurality of receivers, a scattered wave of the wave from the measurement area; and imaging an object in the measurement area using measurement data of the scattered wave, wherein the imaging of the object includes: deriving, using the measurement data and a velocity vector of the object, a scattering field function that receives a transmission position of the wave and a reception position of the scattered wave as input and outputs an amount of the scattered wave at the reception position; deriving an imaging function that receives an imaging target position as input and outputs an image intensity at the imaging target position, and is defined using an amount output from the scattering field function in response to inputting the imaging target position into the scattering field function as the transmission position and the reception position; and imaging the object in the measurement area using the imaging function, and in the deriving of the scattering field function, the scattering field function is derived from the measurement data using a coordinate system that is determined using the velocity vector and in which a position of the object is fixed in at least two directions.

Description

BRIEF DESCRIPTION OF DRAWINGS

[0015] FIG. 1 is a conceptual diagram showing a one-dimensional multistatic array antenna.

[0016] FIG. 2 is a conceptual diagram showing the relationship between a transmission point and a reception point in a plane.

[0017] FIG. 3 is a conceptual diagram showing an example of coordinates relating to a semi-two-dimensional array antenna.

[0018] FIG. 4 is a conceptual diagram illustrating the relationship between a plurality of transmitters, a plurality of receivers, and a flying object.

[0019] FIG. 5 is a conceptual diagram illustrating two coordinate systems when a one-dimensional array transmitting/receiving array antenna is used.

[0020] FIG. 6 is a conceptual diagram illustrating two coordinate systems when a semi-two-dimensional array transmitting array antenna and receiving array antenna are used.

[0021] FIG. 7 is a conceptual diagram showing an analysis region far from the antenna installation position.

[0022] FIG. 8 is a block diagram showing a basic configuration of an imaging device.

[0023] FIG. 9 is a flowchart showing basic operations of an imaging device.

DESCRIPTION OF EMBODIMENTS

[0024] An imaging device according to one aspect of the present disclosure includes: a plurality of transmitters that each transmit a wave to a measurement area; a plurality of receivers that each receive a scattered wave of the wave from the measurement area; and an information processing circuit that images an object in the measurement area using measurement data of the scattered wave. In imaging the object, the information processing circuit: derives, using the measurement data and a velocity vector of the object, a scattering field function that receives a transmission position of the wave and a reception position of the scattered wave as input and outputs an amount of the scattered wave at the reception position; derives an imaging function that receives an imaging target position as input and outputs an image intensity at the imaging target position, and is defined using an amount output from the scattering field function in response to inputting the imaging target position into the scattering field function as the transmission position and the reception position; and images the object in the measurement area using the imaging function. In deriving the scattering field function, the information processing circuit derives the scattering field function from the measurement data using a coordinate system that is determined using the velocity vector and in which a position of the object is fixed in at least two directions.

[0025] This allows the imaging device to image the object by fixing the position of the object. Accordingly, the imaging device is capable of imaging a moving object in the measurement area with high accuracy. The imaging device is capable of transmitting waves and receiving scattered waves from various directions with respect to the object without having to move the plurality of transmitters and the plurality of receivers. Accordingly, the imaging device is capable of obtaining sufficient information and imaging the object with high accuracy even with a small number of transmitters and receivers.

[0026] For example, the plurality of transmitters and the plurality of receivers are arranged along a straight line parallel to a y-axis, [0027] the scattering field function is expressed as:

[00001]$\begin{array}{cc}\left(x,y_1,y_2,z,k\right)=\frac{1}{{\left(2\right)}^3}\underset{-}{\stackrel{}{}}\underset{-}{\stackrel{}{}}\underset{-}{\stackrel{}{}}{e}^{-i\left(k_{x}x+k_{y_1}{y}_1+k_{y_2}{y}_2\right)}\stackrel{~}{}\left({\mathrm{uk}}_x+{\mathrm{wk}}_{y_1}+{\mathrm{wk}}_{y_2},k_{y_1},k_{y_2},k\right).Math.e^{i\left\{\sqrt{{\left(\sqrt{k^2-k_{y_1}^2}+\sqrt{k^2-k_{y_2}^2}\right)}^2-k_x^2}\right\}z}{\mathrm{udk}}_x{\mathrm{dk}}_{y_1}{\mathrm{dk}}_{y_2}& \left[\mathrm{Math}.1\right]\end{array}$ [0028] where x and z of the scattering field function respectively represent an x-coordinate and a z-coordinate of the transmission position and the reception position, y.sub.1 of the scattering field function represents a y-coordinate of the transmission position, y.sub.2 of the scattering field function represents a y-coordinate of the reception position, k represents a wavenumber of the wave, and k.sub.x, k.sub.y1, and k.sub.y2 respectively represent variables corresponding to wavenumbers with respect to x, y.sub.1, and y.sub.2 of the scattering field function,

[00002]$\begin{array}{cc}\stackrel{}{}& \left[\mathrm{Math}.2\right]\end{array}$ [0029] represents the measurement data that has been Fourier transformed, and [0030] u and w are defined as:

[00003]$$\begin{gathered} \begin{array}{cc}u=-v_x{.Math.\sqrt{v_x^2+v_y^2}.Math.}^{-1} \\ w=-v_y{.Math.\sqrt{v_x^2+v_y^2}.Math.}^{-1}& \left[\mathrm{Math}.3\right]\end{array} \end{gathered}$$ [0031] where v.sub.x and v.sub.y respectively represent an x-component and a y-component of the velocity vector.

[0032] This allows the imaging device to image a moving object in the measurement area with high accuracy by using the scattering field function derived from the measurement data obtained by the plurality of transmitters and the plurality of receivers arranged in a straight line.

[0033] For example, the imaging function is expressed as:

[00004]$\begin{array}{cc}\left(x,y,z\right)=\frac{1}{{\left(2\right)}^3}\underset{0}{\stackrel{}{}}\underset{-}{\stackrel{}{}}\underset{-}{\stackrel{}{}}\underset{-}{\stackrel{}{}}{e}^{-i\left(k_{x}x+k_{y_1}y+k_{y_2}y\right)}\stackrel{~}{}\left({\mathrm{uk}}_x+{\mathrm{wk}}_{y_1}+{\mathrm{wk}}_{y_2},k_{y_1},k_{y_2},k\right).Math.e^{{\mathrm{ik}}_{z}z}\left(\frac{\mathrm{dk}}{{\mathrm{dk}}_z}\right)u{\mathrm{dk}}_x{\mathrm{dk}}_{y_1}{\mathrm{dk}}_{y_2}{\mathrm{dk}}_z& \left[\mathrm{Math}.4\right]\end{array}$ [0034] where x, y, and z of the imaging function respectively represent an x-coordinate, a y-coordinate, and a z-coordinate of the imaging target position, and [0035] k.sub.z and dk/dk.sub.z are defined as:

[00005]$\begin{array}{cc}{k}_z=\sqrt{{\left(\sqrt{k^2-k_{y_1}^2}+\sqrt{k^2-k_{y_2}^2}\right)}^2-k_x^2}& \left[\mathrm{Math}.5\right]\end{array}$$\frac{\mathrm{dk}}{{\mathrm{dk}}_z}=\frac{k_z\sqrt{k^2-k_{y_1}^2}\sqrt{k^2-k_{y_2}^2}}{k\left(k_x^2+k_z^2\right)}.$

[0036] This allows the imaging device to image a moving object in the measurement area with high accuracy by using the imaging function derived from the measurement data obtained by the plurality of transmitters and the plurality of receivers arranged in a straight line.

[0037] For example, the plurality of transmitters are arranged along a first straight line parallel to a y-axis, [0038] the plurality of receivers are arranged along a second straight line parallel to the y-axis, [0039] the scattering field function is expressed as:

[00006]$\begin{array}{cc}\left(x_1,y_1,x_2,y_2,z,k\right)=\frac{1}{{\left(2\right)}^3}\underset{-}{\stackrel{}{}}\underset{-}{\stackrel{}{}}\underset{-}{\stackrel{}{}}{e}^{-i\left(k_{x_1}{x}_1+k_{y_1}{y}_1+k_{y_2}{y}_2\right)}.Math.e^{s_3\left(x_2-d\right)}{e}^{s_{4}z}u\stackrel{~}{}\left(k_{x}u+k_{y_1}w+k_{y_2}w,+k_{y_2}w,k_{y_1},k_{y_2},k\right){\mathrm{dk}}_x{\mathrm{dk}}_{y_1}{\mathrm{dk}}_{y_2}& \left[\mathrm{Math}.6\right]\end{array}$ [0040] where z of the scattering field function represents a z-coordinate of the transmission position and the reception position, x.sub.1 and y.sub.1 of the scattering field function respectively represent an x-coordinate and a y-coordinate of the transmission position, x.sub.2 and y.sub.2 of the scattering field function respectively represent an x-coordinate and a y-coordinate of the reception position, k represents a wavenumber of the wave, k.sub.x, k.sub.y1, and ky.sub.2 respectively represent variables corresponding to wavenumbers with respect to x, y.sub.1, and y.sub.2 of the scattering field function, and d represents a distance between the first straight line and the second straight line,

[00007]$\left[\mathrm{Math}.7\right]$$\stackrel{}{}$ [0041] represents the measurement data that has been Fourier transformed, [0042] s.sub.3 and s.sub.4 are defined as:

[00008]$\left[\mathrm{Math}.8\right]$$s_3=\frac{-{\mathrm{ik}}_{x_1}\sqrt{k^2-k_{y_2}^2}}{\sqrt{k^2-k_{y_1}^2}}$$s_4=i\sqrt{{\left(\sqrt{k^2-k_{y_1}^2}+\sqrt{k^2-k_{y_2}^2}\right)}^2-{\left(k_{x_1}+{\mathrm{is}}_3\right)}^2}$ [0043] and u and w are defined as:

[00009]$\left[\mathrm{Math}.9\right]$$u=-v_x{.Math.\sqrt{v_x^2+v_y^2}.Math.}^{-1}$$w=-v_y{.Math.\sqrt{v_x^2+v_y^2}.Math.}^{-1}$ [0044] where v.sub.x and v.sub.y respectively represent an x-component and a y-component of the velocity vector.

[0045] This allows the imaging device to image a moving object in the measurement area with high accuracy by using the scattering field function derived from the measurement data obtained by the plurality of transmitters arranged on a first straight line and the plurality of receivers arranged on a second straight line.

[0046] Moreover, for example, the imaging function is expressed as:

[00010]$\left[\mathrm{Math}.10\right]$$\left(x,y,z\right)=\frac{1}{{\left(2\right)}^3}\underset{0}{\overset{}{}}\mathrm{dk}\underset{-}{\overset{}{}}\underset{-}{\overset{}{}}\underset{-}{\overset{}{}}{e}^{-i\left(k_{x}x+k_{y_1}y+k_{y_2}y\right)}.Math.e^{-s_{3}d}{e}^{s_{4}z}u\stackrel{}{}\left(k_{x}u+k_{y_1}w+k_{y_2}w,k_{y_1},k_{y_2},k\right)d{k}_{x}d{k}_{y_1}{\mathrm{dk}}_{y_2}$ [0047] where x, y, and z of the imaging function respectively represent an x-coordinate, a y-coordinate, and a z-coordinate of the imaging target position.

[0048] This allows the imaging device to image a moving object in the measurement area with high accuracy by using the imaging function derived from the measurement data obtained by the plurality of transmitters arranged on a first straight line and the plurality of receivers arranged on a second straight line.

[0049] For example, the information processing circuit images the object using the imaging function in which an x-coordinate is shifted by x=L+, and [0050] the imaging function in which the x-coordinate is shifted is expressed as:

[00011]$\left[\mathrm{Math}.11\right]$$\left(,y,z\right)=\frac{1}{{\left(2\right)}^3}\underset{0}{\overset{}{}}\mathrm{dk}\underset{-}{\overset{}{}}\underset{-}{\overset{}{}}\underset{-}{\overset{}{}}{e}^{-i\left(k_x+k_{y_1}y+k_{y_2}y\right)}.Math.e^{-s_{3}d}{e}^{s_{4}z}u\{e^{-{\mathrm{ik}}_{x}L}\stackrel{}{}\left(k_{x}u+k_{y_1}w+k_{y_2}w,k_{y_1},k_{y_2},k\right)d{k}_{x}d{k}_{y_1}{\mathrm{dk}}_{y_2}$ [0051] where L represents an amount of shift in the x-coordinate, and represents the x-coordinate shifted by L.

[0052] This allows the imaging device to shift the imaging target area. Accordingly, the imaging device is capable of efficiently imaging a distant object from the plurality of transmitters and the plurality of receivers.

[0053] For example, the information processing circuit: derives a provisional scattering field function using the measurement data without using the velocity vector; derives a provisional imaging function using the provisional scattering field function; and derives the velocity vector using the provisional imaging function.

[0054] This allows the imaging device to derive the velocity vector of the object using the provisional scattering field function and the provisional imaging function derived without using the velocity vector of the object. The imaging device is capable of imaging an object that moves in the measurement area with high accuracy by using the scattering field function and the imaging function derived using the velocity vector.

[0055] For example, in deriving the velocity vector, the information processing circuit: [0056] derives the velocity vector using an arithmetic expression represented as:

[00012]$\left[\mathrm{Math}.12\right]$$\left(\begin{array}{c}{v}_x\\ v_y\\ v_z\end{array}\right)=-{\left(\begin{array}{cccc}2a+& 2& e& g\\ e& & 2b+2& f\\ g& & f& 2c+2\end{array}\right)}^{-1}\left(\begin{array}{c}h\\ m\\ n\end{array}\right)$ [0057] where a, b, c, e, f, g, h, m, and n in the arithmetic expression are defined as:

[00013]$\left[\mathrm{Math}.13\right]$$a=\frac{1}{{\left(2\right)}^3}{k}_x^2{.Math.{}_k.Math.}^2{\mathrm{dk}}^3$$b=\frac{1}{{\left(2\right)}^3}{k}_y^2{.Math.{}_k.Math.}^2{\mathrm{dk}}^3$$c=\frac{1}{{\left(2\right)}^3}{k}_z^2{.Math.{}_k.Math.}^2{\mathrm{dk}}^3$$e=\frac{1}{{\left(2\right)}^3}{k}_x{k}_y{.Math.{}_k.Math.}^2{\mathrm{dk}}^3+\mathrm{cc}.$$f=\frac{1}{{\left(2\right)}^3}{k}_y{k}_x{.Math.{}_k.Math.}^2{\mathrm{dk}}^3+\mathrm{cc}.$$g=\frac{1}{{\left(2\right)}^3}{k}_z{k}_x{.Math.{}_k.Math.}^2{\mathrm{dk}}^3+\mathrm{cc}.$$h=\frac{-i}{{\left(2\right)}^3}{k}_x{}_k.Math.{}_{}{\stackrel{\_}{}}_k{\mathrm{dk}}^3+\mathrm{cc}.$$m=\frac{-i}{{\left(2\right)}^3}{k}_y{}_k.Math.{}_{}{\stackrel{\_}{}}_k{\mathrm{dk}}^3+\mathrm{cc}.$$n=\frac{-i}{{\left(2\right)}^3}{k}_z{}_k.Math.{}_{}{\stackrel{\_}{}}_k{\mathrm{dk}}^3+\mathrm{cc}.$ [0058] v.sub.x, v.sub.y, and v.sub.z respectively represent an x-component, a y-component, and a z-component of the velocity vector, represents a positive value, .sub.k represents the provisional imaging function that includes as one of a plurality of input variables and is Fourier transformed with respect to x, y, and z, and k.sub.x, k.sub.y, and k.sub.z in the arithmetic expression respectively represent wavenumbers with respect to x, y, and z of the provisional imaging function,

[00014]$\left[\mathrm{Math}.14\right]$$k$ [0059] represents a wavenumber vector of k.sub.x, k.sub.y, and k.sub.z in the arithmetic expression, represents a time unit corresponding to a number of data collections, and cc. represents a complex conjugate of a term immediately preceding cc., and

[00015]$\left[\mathrm{Math}.15\right]$${\stackrel{\_}{}}_k$ [0060] represents the complex conjugate of Pk.

[0061] This allows the imaging device to derive the velocity vector of the object from the provisional imaging function using an arithmetic expression.

[0062] An imaging method according to one aspect of the present disclosure includes: transmitting, by each of a plurality of transmitters, a wave to a measurement area; receiving, by each of a plurality of receivers, a scattered wave of the wave from the measurement area; and imaging an object in the measurement area using measurement data of the scattered wave. The imaging of the object includes: deriving, using the measurement data and a velocity vector of the object, a scattering field function that receives a transmission position of the wave and a reception position of the scattered wave as input and outputs an amount of the scattered wave at the reception position; deriving an imaging function that receives an imaging target position as input and outputs an image intensity at the imaging target position, and is defined using an amount output from the scattering field function in response to inputting the imaging target position into the scattering field function as the transmission position and the reception position; and imaging the object in the measurement area using the imaging function. In the deriving of the scattering field function, the scattering field function is derived from the measurement data using a coordinate system that is determined using the velocity vector and in which a position of the object is fixed in at least two directions.

[0063] This allows imaging of the object by fixing the position of the object. Accordingly, it is possible to image a moving object in the measurement area with high accuracy. It is possible to transmit waves and receive scattered waves from various directions with respect to the object without having to move the plurality of transmitters and the plurality of receivers. Accordingly, it is possible to obtain sufficient information and image the object with high accuracy even with a small number of transmitters and receivers.

[0064] Hereinafter, embodiments will be described with reference to the drawings. Each of the following embodiments describes a general or specific example. The numerical values, shapes, materials, elements, the arrangement and connection of the elements, steps, the order of the steps etc., presented in the following embodiments are mere examples, and do not limit the scope of the claims.

[0065] In the following description, in particular the techniques or the like described in PTLs 2 to 5 given above may be referenced to as existing techniques. Although radio waves or electromagnetic waves such as microwaves are primarily assumed as the waves in the following description, the waves are not limited to radio waves or electromagnetic waves such as microwaves, and may be elastic waves or the like. Imaging based on scattering may be expressed as scattering tomography. Thus, the imaging device and imaging method described below may also be expressed as a scattering tomographic device and a scattering tomographic method, respectively.

EMBODIMENT

[0066] An imaging device according to the present embodiment images an object in a measurement area using measurement data of scattered waves. Hereinafter, the imaging device according to the present embodiment, including techniques and theories serving as the basis of the imaging device, will be described in detail.

1. Overview

[0067] Scattering field theory is a theory used for imaging an object in a measurement area. For example, waves are transmitted from each of a plurality of transmission positions to the measurement area, and scattered waves from the measurement area are received at each of a plurality of reception positions. Measurement data of the scattered waves is obtained for each combination of transmission position and reception position. The object in the measurement area is then imaged using the measurement data. At this time, using the scattering field theory, the condition of scattering in the measurement area is calculated from the measurement data, and the object is imaged with high accuracy based on the condition of scattering.

[0068] However, when an object moves in the measurement area, the position of the object becomes uncertain, making it difficult to image the object with high accuracy.

[0069] In view of this, the present disclosure describes a scattering field theory for imaging a moving object in the measurement area with high accuracy. More specifically, first, a scattering field theory that does not take the movement of the object into consideration will be described, and then, a scattering field theory that does take the movement of the object into consideration will be described.

2. Scattering Field Theory Without Consideration of Object Movement

[0070] In this chapter, a scattering field theory that does not take the movement of the object into consideration will be described.

<2-1. One-dimensional Array>

[0071] FIG. 1 is a conceptual diagram showing a one-dimensional multistatic array antenna. As illustrated in FIG. 1, by using two arbitrary elements from among n elements as a transmitting element and a receiving element, it is possible to obtain double the resolution compared to a monostatic environment where the pair of transmitting and receiving elements is fixed. Moreover, singles can be received with a high S/N ratio at a range of distances from short distance to long distance. This considerably improves the quality of an ultimate image. Whereas, as a matter of course, the amount of data is increased to n times, the time required for reconstruction is also shortened dramatically according to the theory described below.

[0072] Here, a situation is examined in which a radio wave radiated from point P.sub.1(x, y.sub.1, z) is reflected at point P(, , ) and received at point P.sub.2(x, y.sub.2, z) as shown in FIG. 1. In the case where point P is assumed to move in entire region D, the signal received at point P.sub.2 is represented by expression (2-1-1) below.

[00016]$\left[\mathrm{Math}.16\right]$$\begin{array}{cc}\left(x,y_1,y_2,z\right)=\underset{D}{}\frac{e^{\mathrm{ik}{}_1}}{{}_1}\frac{e^{\mathrm{ik}{}_2}}{{}_2}\left(,,\right)ddd& \left(2-1-1\right)\end{array}$${}_1=\sqrt{{\left(x-\right)}^2+{\left(y_1-\right)}^2+{\left(z-\right)}^2}$${}_2=\sqrt{{\left(x-\right)}^2+{\left(y_2-\right)}^2+{\left(z-\right)}^2}$

[0073] Here, (, , ) represents the function of the dielectric constant at point P(, , ) and corresponds to the reflectance at point P(, , ). Point P(, , ) corresponds to the reflection point. Note that (, , ) is unknown. It is assumed that the time factor is proportional to exp(it). The kernel function in the integrand term of the above equation is represented as cp in expression (2-1-2) below.

[00017]$\left[\mathrm{Math}.17\right]$$\begin{array}{cc}=\frac{e^{\mathrm{ik}{}_1}}{{}_1}\frac{e^{\mathrm{ik}{}_2}}{{}_2}& \left(2-1-2\right)\end{array}$

[0074] Next, a partial differential equation that has expression (2-1-2) as an asymptotical solution is examined. Thus, calculation is performed while ignoring a high-order term with respect to 1/ obtained as a result of differentiation. Here, an abridged notation for differentiation is defined by expression (2-1-3).

[00018]$\left[\mathrm{Math}.18\right]$$\begin{array}{cc}\frac{}{t}.fwdarw.{}_t,\frac{}{x}.fwdarw.{}_x,\frac{}{y_1}.fwdarw.{}_{y_1},\frac{}{y_2}.fwdarw.{}_{y_2},\frac{}{z}.fwdarw.{}_z& \left(2-1-3\right)\end{array}$

[0075] Here, a partial differential equation that has expression (2-1-2) as an asymptotical solution at short wavelengths (at high frequency or when k is large) is examined. This solution to the partial differential equation may be regarded as almost an exact solution in imaging using microwaves. First, the result of differentiation of each order of is represented by expression (2-1-4) below.

[00019]$\left[\mathrm{Math}.19\right]$$\begin{array}{cc}{}_x=\mathrm{ik}\left(x-\right)\left(\frac{1}{{}_1}+\frac{1}{{}_2}\right)+o\left({}^{-3}\right)& \left(2-1-4\right)\end{array}$${}_{y_1}=\mathrm{ik}\frac{y_1-}{{}_1}+o\left({}^{-3}\right)$${}_{y_2}=\mathrm{ik}\frac{y_2-}{{}_2}+o\left({}^{-3}\right)$${}_z=\mathrm{ik}\left(z-\right)\left(\frac{1}{{}_1}+\frac{1}{{}_2}\right)+o\left({}^{-3}\right)$${}_x{}_x={\left(\mathrm{ik}\right)}^2{\left(x-\right)}^2{\left(\frac{1}{{}_1}+\frac{1}{{}_2}\right)}^2+o\left({}^{-3}\right)$${}_z{}_z={\left(\mathrm{ik}\right)}^2{\left(z-\right)}^2{\left(\frac{1}{{}_1}+\frac{1}{{}_2}\right)}^2+o\left({}^{-3}\right)$${}_{y_1}{}_{y_1}={\left(\mathrm{ik}\right)}^2{\left(\frac{y_1-}{{}_1}\right)}^2+o\left({}^{-3}\right)$${}_{y_2}{}_{y_2}={\left(\mathrm{ik}\right)}^2{\left(\frac{y_2-}{{}_2}\right)}^2+o\left({}^{-3}\right)$

[0076] Hereinafter, the complexity o(*) is omitted. In accordance with the sum of four differential equations of the second order, expression (2-1-5) below is obtained.

[00020]$\left[\mathrm{Math}.20\right]$$\begin{array}{cc}\begin{array}{c}{}_4=\left({}_x^2+{}_{y_1}^2+{}_{y_2}^2+{}_z^2\right)\\ ={\left(\mathrm{ik}\right)}^2\left\{2+2\frac{{\left(x-\right)}^2+{\left(z-\right)}^2}{{}_1{}_2}\right\}\end{array}& \left(2-1-5\right)\end{array}$

[0077] Accordingly, expression (2-1-6) below is obtained from expression (2-1-5).

[00021]$\left[\mathrm{Math}.21\right]$$\begin{array}{cc}\begin{array}{c}\left\{{}_4-2{\left(\mathrm{ik}\right)}^2\right\}=2{\left(\mathrm{ik}\right)}^2\frac{{}_1^2-{\left(y_1-\right)}^2}{{}_1{}_2}\\ =2{\left(\mathrm{ik}\right)}^2\frac{{}_2^2-{\left(y_2-\right)}^2}{{}_1{}_2}\end{array}& \left(2-1-6\right)\end{array}$

[0078] By acting the operation of expression (2-1-6) two times, expression (2-1-7) below is obtained.

[00022]$\left[\mathrm{Math}.22\right]$$\begin{array}{cc}\begin{array}{cc}{\left\{{}_{4}2{\left(ik\right)}^2\right\}}^2& =4{\left(ik\right)}^4\frac{\left\{{}_1^2{\left(y_1\right)}^2\right\}\left\{{}_2^2{\left(y_2\right)}^2\right\}}{{}_1^2{}_2^2}\\ & =4{\left(ik\right)}^4\left\{1{\left(ik\right)}^2{}_{y_1}^2\right\}\left\{1{\left(ik\right)}^2{}_{y_2}^2\right\}\end{array}& \left(2-1-7\right)\end{array}$

[0079] Expression (2-1-7) is summarized to obtain expression (2-1-8) below.

[00023]$\left[\mathrm{Math}.23\right]$$\begin{array}{cc}\left[\frac{1}{4}{\left\{{}_{4}2{\left(ik\right)}^2\right\}}^2{}_{y_1}^2{}_{y_2}^2+{\left(ik\right)}^2\left({}_{y_1}^2+{}_{y_2}^2\right){\left(ik\right)}^4\right]=0& \left(2-1-8\right)\end{array}$

[0080] Although expression (2-1-8) is derived assuming a steady state, it is easy to extend expression (2-1-8) to a non-steady state. Thus, variables are substituted as given by expression (2-1-9) below, using partial differential .sub.t with respect to time t and using propagation velocity c of radio waves.

[00024]$\left[\mathrm{Math}.24\right]$$\begin{array}{cc}-\mathrm{ik}.fwdarw.\frac{1}{c}{}_t& \left(2-1-9\right)\end{array}$

[0081] Through the process described above, an equation represented by expression (2-1-10) below is ultimately obtained.

[00025]$\left[\mathrm{Math}.25\right]$$\begin{array}{cc}\left\{{}_4^2\frac{4}{c^2}\left({}_t^2{}_x^2+{}_t^2{}_z^2\right)4{}_{y_1}^2{}_{y_2}^2\right\}=0& \left(2-1-10\right)\end{array}$${}_4={}_x^2+{}_{y_1}^2+{}_{y_2}^2+{}_z^2$

[0082] Expression (2-1-10) described above is a partial differential equation that has in expression (2-1-2) as a solution. By applying differentiation to the kernel of expression (2-1-1),

[00026]$\left[\mathrm{Math}.26\right]$$$

of expression (2-1-1) also satisfies the partial differential equation described above. This equation is a four-dimensional pseudo wave equation configured by five variables (t, x, y.sub.1, y.sub.2, z).

[0083] Next, this equation is solved by Fourier transform. First,

[00027]$\left[\mathrm{Math}.27\right]$$$

is subjected to multiplex Fourier transform with respect to t, x, y.sub.1, y.sub.2 as given by expression (2-1-11) below.

[00028]$\left[\mathrm{Math}.28\right]$$\begin{array}{cc}\stackrel{~}{}\left(k_x,k_{y_1},k_{y_2},z,\right)=\stackrel{}{\underset{}{}}{e}^{it}\mathrm{dt}\stackrel{}{\underset{}{}}\stackrel{}{\underset{}{}}\stackrel{}{\underset{}{}}{e}^{i\left(k_{x}x+k_{y_1}{y}_1+k_{y_2}{y}_2\right)}\left(x,y_1,y_2,z,t\right){\mathrm{dxdy}}_{1}d{y}_2& \left(2-1-11\right)\end{array}$

[0084] When the differential with respect to z is expressed as D.sub.z, expression (2-1-12) below is obtained from expressions (2-1-10) 5 and (2-1-11).

[00029]$\left[\mathrm{Math}.29\right]$$\begin{array}{cc}\left\{{\left(D_z^2{k}_x^2{k}_{y_1}^2{k}_{y_2}^2\right)}^2+4k^2\left(D_z^2{k}_x^2\right)4k_{y_1}^2{k}_{y_2}^2\right\}\stackrel{~}{}=0& \left(2-1-12\right)\end{array}$

[0085] Here, the relationship of =ck is used. Four basic solutions to this equation are expressed as given by expression (2-1-13) below.

[00030]$\left[\mathrm{Math}.30\right]$$\begin{array}{cc}{E}_1=e^{i\left\{\sqrt{{\left(\sqrt{k^2{k}_{y_1}^2}+\sqrt{k^2{k}_{y_2}^2}\right)}^2{k}_x^2}\right\}z}& \left(2-1-13\right)\end{array}$$E_2=e^{i\left\{\sqrt{{\left(\sqrt{k^2{k}_{y_1}^2}+\sqrt{k^2{k}_{y_2}^2}\right)}^2{k}_x^2}\right\}z}$$E_3=e^{i\left\{\sqrt{{\left(\sqrt{k^2{k}_{y_1}^2}+\sqrt{k^2{k}_{y_2}^2}\right)}^2{k}_x^2}\right\}z}$$E_4=e^{i\left\{\sqrt{{\left(\sqrt{k^2{k}_{y_1}^2}\sqrt{k^2{k}_{y_2}^2}\right)}^2{k}_x^2}\right\}z}$

[0086] Considering the facts that the time factor is e.sup.t, the phase is added using the path of radiated radio waves, and radio waves reflected off the object are bounced off toward a measurement surface (measurement plane), E.sub.1 is the unique meaningful solution. Accordingly, expression (2-1-14) below is obtained.

[00031]$\left[\mathrm{Math}.31\right]$$\begin{array}{cc}\stackrel{~}{}\left(k_x,k_{y_1},k_{y_2},z,k\right)=a\left(k_x,k_{y_1},k_{y_2},k\right)e^{i\left\{\sqrt{{\left(\sqrt{k^2-k_{y_1}^2}+\sqrt{k^2-k_{y_2}^2}\right)}^2-k_x^2}\right\}z}& \left(2-1-14\right)\end{array}$

[0087] By substituting z=0 in expression (2-1-14), a(k.sub.x, k.sub.y1, k.sub.y2, k) is obtained as given by expression (2-1-15) below.

[00032]$\left[\mathrm{Math}.32\right]$$\begin{array}{cc}a\left(k_x,k_{y_1},k_{y_2},k\right)=\stackrel{~}{}\left(k_x,k_{y_1},k_{y_2},0,k\right)& \left(2-1-15\right)\end{array}$

[0088] Ultimately,

[00033]$\left[\mathrm{Math}.33\right]$$$

is obtained as given by expression (2-1-16) below.

[00034]$\left[\mathrm{Math}.34\right]$$\begin{array}{cc}\left(x,y_1,y_2,z,k\right)=\frac{1}{{\left(2\right)}^3}\stackrel{}{\underset{}{}}\stackrel{}{\underset{}{}}\stackrel{}{\underset{}{}}{e}^{i\left(k_{x}x+k_{y_1}{y}_1+k_{y_2}{y}_2\right)}a\left(k_x,k_{y_1},k_{y_2},k\right).Math.e^{i\left\{\sqrt{{\sqrt{k^2{k}_{y_1}^2}+\sqrt{k^2{k}_{y_2}^2})}^2{k}_x^2}\right\}z}d{k}_{x}d{k}_{y_1}d{k}_{y_2}& \left(2-1-16\right)\end{array}$

[0089] By applying a limit operation (y.sub.2.fwdarw.y.sub.1=y) to expression (2-1-16) on condition that k and z are fixed and integrating the result with respect to k, the imaging function is obtained as given by expression (2-1-17) below.

[00035]$\left[\mathrm{Math}.35\right]$$\begin{array}{cc}\left(x,y_1,y_2,z,k\right)=\underset{y_2.fwdarw.y_1=y}{\mathrm{Lim}}\left[\left(x,y,y_2,z,k\right)\right]=\underset{y_2.fwdarw.y_1=y}{\mathrm{Lim}}\left[\frac{1}{{\left(2\right)}^3}\stackrel{}{\underset{}{}}\stackrel{}{\underset{}{}}\stackrel{}{\underset{}{}}{e}^{i\left(k_{x}x+k_{y_1}y+k_{y_2}{y}_2\right)}a\left(k_x,k_{y_1},k_{y_2},k\right).Math.e^{i\left\{\sqrt{{\sqrt{k^2{k}_{y_1}^2}+\sqrt{k^2{k}_{y_2}^2})}^2{k}_x^2}\right\}z}d{k}_{x}d{k}_{y_1}d{k}_{y_2}\right]& \left(2-1-17\right)\end{array}$$\left(x,y,z\right)=\stackrel{}{\underset{0}{}}\left(x,y,y,z,k\right)\mathrm{dk}$

[0090] As described above, it becomes possible to analytically solve a multistatic inverse scattering problem with a one-dimensional array.

<2-2. Two-dimensional Array>

[0091] This section describes the theory for a case pertaining to a two-dimensional array.

[0092] FIG. 2 is a conceptual diagram showing the relationship between a transmission point and a reception point in a plane. As shown in FIG. 2, a microwave radiated from point P.sub.1 is reflected at point P on a target and received at point P.sub.2. Points P.sub.1 and P.sub.2 move to arbitrary points on a grading (two-dimensional array antenna) in a plane. Under this assumption, there are n.sup.4 different microwave paths passing through point P on the target. This large number of paths considerably contributes to an improvement in the quality of an ultimate image. A method for processing such complex data to obtain an image will be described below.

[0093] For example, as shown in FIG. 2, the radio wave radiated from point P.sub.1(x.sub.1, y.sub.1, z) is reflected at point P(, , ) and received at point P.sub.2(x.sub.2, y.sub.2, z). When point P is assumed to move in entire region D, a signal received at P.sub.2 is expressed by the following expression.

[00036]$\left[\mathrm{Math}.36\right]$$\begin{array}{cc}\left(x_1,y_1,x_2,y_2,z\right)=\underset{D}{}\frac{e^{\mathrm{ik}{}_1}}{{}_1}\frac{e^{\mathrm{ik}{}_2}}{{}_2}\left(,,\right)ddd& \left(2-1-1\right)\end{array}$${}_1=\sqrt{{\left(x_1\right)}^2+{\left(y_1\right)}^2+{\left(z\right)}^2}$${}_2=\sqrt{{\left(x_2\right)}^2+{\left(y_2\right)}^2+{\left(z\right)}^2}$

[0094] It is assumed here that the time factor is proportional to exp(it). The kernel function in the integrand term of the above expression is expressed as given by expression (2-2-2) below.

[00037]$\begin{array}{cc}\left[\mathrm{Math}.37\right]& \\ =\frac{e^{\mathrm{ik}{}_1}}{{}_1}\frac{e^{\mathrm{ik}{}_2}}{{}_2}& \left(2-2-2\right)\end{array}$

[0095] Next, a partial differential equation that has expression (2-2-2) as an asymptotical solution at short wavelengths is examined. Thus, calculation is performed while ignoring a high-order term with respect to 1/ obtained as a result of differentiation. Here, an abridged notation for differentiation is defined by expression (2-2-3).

[00038]$\begin{array}{cc}\left[\mathrm{Math}.38\right]& \\ \frac{}{t}.fwdarw.{}_t,\frac{}{x_1}.fwdarw.{}_{x_1},\frac{}{x_2}.fwdarw.{}_{x_2}.fwdarw.\frac{}{y_1}.fwdarw.{}_{y_1},\frac{}{y_2}.fwdarw.{}_{y_2},\frac{}{z}.fwdarw.{}_z& \left(2-2-3\right)\end{array}$

[0096] Using expression (2-2-3), differentiation of each order of the kernel function is expressed as given by expression (2-2-4) below.

[00039]$\begin{array}{cc}\left[\mathrm{Math}.39\right]& \\ x_1=\mathrm{ik}\frac{x_1-}{{}_1}+o\left({}^{-3}\right)x_2=\mathrm{ik}\frac{x_2-}{{}_2}+o\left({}^{-3}\right)y_1=\mathrm{ik}\frac{y_1-}{{}_1}+o\left({}^{-3}\right)y_2=\mathrm{ik}\frac{y_2-}{{}_2}+o\left({}^{-3}\right){}_z=\mathrm{ik}\left(z-\right)\left(\frac{1}{{}_1}+\frac{1}{{}_2}\right)+o\left({}^{-3}\right)x_1{x}_1={\left(\mathrm{ik}\right)}^2{\left(\frac{x_1-}{{}_1}\right)}^2+o\left({}^{-3}\right)y_1{y}_1={\left(\mathrm{ik}\right)}^2{\left(\frac{y_1-}{{}_1}\right)}^2+o\left({}^{-3}\right)x_2{x}_2={\left(\mathrm{ik}\right)}^2{\left(\frac{x_2-}{{}_2}\right)}^2+o\left({}^{-3}\right)y_2{y}_2={\left(\mathrm{ik}\right)}^2{\left(\frac{y_2-}{{}_2}\right)}^2+o\left({}^{-3}\right){}_z{}_z={\left(\mathrm{ik}\right)}^2{\left(z-\right)}^2{\left(\frac{1}{{}_1}+\frac{1}{{}_2}\right)}^2+o\left({}^{-3}\right)& \left(2-2-4\right)\end{array}$