Data processing method, measurement system, and program

Abstract

Provided is a data processing method for analyzing scattered waves of waves emitted to an object, wherein: waves are emitted to an object from a plurality of transmission points p1(x'1, y'1, z'1) arranged two-dimensionally on an xy plane; scattered waves reflected with a reflectivity f(x, y, z) at a reflection point (x, y, z) on the object are received, as measurement values s(x'1, x'2, y'1, y'2, z'1, z'2, k), at a plurality of reception points p2(x'2, y'2, z'2) arranged two-dimensionally on the xy plane; a quadruple Fourier transform is performed on the measurement values s(x'1, x'2, y'1, y'2, z'1, z'2, k) to find S(k'x1, k'x2, k'y1, k'y2, z'1, z'2, k); with k'x1,nyq being defined as the Nyquist frequency in the x'1 direction determined by a measurement interval in the x direction, and k'x2,nyq being defined as the Nyquist frequency in the x'2 direction, variable substitution processing in the x direction is performed in the range of -kx1≤k'x1≤kx1 and -kx2≤k'x2≤kx2 (where k'x1,nyq≤kx1 and k'x2,nyq≤kx2); with k'y1,nyq being defined as the Nyquist frequency in the y'1 direction determined by the measurement interval in the y direction, and k'y2,nyq being defined as the Nyquist frequency in the y'2 direction, variable substitution processing in the y direction is performed in the range of -ky1≤k'y1≤ky1 and -ky2≤k'y2≤ky2 (where k'y1,nyq≤ky1 and k'y2,nyq≤ky2); and a triple inverse Fourier transform is performed to find reflectivity f(x, y, z).

Description

Data processing method, measurement system, and program

[0001] The present invention relates to a data processing method, a measurement system, and a program for processing, using a computer, measurement data of waves whose values ​​are determined by the frequency of waves such as electromagnetic waves generated in space and the spatial coordinates of the space.

[0002] Conventionally, radar devices have been known that nondestructively inspect the interior of nonmetallic structures such as concrete and wood. Conventional radar devices have an array antenna in which multiple antennas are arranged on a flat surface. The array antenna has, for example, a configuration in which antennas such as planar antennas are aligned in one direction, with a transmitting array antenna and a receiving array antenna arranged closely to each other. Furthermore, in order to accurately measure the interior of a structure, the radar device measures the object to be measured using a wide frequency band while changing the frequency of electromagnetic waves at set frequency intervals.

[0003] Regarding radar devices having array antennas, for example, a radar device in which a transmitting array antenna and a receiving array antenna each composed of a plurality of planar antennas are formed on a common dielectric substrate is known (JP 2015-095840 A, hereinafter referred to as "Patent Document 1"). In conventional radar devices, the arrangement direction of the planar antennas of the transmitting array antenna is parallel to the arrangement direction of the planar antennas of the receiving array antenna. The position of the planar antennas of the receiving array antenna in the arrangement direction is midway between the positions of two adjacent planar antennas of the transmitting array antenna.

[0004] Also, a scattering tomography method is known that can quickly and generally analyze an inverse problem and easily visualize information about the interior of an object (Japanese Patent No. 6557747, hereinafter referred to as "Patent Document 2").

[0005] Synthetic aperture processing is used to visualize the inside of a structure from the measured data. Synthetic aperture processing can be broadly divided into two types: addition methods such as the diffraction stacking method, and methods that use Fourier transforms such as the FK migration method. In order to achieve a practical calculation time, synthetic aperture processing that uses Fourier transforms is the most practical.

[0006] In radar devices, high spatial resolution in measurements is desirable to accurately inspect the interior of structures. Generally, the spatial resolution of data obtained by radiating waves having frequencies such as electromagnetic waves is determined by the wavelength of the center frequency of the waves when the distance between the structure to be measured and the measurement surfaces of the transmitting array antenna and the receiving array antenna is relatively short and the measurement interval of the measurement data is small. Here, the distance between the structure to be measured and the measurement surfaces of the transmitting array antenna and the receiving array antenna is, for example, one-fourth or less of the array antenna arrangement length. Furthermore, the spatial resolution is the resolution within the plane on which each antenna of the array antenna is arranged.

[0007] For example, when the measurement interval of measurement data along the plane on which each antenna of the array antenna is arranged is sufficiently small, and the frequency of the radiated wave is swept over a frequency band, taking into account the round-trip path of the electromagnetic wave, the theoretical spatial resolution is one-fourth of the wavelength of the wave at the center frequency of the frequency band. However, when the measurement interval of the measurement data becomes large and exceeds one-fourth of the minimum wavelength of the electromagnetic wave, the spatial resolution in the actual measurement becomes larger than the ideal spatial resolution. In some cases, the spatial resolution in the actual measurement becomes the measurement interval.

[0008] Conventional radar devices measure electromagnetic waves over a wide frequency band, from low to high frequencies, resulting in a long maximum wavelength of the electromagnetic waves. This requires the antennas in an array antenna to be large, resulting in a long antenna arrangement. As a result, the spacing between antennas in a transmitting / receiving array antenna increases, and the measurement interval for measurement data tends to exceed one-quarter of the minimum wavelength of the emitted electromagnetic waves. This reduces the spatial resolution below the theoretical resolution, and the measurement data is prone to aliasing. To achieve the ideal spatial resolution—i.e., one-quarter of the wavelength of the electromagnetic waves at the center frequency—requires increasing the number of antennas in the transmitting / receiving array antenna and shortening the spacing between them. However, as mentioned above, it is difficult to reduce the size of a wideband antenna, making it difficult to shorten the spacing between antennas.

[0009] An object of the present invention is to provide a data processing method, a measurement system, and a program that can improve the spatial resolution in measurement while maintaining the number of antennas arranged constant.

[0010] A first aspect of the present invention is a data processing method for analyzing scattered waves of waves radiated to an object, the method comprising: radiating the waves to the object from a plurality of transmitting points p1(x'1, y'1, z'1) arranged on a y-axis; receiving the scattered waves reflected at a reflecting point (x, y, z) on the object with a reflectivity f(x, y, z) at a plurality of receiving points p2(x'2, y'2, z'2) arranged on the y-axis as measurement values ​​s(x'1, x'2, y'1, y'2, z'1, z'2, k); and performing a triple Fourier transform of the measurement values ​​s(x'1, x'2, y'1, y'2, z'1, z'2, k) using equation (1) to obtain S(k' x1 , k' x2 , k' y1 , k' y2 , z'1, z'2, k) and The Nyquist wave number in the x direction determined by the measurement interval in the x direction is k x,nyq Then, -k x1 ≦k x ≦k x1 (However, k x,nyq ≦k x1 ) variable substitution processing in the x direction is performed within the range, and the Nyquist wave number in the y'1 direction determined by the measurement interval in the y direction is k' y1,nyq and the Nyquist wave number in the y'2 direction is k' y2,nyq Then, -k y1 ≦k' y1 ≦k y1 , and -k y2 ≦k' y2 ≦k y2 (However, k' y1,nyq ≦k y1 , and k' y2,nyq ≦k y2 ) variable substitution processing in the y direction is performed within the range of (x, y, z), and a triple inverse Fourier transform is performed using equation (2) to obtain the reflectance f(x, y, z). This is a data processing method, where x' = x'1 = x'2 z'1 = z'2 = 0 k is the wave number of the propagating wave, k' x1 , k' y1 , k' z1 is the component of the wave vector of the spherical wave of the wave propagating between the transmission point p1 (x'1, y'1, z'1) and the reflection point (x, y, z), k' x2 , k' y2 , k' z2 is the component of the wave vector of the spherical wave of the wave propagating between the reflection point (x, y, z) and the reception point p2 (x'2, y'2, z'2), k x = k' x1 + k' x2 , u = k' x1 - k' x2 , k y = k' y1 + k' y2 , v = k' y1 - k' y2 is.

[0011] A second aspect of the present invention is a measurement system for analyzing scattered waves of waves radiated to an object, the system comprising: a transmitter / receiver unit that radiates the waves to the object from a plurality of transmission points p1(x'1, y'1, z'1) arranged on a y axis; and a receiver unit that receives the scattered waves reflected at reflection points (x, y, z) on the object with reflectivity f(x, y, z) at a plurality of reception points p2(x'2, y'2, z'2) arranged on the y axis as measurement values ​​s(x'1, x'2, y'1, y'2, z'1, z'2, k); and a processing unit that performs a triple Fourier transform of the measurement values ​​s(x'1, x'2, y'1, y'2, z'1, z'2, k) using equation (1) to obtain S(k' x1 , k' x2 , k' y1 , k' y2 , z'1, z'2, k) and The Nyquist wave number in the x direction determined by the measurement interval in the x direction is k x,nyq Then, -k x1 ≦kx ≦k x1 (However, k x,nyq ≦k x1 ) and the procedure for performing variable substitution processing in the x direction within the range of the measurement interval in the y direction. y1,nyq and the Nyquist wave number in the y'2 direction is k' y2,nyq Then, -k y1 ≦k' y1 ≦k y1 , and -k y2 ≦k' y2 ≦k y2 (However, k' y1,nyq ≦k y1 , and k' y2,nyq ≦k y2 ) a procedure for performing variable substitution processing in the y direction within the range of (x, y, z); a procedure for performing a triple inverse Fourier transform using equation (2) to obtain the reflectance f(x, y, z); and a processing unit for executing the above equation: where x' = x'1 = x'2, z'1 = z'2 = 0, k is the wave number of the propagating wave, and k' x1 , k' y1 , k' z1 is the component of the wave vector of the spherical wave of the wave propagating between the transmission point p1 (x'1, y'1, z'1) and the reflection point (x, y, z), k' x2 , k' y2 , k' z2 is the component of the wave vector of the spherical wave of the wave propagating between the reflection point (x, y, z) and the reception point p2 (x'2, y'2, z'2), k x = k' x1 + k' x2 , u = k' x1 - k' x2 , k y = k' y1 + k' y2 , v = k' y1 - k' y2 is.

[0012] A third aspect of the present invention is a program for analyzing scattered waves of waves radiated to an object, the program comprising: performing a triple Fourier transform on a measurement value s(x'1, x'2, y'1, y'2, z'1, z'2, k) using equation (1) to obtain S(k' x1 , k' x2 , k' y1 , k' y2 , z'1, z'2, k) and The Nyquist wave number in the x direction determined by the measurement interval in the x direction is k x,nyq Then, -k x1 ≦k x ≦k x1 (However, k x,nyq ≦k x1 ) and the procedure for performing variable substitution processing in the x direction within the range of the measurement interval in the y direction. y1,nyq and the Nyquist wave number in the y'2 direction is k' y2,nyq Then, -k y1 ≦k' y1 ≦k y1 , and -k y2 ≦k' y2 ≦k y2 (However, k' y1,nyq ≦k y1 , and k' y2,nyq ≦k y2 ) and a procedure for performing variable substitution processing in the y direction within the range of (x, y, z). A procedure for calculating the reflectance f(x, y, z) by performing a triple inverse Fourier transform using equation (2). The program is executed by a computer, where x' = x'1 = x'2 z'1 = z'2 = 0 k is the wave number of the propagating wave, k' x1 , k' y1 , k' z1 is the component of the wave vector of the spherical wave of the wave propagating between the transmission point p1 (x'1, y'1, z'1) and the reflection point (x, y, z), k' x2 , k' y2 , k' z2 is the component of the wave vector of the spherical wave of the wave propagating between the reflection point (x, y, z) and the reception point p2 (x'2, y'2, z'2), k x = k'x1 + k' x2 , u = k' x1 - k' x2 , k y = k' y1 + k' y2 , v = k' y1 - k' y2 is.

[0013] A fourth aspect of the present invention is a data processing method for analyzing scattered waves of waves radiated to an object, the method comprising: radiating the waves to the object from a plurality of transmission points p1(x'1, y'1, z'1) arranged two-dimensionally on an xy plane; receiving the scattered waves reflected at reflection points (x, y, z) on the object with reflectivity f(x, y, z) at a plurality of reception points p2(x'2, y'2, z'2) arranged two-dimensionally on the xy plane as measurement values ​​s(x'1, x'2, y'1, y'2, z'1, z'2, k); and performing a quadruple Fourier transform on the measurement values ​​s(x'1, x'2, y'1, y'2, z'1, z'2, k) using equation (1) to obtain S(k' x1 , k' x2 , k' y1 , k' y2 , z'1, z'2, k) and The Nyquist wave number in the x'1 direction determined by the measurement interval in the x direction is k' x1,nyq and the Nyquist wave number in the x'2 direction is k' x2,nyq Then, -k x1 ≦k' x1 ≦k x1 , and -k x2 ≦k' x2 ≦k x2 (However, k' x1,nyq ≦k x1 , and k' x2,nyq ≦k x2 ) variable substitution processing in the x direction is performed within the range, and the Nyquist wave number in the y'1 direction determined by the measurement interval in the y direction is k' y1,nyq and the Nyquist wave number in the y'2 direction is k' y2,nyq Then, -k y1 ≦k' y1 ≦k y1 , and -k y2 ≦k'y2 ≦k y2 (However, k' y1,nyq ≦k y1 , and k' y2,nyq ≦k y2 ) variable substitution processing in the y direction is performed within the range of (x, y, z), and a triple inverse Fourier transform is performed using equation (2) to obtain the reflectance f(x, y, z). This is a data processing method, where z'1 = z'2 = 0, k is the wave number of the propagating wave, and k' x1 , k' y1 , k' z1 is the component of the wave vector of the spherical wave of the wave propagating between the transmission point p1 (x'1, y'1, z'1) and the reflection point (x, y, z), k' x2 , k' y2 , k' z2 is the component of the wave vector of the spherical wave of the wave propagating between the reflection point (x, y, z) and the reception point p2 (x'2, y'2, z'2), k x = k' x1 + k' x2 , u = k' x1 - k' x2 , k y = k' y1 + k' y2 , v = k' y1 - k' y2 is.

[0014] A fifth aspect of the present invention is a measurement system for analyzing scattered waves of waves radiated to an object, the system comprising: a transmitter / receiver unit that radiates the waves to the object from a plurality of transmission points p1(x'1, y'1, z'1) arranged two-dimensionally on an xy plane; and a receiver unit that receives the scattered waves reflected at reflection points (x, y, z) on the object with reflectivity f(x, y, z) at a plurality of reception points p2(x'2, y'2, z'2) arranged two-dimensionally on the xy plane as measurement values ​​s(x'1, x'2, y'1, y'2, z'1, z'2, k), and a processing unit that performs a quadruple Fourier transform of the measurement values ​​s(x'1, x'2, y'1, y'2, z'1, z'2, k) using equation (1) to obtain S(k'x1 , k' x2 , k' y1 , k' y2 , z'1, z'2, k) and The Nyquist wave number in the x'1 direction determined by the measurement interval in the x direction is k' x1,nyq and the Nyquist wave number in the x'2 direction is k' x2,nyq Then, -k x1 ≦k' x1 ≦k x1 , and -k x2 ≦k' x2 ≦k x2 (However, k' x1,nyq ≦k x1 , and k' x2,nyq ≦k x2 ) and the procedure for performing variable substitution processing in the x direction within the range of the measurement interval in the y direction. y1,nyq and the Nyquist wave number in the y'2 direction is k' y2,nyq Then, -k y1 ≦k' y1 ≦k y1 , and -k y2 ≦k' y2 ≦k y2 (However, k' y1,nyq ≦k y1 , and k' y2,nyq ≦k y2 ) a procedure for performing variable substitution processing in the y direction within the range of (x, y, z); a procedure for performing a triple inverse Fourier transform using equation (2) to obtain the reflectance f(x, y, z); and a processing unit for executing the above equation, where z'1 = z'2 = 0, k is the wave number of the propagating wave, and k' x1 , k' y1 , k' z1 is the component of the wave vector of the spherical wave of the wave propagating between the transmission point p1 (x'1, y'1, z'1) and the reflection point (x, y, z), k' x2 , k' y2 , k' z2 is the component of the wave vector of the spherical wave of the wave propagating between the reflection point (x, y, z) and the reception point p2 (x'2, y'2, z'2), k x = k'x1 + k' x2 , u = k' x1 - k' x2 , k y = k' y1 + k' y2 , v = k' y1 - k' y2 is.

[0015] A sixth aspect of the present invention is a program for analyzing scattered waves of waves radiated to an object, the program comprising: performing a quadruple Fourier transform on a measurement value s(x'1, x'2, y'1, y'2, z'1, z'2, k) using equation (1) to obtain S(k' x1 , k' x2 , k' y1 , k' y2 , z'1, z'2, k) and The Nyquist wave number in the x'1 direction determined by the measurement interval in the x direction is k' x1,nyq and the Nyquist wave number in the x'2 direction is k' x2,nyq Then, -k x1 ≦k' x1 ≦k x1 , and -k x2 ≦k' x2 ≦k x2 (However, k' x1,nyq ≦k x1 , and k' x2,nyq ≦k x2 ) and the procedure for performing variable substitution processing in the x direction within the range of the measurement interval in the y direction. y1,nyq and the Nyquist wave number in the y'2 direction is k' y2,nyq Then, -k y1 ≦k' y1 ≦k y1 , and -k y2 ≦k' y2 ≦k y2 (However, k' y1,nyq ≦k y1 , and k' y2,nyq ≦k y2 ) and a procedure for performing variable substitution processing in the y direction within the range of (x, y, z). A procedure for calculating the reflectance f(x, y, z) by performing a triple inverse Fourier transform using equation (2). The program is executed by a computer. However, z'1 = z'2 = 0, k is the wave number of the propagating wave, and k' x1 , k' y1 , k' z1 is the component of the wave vector of the spherical wave of the wave propagating between the transmitting point p1 (x'1, y'1, z'1) and the reflecting point (x, y, z), k' x2 , k' y2 , k' z2 is the component of the wave vector of the spherical wave of the wave propagating between the reflection point (x, y, z) and the receiving point p2 (x'2, y'2, z'2), k x = k' x1 + k' x2 , u = k' x1 - k' x2 , k y = k' y1 + k' y2 , v = k' y1 - k' y2 is.

[0016] According to the data processing method, measurement system, and program of the present invention, it is possible to improve the spatial resolution in measurement while maintaining the number of antennas arranged constant.

[0017] FIG. 1 shows the configuration of a radar device according to a first embodiment. FIG. 2 shows the configuration of the array antenna shown in FIG. 1. FIG. 3 explains the positional relationship between the array antenna according to the first embodiment and a measurement target. １ axis, x' ２ Fourier transform about the axis y' in the second embodiment １ axis, y' ２Fourier transform about the x-axis. A diagram for explaining the positional relationship between the array antenna and the object to be measured in the third embodiment. Flowchart showing the data processing method in the third embodiment. Fourier transform about the x-axis in the third embodiment. y' in the third embodiment. １ axis, y' ２ axial Fourier transform xz plane image of point target 1 simulated by the data processing method of Comparative Example 1 xz plane image of point target 1 simulated by the data processing method of Example 1 yz plane image of point target 1 simulated by the data processing method of Comparative Example 1 yz plane image of point target 1 simulated by the data processing method of Example 1 Waveforms in the x-axis direction of point target 1 of Comparative Example 1 and Example 1 Waveforms in the y-axis direction of point target 1 of Comparative Example 1 and Example 1 Waveforms in the z-axis direction of point target 1 of Comparative Example 1 and Example 1 xz plane image of point target 2 simulated by the data processing method of Comparative Example 1 xz plane image of point target 2 simulated by the data processing method of Example 1 yz plane image of point target 2 simulated by the data processing method of Comparative Example 1 yz plane image of point target 2 simulated by the data processing method of Example 1 Waveforms in the x-axis direction of point target 2 of Comparative Example 1 and Example 1 Waveforms in the y-axis direction of point target 2 of Comparative Example 1 and Example 1 Waveforms in the z-axis direction of point target 2 of Comparative Example 1 and Example 1 Cross-shaped k' y1 -k' y2 Conceptual diagram of the filter Simulation results for four point targets Recovery rate of point targets Layout of transmitting and receiving antennas 3D image of test object without super-resolution processing 3D image of test object with super-resolution processing Cross-shaped k' y1 -k' y2 3D images of a test specimen subjected to super-resolution processing using a filter Comparative Example 2 and Example 2 Waveform in the x-axis direction of point target 1 Comparative Example 2 and Example 2 Waveform in the y-axis direction of point target 1 Comparative Example 2 and Example 2 Waveform in the z-axis direction of point target 1 Comparative Example 2 and Example 2 Waveform in the x-axis direction of point target 2 Comparative Example 2 and Example 2 Waveform in the y-axis direction of point target 2 Comparative Example 2 and Example 2 Waveform in the z-axis direction of point target 2 Cross-shaped k' ｘ1 -k' ｘ2 Conceptual diagram of a filter

[0018] <First Embodiment> A data processing method, a measurement system, and a program according to a first embodiment will be described in detail below. Fig. 1 shows the configuration of a radar device according to this embodiment. Fig. 2 shows the configuration of the array antenna shown in Fig. 1. Fig. 3 is a diagram illustrating the positional relationship between the array antenna according to this embodiment and an object to be measured. In this embodiment, electromagnetic waves are described as waves that are radiated into space, but waves that propagate through space, such as X-rays or ultrasound, may also be used instead of electromagnetic waves.

[0019] The measurement system 1 of this embodiment includes a transceiver and a processing device. The processing device may be provided integrally with the transceiver, or may be provided in a separate location connected to the transceiver via a network. In the following embodiment, an example in which the processing device is provided integrally with the transceiver will be described.

[0020] 1 uses a transmitting array antenna and a receiving array antenna (transmitter / receiver) to emit electromagnetic waves from the transmitting antenna while sweeping the frequency of the electromagnetic waves. The radar device 60 then receives the waves reflected from the object to be measured with the receiving antenna to obtain measurement data s(x', y', z', k). The measurement data s(x', y', z', k) is data whose variables are the x-coordinate component, the y-coordinate component, the z-coordinate component, and the frequency of the electromagnetic waves.

[0021] The radar device 60 has a measurement unit 61, a data processing unit (processing device) 66, and an image display unit 68. The measurement unit 61 has a transmitting array antenna 50, a receiving array antenna 52, high-frequency switches 58 and 59, a high-frequency circuit 62, and a system control circuit 64. The radar device 60 emits electromagnetic waves of 10 MHz or higher, for example, 10 to 20 GHz, but the frequency of the electromagnetic waves is not particularly limited.

[0022] As shown in Figure 2, the transmitting array antenna 50 has multiple transmitting antennas 10a arranged in one direction. Each transmitting antenna 10a radiates electromagnetic waves toward the object to be measured. The receiving array antenna 52 has multiple receiving antennas 10b arranged along the direction in which the transmitting antennas 10a are arranged. Each receiving antenna 10b receives electromagnetic waves reflected from the object to be measured. The transmitting antennas 10a of the transmitting array antenna 50 and the receiving antennas 10b of the receiving array antenna 52 are arranged on the same plane. The transmitting array antenna 50 and the receiving array antenna 52 are arranged so that the object to be measured faces this plane.

[0023] The data processing unit 66 processes a plurality of measurement data obtained by transmission to the measurement object by the plurality of transmitting antennas 10 a and reception by the plurality of receiving antennas 10 b, and calculates image data relating to the measurement object. The transmitting antenna 10 a and the receiving antenna 10 b in this embodiment are planar antennas in which antenna patterns are formed in a plane on a substrate, but are not limited to planar antennas.

[0024] The transmitting array antenna 50 and the receiving array antenna 52 move parallel to the surface of the object to be measured. That is, the transmitting array antenna 50 and the receiving array antenna 52 perform measurements while scanning along the surface of the object to be measured. When the transmitting array antenna 50 and the receiving array antenna 52 move, the system control circuit 64 controls the operation of the high-frequency circuit 62. Specifically, the system control circuit 64 controls the operation of the high-frequency circuit 62 so that the transmitting antenna 10a is switched by the high-frequency switch 58 for each unit length of movement of the transmitting array antenna 50 and the receiving array antenna 52, and electromagnetic waves are radiated.

[0025] The radar device 60 has an encoder 69. The encoder 69 generates a pulse signal every time a certain distance is moved. The encoder 69 senses the movement of the transmitting array antenna 50 and the receiving array antenna 52. At this time, every time an electromagnetic wave is radiated from each transmitting antenna 10a, the high-frequency switch 59 sequentially switches among the multiple receiving antennas 10b, causing each receiving antenna 10b to receive the radiation.

[0026] The frequency of the electromagnetic waves radiated from the transmitting array antenna 50 is swept at set frequency intervals, for example, within a range of 10 to 20 GHz, over a fixed period of time. Therefore, the measurement data obtained from the high-frequency circuit 62 is determined by the position from which the transmitting antenna 10a transmits, the position from which the receiving antenna 10b receives the signal, the frequency, and the target position. The operation of the high-frequency switch 59 is controlled so that the reflected electromagnetic waves radiated from the transmitting antenna 10a and reflected by the object being measured are received by the receiving antenna 10b closest to the transmitting antenna 10a that emitted the electromagnetic waves. The receiving microwave amplifier (RF amplifier) ​​may be configured to change its gain for each pair of transmitting antenna 10a and receiving antenna 10b. In this case, the high-frequency circuit 62 has a variable gain amplification function that switches the gain depending on the selected pair of transmitting antenna 10a and receiving antenna 10b. This allows for a wider inspection depth for defects, etc., in the object being measured.

[0027] In this embodiment, the arrangement directions of the transmitting antennas 10a and the receiving antennas 10b are parallel, and as shown in Figure 2, the arrangement direction is defined as the y direction. Meanwhile, the movement direction (scanning direction) of the transmitting array antenna 50 and the receiving array antenna 52 is defined as the x direction. The direction toward the object to be measured (the transmission direction of electromagnetic waves) as viewed from the transmitting array antenna 50 and the receiving array antenna 52 is defined as the z direction.

[0028] The movement direction (scanning direction) of the transmitting array antenna 50 and the receiving array antenna 52 may be the y direction. That is, they may move (scan) in the same direction as the arrangement direction of the transmitting antennas 10a and the receiving antennas 10b. Also, the transmitting array antenna 50 may have only one transmitting antenna 10a, and the receiving array antenna 52 may have multiple receiving antennas 10b. In this case, too, the movement direction (scanning direction) of the transmitting array antenna 50 and the receiving array antenna 52 may be the y direction. That is, they may move (scan) in the same direction as the arrangement direction of the receiving antennas 10b.

[0029] The data processing unit 66 processes the measurement data s(x', y', z', k) obtained by transmitting and receiving electromagnetic waves using the transmitting array antenna 50 and the receiving array antenna 52, and creates image data representing the interior of the object to be measured. The data processing unit 66 is configured, for example, by a computer, and invokes and starts a program stored in the storage unit 66a. This enables the functions of the data processing unit 66 to be fulfilled. In other words, the data processing unit 66 is configured by software modules. The image display unit 68 displays an image of the interior of the object to be measured using the created image data.

[0030] 2 shows a schematic diagram of a transmitting array antenna 50 and a receiving array antenna 52. The transmitting antenna 10a and the receiving antenna 10b are offset in the x direction by ΔL, but in the following description, the x direction positions of the transmitting antenna 10a and the receiving antenna 10b are assumed to be at the circular point midway between the transmitting antenna 10a and the receiving antenna 10b. This circular point is called the transmitting and receiving point. Note that there may be no offset in the y direction between the transmitting antenna 10a and the receiving antenna 10b. That is, there may be cases where Δy = 0. There may also be cases where the transmitting antenna 10a and the receiving antenna 10b are shared. That is, there may be cases where Δy = 0 and ΔL = 0.

[0031] Therefore, the positional relationship between the object to be measured, the transmitting array antenna 50, and the receiving array antenna 52 can be expressed as shown in Figure 3. Here, the coordinates of the transmitting and receiving points are assumed to be p(x', y', z'). The reflectance at the reflection point (x, y, z) of the object to be measured is assumed to be f(x, y, z). The measurement data at the transmitting and receiving point p(x', y', z') is assumed to be s(x', y', z', k). The propagation wavelength of the electromagnetic wave in a vacuum is assumed to be λ ０ The relative dielectric constant of the medium is ε ｒ Let k be the wave number of the propagating electromagnetic wave.

[0032] In this case, the measurement data s(x', y', z', k) at the transmitting / receiving point p(x', y', z') can be expressed by the following equation. however, is.

[0033] In equation (1-1), electromagnetic waves are expressed as spherical waves, and distance attenuation is omitted. This distance attenuation is omitted because it has little effect on subsequent processing. When the exponent part of the integrand function in the second equation in equation (1-1) is expressed in Fourier transform notation, it becomes the following equation. This is equivalent to decomposing the round-trip spherical wave of equation (1-1) into three-dimensional plane waves. Here, (k ｘ , k ｙ , k ｚ ) are components of the wave vector of the round-trip spherical wave of the wave propagating between the transmitting and receiving point p(x', y', z') and the reflecting point (x, y, z). Meet the following.

[0034] Hereinafter, the reflectance f(x, y, z) is derived from the measurement data s(x', y', z', k) based on the formula (1-3). First, the formula (1-3) is rearranged as follows:

[0035] Here, the integral inside {} is the triple Fourier transform with respect to (x, y, z). Also, the integral inside [ ] is (k ｘ , k ｙ ) is a double inverse Fourier transform with respect to (x', y'). Therefore, a double Fourier transform is performed on both sides of equation (1-5) with respect to (x', y'). The function after triple Fourier transform of the function f(x, y, z) is F(k ｘ , k ｙ , k ｚ ) The function after double Fourier transform of the measurement data s(x', y', z', k) is S(k ｘ , k ｙ , z', k). In this case, equation (1-5) is expressed as follows:

[0036] Both sides of the second line of the equation (1-6) are (k ｘ , k ｙ , k ｚ ), the reflectance f(x, y, z) is obtained as follows:

[0037] In this embodiment, as shown in FIG. 3, the transmitting and receiving point p(x', y', z') is arranged on the xy plane, and z'=0, so equation (1-7) can be expressed as follows:

[0038] As described above, the data processing unit 66 obtains the reflectance f(x, y, z) based on the measurement data s(x', y', z', k).

[0039] The data processing method and program of this embodiment will be described below with reference to Fig. 4. Fig. 4 is a flowchart showing the data processing method of this embodiment. First, the measurement unit 61 acquires measurement data s(x', y', 0, k) (step S1-1). Then, the data processing unit 66 performs a Hilbert transform on the measurement data s(x', y', 0, k) (step S1-2). This allows the imaginary component of the frequency data at each transmitting and receiving point to be obtained.

[0040] Next, the data processing unit 66 performs a double Fourier transform of the measurement data s(x', y', 0, k) with respect to (x', y') (step S1-3). As a result, as shown in equation (1-6), S(k ｘ , k ｙ , 0, k) is obtained. Then, the data processing unit 66 obtains S(k ｘ , k ｙ , 0, k) is subjected to variable substitution (step S1-4). Specifically, using equation (1-4), ｘ , k ｙ , k) function (k ｘ , k ｙ , k ｚ ) function. ｘ , k ｙ , k ｚ ) is obtained. Then, the data processing unit 66 obtains S(k ｘ , k ｙ , k ｚ ) for (k ｘ , k ｙ , k ｚ) is subjected to a triple inverse Fourier transform (step S1-5), thereby obtaining the reflectance f(x, y, z) as shown in equation (1-8).

[0041] The storage unit 66a stores a program for executing the data processing method of this embodiment. The program stored in the storage unit 66a causes the data processing unit 66 to execute the data processing method of this embodiment.

[0042] Here, we will explain alias data, which is considered to be aliasing noise in Fourier transform processing. When measuring with an array antenna, aliasing occurs in the measurement data unless the antenna size and scanning interval are set to below the Nyquist criterion. Aliasing refers to the inability to distinguish continuous signals with different frequency components due to sampling.

[0043] Figure 5 shows a conceptual diagram of aliasing. The Nyquist wave number k determined from the spatial sampling interval is ｎｙｑ Defined as: Nyquist wave number k ｎｙｑ is expressed by the following equation (1-10). Here, Δx and Δy are the measurement intervals in the x-axis and y-axis directions, respectively. ｎｙｑ Signals exceeding this are called aliases and are treated as aliasing noise in signal processing.

[0044] 6 shows an example of a measured waveform in the first embodiment. Here, the frequency of the electromagnetic wave is 15 GHz, the relative dielectric constant is 1, the measurement interval Δy = 7.5 mm, the measurement width is 562.5 mm, and the depth of the point target is 120 mm directly below the origin. Here, the wave number k y of the waveform measured at a position a distance y from the origin is expressed by the following equation (1-11). where θ ty is the angle between the line connecting the outermost transmitting / receiving antenna and the target and the z-axis. The Nyquist wave number is expressed by the following equation (1-12).

[0045] From this, the Nyquist criterion is expressed by the following equation (1-13). Any waveform of the y coordinate that does not satisfy this condition, that is, data in the aliasing region in FIG. 6, will be alias data (aliasing noise).

[0046] From the above equation, we can see that the alias data increases as the wavelength of the electromagnetic wave becomes shorter, the measurement interval becomes wider, or the target gets closer to the measurement surface. Normally, alias data is also treated as aliasing noise in signal processing. Therefore, an increase in alias data causes a deterioration in image resolution, a decrease in image intensity of shallow targets, and a deterioration in the S / N ratio.

[0047] Hereinafter, the measurement interval Δx = 10 mm, the measurement interval Δy = 19.25 mm, and the wavelength λ of the highest frequency in the medium ｆｍａｘ = 21.24 mm. In this case, the following formula (1-14) is obtained. In this case, the Nyquist criterion is not satisfied, and aliasing occurs in both the x-axis and y-axis directions. The Nyquist wave number in this case is expressed by the following equation (1-15).

[0048] The maximum wave number obtained from the highest frequency is expressed by the following equation (1-16).

[0049] Here, the Fourier transform for the x-axis in step S1-3 is shown in FIG. 7. The Fourier transform for the y-axis in step S1-3 is shown in FIG. 8. From FIG. 7, in the Fourier transform for the x-axis, ±k ｘ，ｎｙｑ The upper limit wave number is output in range A. ｘ，ｎｙｑ Data for wavenumbers exceeding 2k are treated as alias data. ｘ，ｎｙｑ Therefore, in the variable substitution process in step S1-4, the range is not A but ±k ｘｍａｘ Variable substitution processing is performed within range A', with the upper limit set to . This allows the alias data to be added to the variable substitution processing as meaningful data rather than discarded as aliasing noise, thereby improving the resolution in the x-axis direction and further increasing the image intensity of shallow targets. Similarly, variable substitution processing is performed for the y-axis within range B' instead of range B, as shown in FIG.

[0050] <Second embodiment> A data processing method, a measurement system, and a program according to a second embodiment will be described in detail below. The transmitting array antenna 50 and the receiving array antenna 52 of the first embodiment are arranged in one direction (the y direction in FIG. 3), but in this embodiment, the arrangement of the transmitting array antenna 50 and the receiving array antenna 52 is different. The transmitting array antenna 50 and the receiving array antenna 52 of this embodiment are arranged in a plane. Furthermore, in the first embodiment, the coordinates of the transmitting point and the receiving point are both p(x', y', z'), but in this embodiment, the coordinates of the transmitting point and the receiving point are different. In this embodiment, as shown in FIG. 9, the transmitting point p １ (x' １ , y′ １ , z' １ ), reception point p ２ (x' ２ , y′ ２ , z' ２ ) are arranged in the xy plane.

[0051] Here, the reflectance at the reflection point (x, y, z) of the object to be measured is assumed to be f(x, y, z). １ (x' １ , y′ １ , z' １ ) and receiving point p ２ (x' ２ , y′ ２ , z' ２ ) is measured data s(x' １ , x' ２ , y′ １ , y′ ２ , z' １ , z' ２ , k). The propagation wavelength of an electromagnetic wave in a vacuum is λ ０ The relative dielectric constant of the medium is ε ｒ Let k be the wave number of the propagating electromagnetic wave.

[0052] At this time, the measurement data s(x' １ , x' ２ , y′ １ , y′ ２ , z' １ , z' ２ , k) can be expressed by the following formula: however, is.

[0053] In equation (2-1), electromagnetic waves are expressed as spherical waves, and distance attenuation is omitted. This distance attenuation is omitted because it has little effect on subsequent processing. When the exponent part of the integrand in equation (2-1) is expressed in Fourier transform notation, it becomes the following equation. This is equivalent to decomposing the spherical wave in equation (2-1) into three-dimensional plane waves.

[0054] Here, (k' ｘ１ , k' ｙ１ , k' ｚ１ ) is a component of the wave vector of the spherical wave of the wave propagating between the transmitting point and the reflecting point. ｘ２ , k' ｙ２ , k' ｚ２ ) is the component of the wave vector of the spherical wave of the wave propagating from the reflection point to the receiving point. Meet the following.

[0055] Hereinafter, based on the formula (2-3), s(x' １ , x' ２ , y′ １ , y′ ２ , z' １ , z' ２ , k), the reflectance f(x, y, z) is derived. First, both sides of equation (2-3) are converted into x' １ , x' ２ , y′ １ , y′ ２ A quadruple Fourier transform is performed on

[0056] The left side of equation (2-5) is rewritten and rearranged as equation (2-6) below. Then, equation (2-5) is expressed as equation (2-7).

[0057] The following integral is performed on both sides of equation (2-7).

[0058] Here, the following variable substitutions are made:

[0059] Here, the following equation is obtained from equations (2-9) and (2-10).

[0060] The absolute value |J| of the Jacobian in this variable substitution is given by the following equations from equations (2-12) and (2-13).

[0061] Substituting equations (2-9), (2-10), (2-14), and (2-15) into equation (2-8) and performing variable substitution, the following equation is obtained.

[0062] Here, the integral for (u, v) on the right side of the second line of equation (2-16) is a constant, so it has been omitted. ｘ , k ｙ , k ｚ ), the reflectance f(x, y, z) is obtained as follows:

[0063] To solve equation (2-17), (k' ｚ１ , k' ｚ２ , k) to (k' ｘ１ , k' ｘ２ , k' ｙ１ , k' ｙ２ , k ｚ ) or (k ｘ , u, k ｙ , v, k ｚ ) can be expressed by using equations (2-4) and (2-11). If the transmitting point and receiving point are both located on the xy plane passing through the origin, z' １ = z' ２ = 0, so equation (2-17) can be expressed as follows:

[0064] As described above, the data processing unit 66 processes the measurement data s(x' １ , x' ２ , y′ １ , y′ ２ , z' １ , z' ２ , k), the reflectance f(x, y, z) is calculated.

[0065] The data processing method and program of this embodiment will be described below with reference to FIG. 10. FIG. 10 is a flowchart showing the data processing method of this embodiment. First, the measurement unit 61 measures the measurement data s(x' １ , x' ２ , y′ １ , y′ ２ , z' １ , z' ２ , k) (step S2-1). Then, the data processing unit 66 acquires the measurement data s(x' １ , x' ２ , y′ １ , y′ ２ , z' １ , z' ２ , k) (step S2-2), thereby obtaining the imaginary component of the frequency data at each measurement point.

[0066] Next, the data processing unit 66 processes the measurement data s(x' １ , x' ２ , y′ １ , y′ ２ , z' １ , z' ２ , k), (x' １ , x' ２ , y′ １ , y′ ２ ) is subjected to a quadruple Fourier transform (step S2-3). As a result, as shown in equation (2-6), S(k' ｘ１ , k' ｘ２ , k' ｙ１ , k' ｙ２ , 0, 0, k). Then, the data processing unit 66 obtains S(k' ｘ１ , k' ｘ２ , k' ｙ１ , k' ｙ２ , 0, 0, k) is subjected to variable substitution (step S2-4). Specifically, using equations (2-9), (2-10), (2-14), and (2-15), (k' ｘ１ , k' ｘ２ , k' ｙ１ , k' ｙ２ , k) function (k ｘ , kｙ , k ｚ ) function. ｘ , k ｙ , k ｚ ) is obtained. Then, the data processing unit 66 obtains S(k ｘ , k ｙ , k ｚ ) for (k ｘ , k ｙ , k ｚ ) is subjected to a triple inverse Fourier transform (step S2-5), thereby obtaining the reflectance f(x, y, z) as shown in equation (2-18).

[0067] The storage unit 66a stores a program for executing the data processing method of this embodiment. The program stored in the storage unit 66a causes the data processing unit 66 to execute the data processing method of this embodiment.

[0068] Here, the frequency of the electromagnetic wave is 4.46484 GHz, the relative dielectric constant is 10, and the measurement interval is Δx′. １ = 38.5 mm, measurement interval Δx' ２ = 38.5 mm, measurement interval Δy' １ = 38.5 mm, measurement interval Δy' ２ = 38.5 mm, measurement width 616 mm, wavelength λ of the highest frequency in the medium ｆｍａｘ = 21.24 mm. In this case, the following formula (2-20) is obtained. In this case, the Nyquist criterion is not satisfied, and x' １ Axial direction, x' ２ axial direction, y' １ axial direction, y' ２ Aliasing occurs in each of the axial directions. In this case, the Nyquist wave number is expressed by the following equation (2-21).

[0069] The maximum wave number obtained from the highest frequency is expressed by the following equation (2-22).

[0070] Here, x' in step S2-3 １ axis, x' ２The Fourier transform for the axis is shown in FIG. １ axis, y' ２ The Fourier transform for the x' axis is shown in Figure 12. １ axis, x' ２ In the Fourier transform of the axes, ±k' are respectively ｘ１，ｎｙｑ , ±k′ ｘ２，ｎｙｑ The wave numbers with an upper limit of ±k' are output in range A. ｘ１，ｎｙｑ , ±k′ ｘ２，ｎｙｑ The data for wavenumbers exceeding 2k' are considered alias data. ｘ１，ｎｙｑ , 2k' ｘ２，ｎｙｑ Therefore, in the variable substitution process in step S2-4, the range is not A, but ±k'. ｘ１ｍａｘ , ±k′ ｘ２ｍａｘ In this way, the alias data is not discarded as aliasing noise, but is added to the variable substitution process as actually meaningful data, so that x' １ axis, x' ２ This allows for improved axial resolution and even stronger image intensity for shallow targets. １ axis, y' ２ Similarly, as shown in FIG. 12, the axes are not in the range B but in the ranges ±k′. ｙ１ｍａｘ , ±k′ ｙ２ｍａｘ The variable substitution process is performed within a range B' having an upper limit of:

[0071] Third Embodiment A data processing method, a measurement system, and a program according to a third embodiment will be described in detail below. In the second embodiment, the transmitting array antenna 50 and the receiving array antenna 52 are arranged in a plane, but in this embodiment, the arrangement of the transmitting array antenna 50 and the receiving array antenna 52 is different. In this embodiment, the transmitting array antenna 50 and the receiving array antenna 52 are arranged in a line. Specifically, in this embodiment, the transmitting antenna 10a and the receiving antenna 10b are arranged in the y direction, as shown in FIG. 13 . The movement direction (scanning direction) of the transmitting array antenna 50 and the receiving array antenna 52 is defined as the x direction. The direction toward the object to be measured (the direction of electromagnetic wave transmission) as viewed from the transmitting array antenna 50 and the receiving array antenna 52 is defined as the z direction. The movement direction (scanning direction) of the transmitting array antenna 50 and the receiving array antenna 52 may also be defined as the y direction.

[0072] Therefore, the positional relationship between the object to be measured, the transmitting array antenna 50, and the receiving array antenna 52 can be expressed as shown in Figure 13. Here, the coordinates of the transmitting point are p １ (x' １ , y′ １ , z' １ ), the coordinates of the receiving point are p ２ (x' ２ , y′ ２ , z' ２ ) The reflectance at the reflection point (x, y, z) of the object to be measured is f(x, y, z). ２ (x' ２ , y′ ２ , z' ２ ) is measured data s(x' １ , x' ２ , y′ １ , y′ ２ , z' １ , z' ２ , k). The propagation wavelength of an electromagnetic wave in a vacuum is λ ０ The relative dielectric constant of the medium is ε ｒ Let k be the wave number of the propagating electromagnetic wave.

[0073] At this time, the measurement data s(x' １ , x'２ , y′ １ , y′ ２ , z' １ , z' ２ , k) can be expressed by the following formula: however is.

[0074] In equation (3-1), electromagnetic waves are expressed as spherical waves, and distance attenuation is omitted. This distance attenuation is omitted because it has little effect on subsequent processing. When the exponent part of the integrand function in the second equation in equation (3-1) is expressed in Fourier transform notation, it becomes the following equation. This is equivalent to decomposing the spherical wave in equation (3-1) into three-dimensional plane waves.

[0075] Here, (k' ｘ１ , k' ｙ１ , k' ｚ１ ) is a component of the wave vector of the spherical wave of the wave propagating between the transmitting point and the reflecting point. ｘ２ , k' ｙ２ , k' ｚ２ ) is the component of the wave vector of the spherical wave of the wave propagating from the reflection point to the receiving point. Meet the following.

[0076] Here, the transmitting point p １ (x' １ , y′ １ , z' １ ) and receiving point p ２ (x' ２ , y′ ２ , z' ２ ) have the same x-coordinate, so x' = x' １ = x' ２ Then, equation (3-3) is expressed as follows:

[0077] Here, the following variable substitutions are made: The following equation is obtained from equation (3-6):

[0078] Calculating the absolute value of the Jacobian from equation (3-7) gives the following equation:

[0079] Substituting equations (3-6) and (3-8) into equation (3-5) and performing variable substitution, the following equation is obtained.

[0080] Here, the integral for u in the second line of equation (3-9) is omitted because it is a constant. １ , y ２ ), the following equation is obtained:

[0081] The left side of equation (3-10) is rewritten and rearranged as follows: Then, equation (3-10) is expressed as follows:

[0082] By performing the following integral on both sides of equation (3-12), the following equation is obtained:

[0083] Here, (k' ｙ１ , k' ｙ２ ), (k' ｚ１ , k' ｚ２ ), we define the following variable substitutions:

[0084] Here, the following equation is obtained from equation (3-14): The absolute value of the Jacobian under this variable substitution is given by the following formula:

[0085] Substituting equations (3-1), (3-15), and (3-17) into equation (3-13) and performing variable substitution, the following equation is obtained.

[0086] Here, the integral for v on the right side of the second line of equation (3-18) is omitted because it is a constant. ｘ , k ｙ , k ｚ ), the reflectance f(x, y, z) is obtained as follows: Here, if z'=0 is set in order to align the measurement surface with the x'-y' plane passing through the origin, equation (3-19) can be expressed as follows:

[0087] To solve equation (3-20), let k be (k' ｙ１ , k' ｙ２ , k ｚ ) or (k ｙ , v, k ｚ ) The simultaneous equations of four equations, equation (3-4), equation (3-6), equation (3-15), and the following equation (3-21) obtained from the assumption, are solved.

[0088] From this, k is expressed by the following formula.

[0089] As described above, the data processing unit 66 processes the measurement data s(x' １ , x' ２ , y′ １ , y′ ２ , z' １ , z' ２ , k), the reflectance f(x, y, z) is calculated.

[0090] The data processing method and program of this embodiment will be described below with reference to FIG. 14. FIG. 14 is a flowchart showing the data processing method of this embodiment. First, the measurement unit 61 measures the measurement data s(x', y' １ , y′ ２ , 0, 0, k) (step S3-1). Then, the data processing unit 66 acquires the measurement data s(x', y' １ , y′ ２ , 0, 0, k) is subjected to a Hilbert transform (step S3-2). This provides the imaginary component of the frequency data at each measurement point.

[0091] Next, the data processing unit 66 processes the measurement data s(x', y' １ , y′ ２ , 0, 0, k), (x', y' １ , y′ ２ ) is subjected to triple Fourier transform (step S3-3). As a result, as shown in equation (3-11), S(k ｘ , k' ｙ１ , k' ｙ２, 0, 0, k) is obtained. Then, the data processing unit 66 obtains S(k ｘ , k' ｙ１ , k' ｙ２ , 0, 0, k) is subjected to variable substitution (step S3-4). Specifically, using equations (3-14) and (3-15), ｘ , k' ｙ１ , k' ｙ２ , k) function (k ｘ , k ｙ , v, k). ｘ , k ｙ , v, 0, 0, k) is obtained. Then, the data processing unit 66 obtains S(k ｘ , k ｙ , v, 0, 0, k), for (k ｘ , k ｙ , k ｚ ) is subjected to a triple inverse Fourier transform (step S3-5), thereby obtaining the reflectance f(x, y, z) as shown in equation (3-20).

[0092] The storage unit 66a stores a program for executing the data processing method of this embodiment. The program stored in the storage unit 66a causes the data processing unit 66 to execute the data processing method of this embodiment.

[0093] Here, the frequency of the electromagnetic wave is 4.46484 GHz, the relative dielectric constant is 10, the measurement interval Δx=10 mm, and the measurement interval Δy′ １ = 38.5 mm, measurement interval Δy' ２ = 38.5 mm, measurement width 616 mm, wavelength λ of the highest frequency in the medium ｆｍａｘ = 21.24 mm. In this case, the following equation (3-23) is obtained. In this case, the Nyquist criterion is not satisfied, and the x-axis and y' １ axial direction, y' ２ Aliasing occurs in each of the axial directions. In this case, the Nyquist wave number is expressed by the following equation (3-24).

[0094] The maximum wave number obtained from the highest frequency is expressed by the following equation (3-25).

[0095] Here, the Fourier transform for the x-axis in step S3-3 is shown in FIG. １ axis, y' ２ The Fourier transform for the x-axis is shown in Fig. 16. From Fig. 15, in the Fourier transform for the x-axis, ±k ｘ，ｎｙｑ , ±k′ ｘ２，ｎｙｑ Wave numbers with an upper limit of ±k are output in range A. ｘ，ｎｙｑ Data for wavenumbers exceeding 2k are considered alias data. ｘ，ｎｙｑ Therefore, in the variable substitution process in step S3-4, instead of the range A, ±k ｘｍａｘ The variable substitution process is performed within a range A' with an upper limit of y'. This allows the alias data to be added to the variable substitution process as meaningful data rather than discarded as aliasing noise, thereby improving the resolution in the x-axis direction and further increasing the image intensity of shallow targets. １ axis, y' ２ As for the axes, as in the second embodiment, as shown in FIG. 16, the range is not B but ±k′. ｙ１ｍａｘ , ±k′ ｙ２ｍａｘ The variable substitution process is performed within a range B' having an upper limit of:

[0096] (Simulation Results) The results of a computer simulation of the data processing method of the third embodiment will be described below. The simulation conditions are as follows.

[0097] ・Frequency band used f ｍｉｎ ~f ｍａｘ : DC to 4.46484 GHz Center frequency fc: 2.23242 GHz Wavelength λ of center frequency in vacuum ０ｃ : 134.38 mm Relative dielectric constant ε of medium ｒ : 10 Wavelength λc of the center frequency in the medium: 42.50 mm Wavelength λ of the highest frequency in the medium ｆｍａｘ : 21.24 mm ・Number of measurements Nx: 128pt ・Measurement interval Δx: 10 mm ・Measurement width: 1280 mm ・Number of measurements N ｙ’１: 16pt Measurement interval Δy'1: 38.5mm Measurement width: 616mm Number of measurements N ｙ’２ : 16pt Measurement interval Δy'2: 38.5 mm Measurement width: 616 mm Coordinates of point target 1 (unit: mm): (640, 307, 20) Coordinates of point target 2 (unit: mm): (640, 307, 200)

[0098] FIG. 17 shows an xz plane image of the point target 1 simulated using the data processing method of Comparative Example 1. FIG. 18 shows an xz plane image of the point target 1 simulated using the data processing method of Example 1. FIG. 19 shows a yz plane image of the point target 1 simulated using the data processing method of Comparative Example 1. FIG. 20 shows a yz plane image of the point target 1 simulated using the data processing method of Example 1. FIG. 21 shows the waveforms in the x-axis direction of the point target 1 of Comparative Example 1 and Example 1. FIG. 22 shows the waveforms in the y-axis direction of the point target 1 of Comparative Example 1 and Example 1. FIG. 23 shows the waveforms in the z-axis direction of the point target 1 of Comparative Example 1 and Example 1. In all of FIGS. 21 to 23, Example 1 is shown by a solid line, and Comparative Example 1 is shown by a dashed line.

[0099] FIG. 24 shows an xz plane image of the point target 2 simulated using the data processing method of Comparative Example 1. FIG. 25 shows an xz plane image of the point target 2 simulated using the data processing method of Example 1. FIG. 26 shows a yz plane image of the point target 2 simulated using the data processing method of Comparative Example 1. FIG. 27 shows a yz plane image of the point target 2 simulated using the data processing method of Example 1. FIG. 28 shows the waveforms in the x-axis direction of the point target 2 of Comparative Example 1 and Example 1. FIG. 29 shows the waveforms in the y-axis direction of the point target 2 of Comparative Example 1 and Example 1. FIG. 30 shows the waveforms in the z-axis direction of the point target 2 of Comparative Example 1 and Example 1. In all of FIGS. 28 to 30, Example 1 is shown by a solid line, and Comparative Example 1 is shown by a dashed line.

[0100] In Comparative Example 1, in the variable substitution process of step S3-4 in this embodiment, the variable substitution process was performed in the range A. In Example 1, in the variable substitution process of step S3-4 in this embodiment, the range A was not used, but ±kｘｍａｘ The variable substitution process was performed within a range A' with the upper limit being ±k ｘｍａｘ The variable substitution process in the range A' having an upper limit of is called super-resolution process.

[0101] 17 to 30, it was confirmed that, compared to Comparative Example 1, Example 1 in which super-resolution processing is performed improves the resolution in the x-axis direction and the y-axis direction for both point targets 1 and 2.

[0102] Fourth Embodiment In this embodiment, noise that may occur in the third embodiment is reduced. Measurement data s in the y-axis direction is expressed by the following equation (4-1), where t is the amplitude of the electromagnetic wave propagating from the transmitting antenna to the target, r is the reflectivity of the target, and r is the amplitude of the electromagnetic wave propagating from the target to the receiving antenna.

[0103] At this time, the amplitude component below the Nyquist frequency is t ０ , r ０ The n-times aliased amplitude component (alias data) is defined as t ｎ , r ｎ Then, t and r are respectively expressed by the following equation (4-2).

[0104] Substituting equation (4-2) into equation (4-1), the following equation (4-3) is obtained.

[0105] The super-resolution processing shown in the third embodiment is carried out by, for example, substituting the variables in step S3-4 using the (k' y1 , k' y2 ) space, k' due to the switching of the transmit antenna is expressed as the following equation (4-4): y1 and k' due to the receiving antenna y2 If both are within a range equal to or less than the Nyquist wave number, all components including alias data (aliasing noise components) become noise components in the signal processing.

[0106] In this case, equation (4-3) is expressed as equation (4-5). Here, the r surrounded by a solid line ０t ０ is the signal component, and the part surrounded by the dashed line is the noise component.

[0107] Next, k' y1 When only the signal .gamma. , .gamma. , .gamma. , .gamma. , falls within the aliasing noise region once, the condition of equation (4-6) is satisfied, and equation (4-3) is expressed by equation (4-7). Here, the r surrounded by a solid line ０ t １ is the signal component, and the part surrounded by the dashed line is the noise component.

[0108] Next, k' y２ When falls within the double aliasing noise region, the condition of equation (4-8) is satisfied, and equation (4-3) is expressed by equation (4-9). Here, the r surrounded by a solid line ２ t １ is the signal component, and the part surrounded by the dashed line is the noise component.

[0109] As shown in Figure 5, when the y'1 coordinate of the transmitting antenna measurement point and the y'2 coordinate of the receiving antenna measurement point are close to the target, i.e., when they are located directly above the target, the spatial frequencies of the measurement data in the y'1 and y'2 axis directions are low. Conversely, as the y'1 and y'2 coordinates move farther from the target, the spatial frequencies of the measurement data in the y'1 and y'2 axis directions are higher. When the wave number at that point exceeds the Nyquist wave number, it is included in the measurement data as an aliasing noise component. In this way, alias data components (aliasing noise components) tend to occur at points relatively far from the target. Therefore, due to distance attenuation, their amplitudes are smaller than those of data components below the Nyquist wave number. Therefore, the following equation (4-10) generally holds true.

[0110] From this, r ｎ and ｎ The magnitude of the second-order term with respect to r ０ Or t ０ This is expressed by the following equation (4-11).

[0111] Furthermore, unlike air, where the attenuation rate is small, when electromagnetic waves propagate through concrete or underground, where the attenuation rate is large, the attenuation rate due to the medium in addition to the distance attenuation is also a factor. ｎ and ｎ In this case, the formula (4-11) can be expressed as the following formula (4-12).

[0112] Therefore, from equations (4-5), (4-7), (4-9), and (4-12), t ｎ and ｎ The second-order term for r is data with a relatively poor S / N ratio compared to the other terms. ｎ and ｎ In this case, the second-order term related to (4-13) can be expressed as follows: Here, only the components enclosed by solid lines are used in the variable substitution process, and the parts enclosed by dashed lines are not used in the variable substitution process.

[0113] In this way, when electromagnetic waves propagate through a medium with a large attenuation rate, the degradation of the S / N ratio of a 3D image can be suppressed by using the filter expressed by equation (4-13) during variable substitution processing. This is because (k' y1 , k' y2 ) "Cross-shaped k'" in space y1 -k' y2 FIG. 31 shows a cross-shaped k′ ​​filter. y1 -k' y2 FIG. 1 is a conceptual diagram of a filter.

[0114] The k shown in Figs. ｘｍａｘ , k' ｙ１ｍａｘ , k' ｙ２ｍａｘ depends on the target position or the antenna directivity. If targets near the measurement plane (z = 0) are included in the measurement target, the antenna directivity θ ｂｘ , θ ｂｙ The maximum value may be determined by: however, is.

[0115] (Simulation Result 1) Below, the results of a computer simulation of the data processing method of the fourth embodiment will be described. First, the effect of improving the image intensity of shallow depth targets will be verified under the same simulation conditions as in the third embodiment. In the super-resolution processing, the lost alias data is used. Therefore, the target image intensity (image amplitude) increases compared to when the super-resolution processing is not performed. This effect is greater for shallow depth targets with a large amount of alias data than for deep depth targets. In other words, the image intensity of shallow depth targets that has been reduced by aliasing is largely restored by the super-resolution processing.

[0116] Figure 32 shows the simulation results for four point targets. The horizontal position of each point target is the center of the measurement surface, and the depths are 20 mm, 200 mm, 400 mm, and 600 mm. In Figure 32, Example 1, in which super-resolution processing was performed, is shown by a solid line, and Comparative Example 1, in which super-resolution processing was not performed, is shown by a dashed line. The magnitude of the 3D image amplitude is normalized, with the amplitude of the 20 mm deep target without super-resolution processing being set to 1.

[0117] The shallower the target depth, the stronger the target image amplitude due to the effect of super-resolution processing. Because distance attenuation is taken into account in the simulation data, the amplitude of deep targets becomes very small, making it difficult to directly compare the restoration effects. Therefore, for the amplitude of targets at the same depth, the ratio of the 3D image amplitude after super-resolution processing to the 3D image amplitude without super-resolution processing is defined as the restoration rate. Figure 33 shows the restoration rate for each point target. Figure 33 shows that the shallower the target, the greater the restoration rate due to super-resolution processing, since there is more alias data. At a depth of 20 mm, the restoration rate reaches 6 times. On the other hand, the deeper the target, the less alias data there is, and the smaller the restoration rate becomes. At a depth of 600 mm, the restoration rate is approximately 1.2 times.

[0118] From the above, it was confirmed that the image amplitude intensity can be improved by using alias data through super-resolution processing. In particular, a significant improvement is observed in images of shallow targets with a large amount of alias data. This is mainly reflected in the 3D image as an improvement in the S / N ratio of shallow target images.

[0119] (Simulation Results 2) The results of computer simulation of the data processing methods of the third and fourth embodiments will be described below. The simulation conditions are as follows: Used frequency band f ｍｉｎ ~f ｍａｘ : DC to 4.46484 GHz Center frequency fc: 2.23242 GHz Wavelength λ of center frequency in vacuum ０ｃ : 134.38 mm Relative dielectric constant ε of medium ｒ : 5 Wavelength λc of the center frequency in the medium: 60.1 mm Wavelength λ of the highest frequency in the medium ｆｍａｘ : 30.05 mm Measurement interval in scanning direction Δx: 10 mm Number of measurements Nx: 115 pt Measurement width: 1150 mm Measurement interval of transmitting antenna Δy1: 38.5 mm Number of transmitting antennas N ｙ１ : 12 Measurement interval of receiving antenna Δy2: 38.5 mm Number of receiving antennas N ｙ２ : 12pt ・Measurement width: 481.25mm

[0120] The results of measurements on a reinforced concrete specimen are shown below. The layout of the transmitting and receiving antennas used for the measurements is shown in Figure 34. The specimen contains multiple rebars with different z-coordinates. Figure 35 shows a 3D image of the specimen without super-resolution processing. Figure 36 shows a 3D image of the specimen with super-resolution processing. The z-coordinates of multiple rebars are shown in Figures 35 and 36. Comparing Figures 35 and 36, the image of the rebars in Figure 36, which underwent super-resolution processing, is thinner overall, confirming improved image resolution. Furthermore, the image intensity of the rebars at depths of 35 mm and 45 mm was enhanced, confirming the recovery effect of the image amplitude of shallow targets. However, it was also confirmed that new unwanted noise (aliasing noise) such as interference fringes was generated between the rebars.

[0121] Next, the fourth embodiment, the cross-shaped k′ ​​described in FIG. y1 -k' y2 A filter was applied. Figure 37 shows the cross k' y1 -k' y2 This is a 3D image of the test specimen after applying a filter and super-resolution processing. y1 -k' y2 FIG. 36 is a three-dimensional image of the test specimen subjected to super-resolution processing without applying a filter, and FIG. 37 is a cross-shaped image of the test specimen subjected to super-resolution processing without applying a filter. y1 -k' y2 Comparing Fig. 37, where a filter is applied and super-resolution processing is performed, the cross-shaped k' y1 -k' y2 It was confirmed that the filter can reduce interference fringe noise (aliasing noise) while suppressing degradation of the resolution of rebar images and the intensity of shallow-depth rebar images.

[0122] Next, cross k' y1 -k' y2 The effect of the filter on the resolution was evaluated using simulation data. The simulation conditions were the same as those in the third embodiment. FIG. 38 shows the waveforms in the x-axis direction of the point target 1 of Comparative Example 2 and Example 2. FIG. 39 shows the waveforms in the y-axis direction of the point target 1 of Comparative Example 2 and Example 2. FIG. 40 shows the waveforms in the z-axis direction of the point target 1 of Comparative Example 2 and Example 2. FIG. 41 shows the waveforms in the x-axis direction of the point target 2 of Comparative Example 2 and Example 2. FIG. 42 shows the waveforms in the y-axis direction of the point target 2 of Comparative Example 2 and Example 2. FIG. 43 shows the waveforms in the z-axis direction of the point target 2 of Comparative Example 2 and Example 2. In all of FIGS. 38 to 43, Example 2 is shown by a solid line, and Comparative Example 2 is shown by a dashed line.

[0123] In Comparative Example 2, in the variable substitution process of step S3-4 in the third embodiment, the cross k' y1 -k' y2 Without applying a filter, ｘｍａｘ In Example 2, in the variable substitution process of step S3-4 in the third embodiment, the cross k' y1 -k' y2 Applying the filter,ｘｍａｘ The variable substitution process was performed within the range A' with the upper limit of k'. y1 -k' y2 It was confirmed that applying a filter to perform super-resolution processing had almost no effect on image resolution in any of the x-axis, y-axis, and z-axis.

[0124] Fifth Embodiment In this embodiment, noise that may occur in the second embodiment is reduced. In a fourth embodiment, the cross-shaped k' in the third embodiment is reduced. y1 -k' y2 In this embodiment, a cross-shaped k' filter is applied to the second embodiment. ｘ1 -k' ｘ2 Filter and cross k' y1 -k' y2 A filter is applied to perform super-resolution processing. ｘ1 -k' ｘ2 44 is a conceptual diagram of a filter. That is, in this embodiment, the cross-shaped filter k′ shown in FIG. ｘ1 -k' ｘ2 The filter and the cross-shaped k' shown in FIG. y1 -k' y2 Both filters are applied for super-resolution processing.

[0125] k' ｘ１ｍａｘ , k' ｘ２ｍａｘ , k' ｙ１ｍａｘ , k' ｙ２ｍａｘ depends on the position of the target or the directivity of the antenna. If targets near the measurement plane (z = 0) are included in the measurement target, the directivity θ of the antenna is ｂｘ , θ ｂｙ The maximum value may be determined by: however, is.

[0126] 10a Transmitting antenna 10b Receiving antenna 50 Transmitting array antenna 52 Receiving array antenna 60 Radar device 61 Measurement unit 64 System control circuit 66 Data processing unit 68 Image display unit 58, 59 High frequency switch 62 High frequency circuit 69 Encoder

Claims

1. A data processing method for analyzing scattered waves of a wave radiated to an object, comprising the steps of: Multiple transmission points p are arranged two-dimensionally on the xy plane. 1 (x' 1 , y' 1 , z' 1 ) to radiate the wave to the object; The scattered wave reflected at the reflection point (x, y, z) on the object with a reflectance f(x, y, z) is received by multiple receiving points p 2 (x' 2 , y' 2 , z' 2 ) and the measured value s(x' 1 , x' 2 , y' 1 , y' 2 , z' 1 , z' 2 , k) and The measured value s(x' 1 , x' 2 , y' 1 , y' 2 , z' 1 , z' 2 , k) is transformed by a quadruple Fourier transform using equation (1) to obtain S(k' x1 , k' x2 , k' y1 , k' y2 , z' 1 , z' 2 , k) and x' determined by the measurement interval in the x direction 1 The Nyquist wave number in the direction k' x1,nyq Let x' 2 The Nyquist wave number in the direction k' x2,nyq Then, -k x1 ≦k' x1 ≦k x1 , and -k x2 ≦k' x2 ≦k x2 (However, k' x1,nyq ≦k x1 , and k' x2,nyq ≦k x2 ) variable substitution processing in the x direction is performed in the range y' determined by the measurement interval in the y direction 1 The Nyquist wave number in the direction k' y1,nyq Let y' 2 The Nyquist wave number in the direction k' y2,nyq Then, -k y1 ≦k' y1 ≦k y1 , and -k y2 ≦k' y2 ≦k y2 (However, k' y1,nyq ≦k y1 , and k' y2,nyq ≦k y2 ) variable substitution processing in the y direction is performed in the range The reflectance f(x, y, z) is calculated by performing a triple inverse Fourier transform using the formula (2). Data processing methods. however, With' 1 = from' 2 = 0 k is the wave number of the propagating wave, k' x1 , k' y1 , k' z1 is the transmission point p 1 (x' 1 , y' 1 , z' 1 ) to the reflection point (x, y, z), the components of the wave vector of the spherical wave of the wave motion propagating between the reflection point (x, y, z), k' x2 , k' y2 , k' z2 is the distance from the reflection point (x, y, z) to the reception point p 2 (x' 2 , y' 2 , z' 2 ) the components of the wave vector of the spherical wave of the wave propagating between k x = k’ x1 + k’ x2 , u = k’ x1 - k’ x2 , k y = k’ y1 + k’ y2 , v = k’ y1 - k’ y2 It is.

2. x' determined by equation (3) 1 The maximum wave number in the direction k' x1max Let x' 2 The maximum wave number in the direction k' x2max Then, -k' x1max ≦k' x1 ≦k' x1max , and -k' x2max ≦k' x2 ≦k' x2max Perform variable substitution processing in the x direction in the range of y' determined by equation (4) 1 The maximum wave number in the direction k' y1max Let y' 2 The maximum wave number in the direction k' y2max Then, -k' y1max ≦k' y1 ≦k' y1max , and -k' y2max ≦k' y2 ≦k' y2max Perform variable substitution processing in the y direction in the range of 2. The data processing method according to claim 1.

3. In the variable substitution process in the x direction, k' x1,nyq ≦|k' x1 |≦k' x1max , and k' x2,nyq ≦|k' x2 |≦k' x2max and performing the variable substitution process in the x direction by selectively excluding the range of In the variable substitution process in the y direction, k' y1,nyq ≦|k' y1 |≦k' y1max , and k' y2,nyq ≦|k' y2 |≦k' y2max and selectively excluding the range of The data processing method according to claim 2.

4. The components of the wave number and wave vector of the wave satisfy the formula (5). The data processing method according to any one of claims 1 to 3.

5. A measurement system for analyzing scattered waves of a wave radiated to an object, comprising: A transceiver unit, Multiple transmission points p are arranged two-dimensionally on the xy plane. 1 (x' 1 , y' 1 , z' 1 A transmitting unit that radiates the wave from the object to the object; The scattered wave reflected at the reflection point (x, y, z) on the object with a reflectance f(x, y, z) is received by multiple receiving points p 2 (x' 2 , y' 2 , z' 2 ) and the measured value s(x' 1 , x' 2 , y' 1 , y' 2 , z' 1 , z' 2 , k) as a receiving unit; A transceiver unit having A processing device comprising: The measured value s(x' 1 , x' 2 , y' 1 , y' 2 , z' 1 , z' 2 , k) is transformed by a quadruple Fourier transform using equation (1) to obtain S(k' x1 , k' x2 , k' y1 , k' y2 , z' 1 , z' 2 , k) and x' determined by the measurement interval in the x direction 1 The Nyquist wave number in the direction k' x1,nyq Let x' 2 The Nyquist wave number in the direction k' x2,nyq Then, -k x1 ≦k' x1 ≦k x1 , and -k x2 ≦k' x2 ≦k x2 (However, k' x1,nyq ≦k x1 , and k' x2,nyq ≦k x2 ) in the x direction; y' determined by the measurement interval in the y direction 1 The Nyquist wave number in the direction k' y1,nyq Let y' 2 The Nyquist wave number in the direction k' y2,nyq Then, -k y1 ≦k' y1 ≦k y1 , and -k y2 ≦k' y2 ≦k y2 (However, k' y1,nyq ≦k y1 , and k' y2,nyq ≦k y2 ) in the y-direction, A step of calculating the reflectance f(x, y, z) by performing a triple inverse Fourier transform using the formula (2); A processing unit for executing the above-mentioned A measurement system having the following features: however, With' 1 = from' 2 = 0 k is the wave number of the propagating wave, k' x1 , k' y1 , k' z1 is the transmission point p 1 (x' 1 , y' 1 , z' 1 ) to the reflection point (x, y, z), the components of the wave vector of the spherical wave of the wave motion propagating between the reflection point (x, y, z), k' x2 , k' y2 , k' z2 is the distance from the reflection point (x, y, z) to the reception point p 2 (x' 2 , y' 2 , z' 2 ) the components of the wave vector of the spherical wave of the wave propagating between k x = k’ x1 + k’ x2 , u = k’ x1 - k’ x2 , k y = k’ y1 + k’ y2 , v = k’ y1 - k’ y2 It is.

6. The processing device includes: x' determined by equation (3) 1 The maximum wave number in the direction k' x1max Let x' 2 The maximum wave number in the direction k' x2max Then, -k' x1max ≦k' x1 ≦k' x1max , and -k' x2max ≦k' x2 ≦k' x2max Perform variable substitution processing in the x direction in the range of y' determined by equation (4) 1 The maximum wave number in the direction k' y1max Let y' 2 The maximum wave number in the direction k' y2max Then, -k' y1max ≦k' y1 ≦k' y1max , and -k' y2max ≦k' y2 ≦k' y2max Perform variable substitution processing in the y direction in the range of The measurement system of claim 5 .

7. The processing device includes: In the variable substitution process in the x direction, k' x1,nyq ≦|k' x1 |≦k' x1max , and k' x2,nyq ≦|k' x2 |≦k' x2max performing said variable substitution process in the x direction, selectively excluding the range of In the variable substitution process in the y direction, k' y1,nyq ≦|k' y1 |≦k' y1max , and k' y2,nyq ≦|k' y2 |≦k' y2max performing said variable substitution process in the y direction, selectively excluding the range of Execute The measurement system of claim 6.

8. The components of the wave number and wave vector of the wave satisfy the formula (5). The measurement system according to any one of claims 5 to 7.

9. A program for analyzing scattered waves of a wave radiated to an object, Measurement value s(x' 1 , x' 2 , y' 1 , y' 2 , z' 1 , z' 2 , k) is transformed by a quadruple Fourier transform using equation (1) to obtain S(k' x1 , k' x2 , k' y1 , k' y2 , z' 1 , z' 2 , k) and x' determined by the measurement interval in the x direction 1 The Nyquist wave number in the direction k' x1,nyq Let x' 2 The Nyquist wave number in the direction k' x2,nyq Then, -k x1 ≦k' x1 ≦k x1 , and -k x2 ≦k' x2 ≦k x2 (However, k' x1,nyq ≦k x1 , and k' x2,nyq ≦k x2 ) in the x direction; y' determined by the measurement interval in the y direction 1 The Nyquist wave number in the direction k' y1,nyq Let y' 2 The Nyquist wave number in the direction k' y2,nyq Then, -k y1 ≦k' y1 ≦k y1 , and -k y2 ≦k' y2 ≦k y2 (However, k' y1,nyq ≦k y1 , and k' y2,nyq ≦k y2 ) in the y-direction, A procedure for calculating the reflectance f(x, y, z) by performing a triple inverse Fourier transform using equation (2); A program that causes a computer to execute the following. however, With' 1 = from' 2 = 0 k is the wave number of the propagating wave, k' x1 , k' y1 , k' z1 is the transmission point p 1 (x' 1 , y' 1 , z' 1 ) to the reflection point (x, y, z), the components of the wave vector of the spherical wave of the wave motion propagating between the reflection point (x, y, z), k' x2 , k' y2 , k' z2 is the distance from the reflection point (x, y, z) to the receiving point p 2 (x' 2 , y' 2 , z' 2 ) the components of the wave vector of the spherical wave of the wave propagating between k x = k’ x1 + k’ x2 , u = k’ x1 - k’ x2 , k y = k’ y1 + k’ y2 , v = k’ y1 - k’ y2 It is.

10. x' determined by equation (3) 1 The maximum wave number in the direction k' x1max Let x' 2 The maximum wave number in the direction k' x2max Then, -k' x1max ≦k' x1 ≦k' x1max , and -k' x2max ≦k' x2 ≦k' x2max A procedure for performing a variable substitution process in the x direction within the range of y' determined by equation (4) 1 The maximum wave number in the direction k' y1max Let y' 2 The maximum wave number in the direction k' y2max Then, -k' y1max ≦k' y1 ≦k' y1max , and -k' y2max ≦k' y2 ≦k' y2max A procedure for performing a variable substitution process in the y direction in the range of The program according to claim 9, further comprising:

11. In the variable substitution process in the x direction, k' x1,nyq ≦|k' x1 |≦k' x1max , and k' x2,nyq ≦|k' x2 |≦k' x2max performing said variable substitution process in the x direction, selectively excluding the range of In the variable substitution process in the y direction, k' y1,nyq ≦|k' y1 |≦k' y1max , and k' y2,nyq ≦|k' y2 |≦k' y2max performing said variable substitution process in the y direction, selectively excluding the range of The program according to claim 10, further comprising:

12. The measured value s(x' 1 , x' 2 , y' 1 , y' 2 , z' 1 , z' 2 , k) are multiple transmission points p 1 (x' 1 , y' 1 , z' 1 ) to the object, the wave is reflected at the reflection point (x, y, z) on the object with the reflectivity f(x, y, z), and the scattered wave is received by a plurality of receiving points p 2 (x' 2 , y' 2 , z' 2 ), The program according to any one of claims 9 to 11.

13. The components of the wave number and wave vector of the wave satisfy the formula (5). The program according to any one of claims 9 to 11.

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