Frequency domain gradient weighted least square phase unwrapping method

Abstract

The invention discloses a frequency domain gradient weighted least square phase unwrapping method. The method comprises the following steps: collecting a complex voltage signal of an electromagnetic induction signal; performing fast Fourier transform on the complex voltage signal to obtain a frequency domain signal, and calculating a frequency domain phase spectrum of electromagnetic induction according to the frequency domain signal; calculating a frequency domain phase gradient between adjacent frequency points of the complex voltage signal according to the frequency domain phase spectrum; packaging the obtained frequency domain phase gradient to obtain a packaging phase gradient; constructing an adaptive weighting function based on the amplitude and noise variance of the frequency domain signal, and obtaining the weight of the frequency domain signal; and constructing a weighted least square objective function according to the packaging phase gradient and the weight of the frequency domain signal, and solving an unwrapping phase based on the objective function.

Description

Technical Field

[0001] This invention relates to the fields of signal processing and electromagnetic induction detection technology, and more specifically to a frequency domain gradient-weighted least squares phase unwrapping method. Background Technology

[0002] Phase unwrapping is an important technique in signal processing, widely used in radar, acoustics, medical imaging and electromagnetic detection. In electromagnetic induction detection, phase information contains rich information about position and material properties. However, due to the periodicity of the phase function, there is a 2π jump problem, which requires phase unwrapping technology to obtain continuous absolute phase.

[0003] Traditional phase unwrapping methods are mainly performed in the spatial domain, including path integral, minimum norm, and region growing methods. These traditional methods face many technical bottlenecks in practical applications, the most prominent of which is the error accumulation problem. In the spatial path integral method, local errors will continue to propagate along the integration path, leading to global error accumulation, which seriously affects the unwrapping accuracy. This cumulative effect means that even small local errors may be amplified during the processing, ultimately leading to serious distortion of the unwrapping results.

[0004] Noise sensitivity is another key factor limiting the performance of traditional phase unwrapping methods. In actual electromagnetic induction detection environments, signals are often subject to various noise interferences. The sensitivity of traditional methods to noise causes their performance to drop sharply in low signal-to-noise ratio environments, making it difficult to meet the reliability requirements of practical applications. This sensitivity not only affects the accuracy of unwrapping but also leads to a decrease in the stability of the algorithm, which is particularly evident in complex detection environments.

[0005] Excessive computational complexity is also a significant drawback of traditional spatial processing methods. These methods typically have a computational complexity of O(N²), and the computation time increases exponentially with the increase in data volume, making it difficult to meet the urgent needs of modern applications for real-time processing. In industrial inspection scenarios that require rapid response, this computational burden often becomes a bottleneck restricting the application of the technology.

[0006] Traditional methods based on local search are prone to getting stuck in local optima, affecting the quality and stability of unwrapping. This local optima problem makes it impossible for the algorithm to obtain the globally optimal unwrapping result, which is particularly serious in the case of complex phase distribution, and directly affects the overall performance and reliability of the detection system.

[0007] In the existing technology, although some research has attempted to perform signal processing in the frequency domain, it has mainly focused on filtering and feature extraction. Research on systematically introducing frequency domain processing into the field of phase unwrapping is still insufficient. The lack of a complete theoretical framework and practical technical solutions has greatly limited the development of frequency domain phase unwrapping technology. There is an urgent need to develop new technical paths to solve these key problems. Summary of the Invention

[0008] In view of this, the present invention provides a frequency domain gradient weighted least squares phase unwrapping algorithm and an electromagnetic induction inversion method to at least solve some of the technical problems in the background art.

[0009] To achieve the above objectives, the present invention adopts the following technical solution:

[0010] A frequency-domain gradient-weighted least squares phase unwrapping algorithm and an electromagnetic induction inversion method include the following steps:

[0011] S1. Acquire complex voltage signals from electromagnetic induction signals;

[0012] S2. Obtain the frequency domain signal by performing a fast Fourier transform on the complex voltage signal, and calculate the frequency domain phase spectrum of electromagnetic induction based on the frequency domain signal;

[0013] S3. Calculate the frequency domain phase gradient between adjacent frequency points of the complex voltage signal based on the frequency domain phase spectrum;

[0014] S4. The obtained frequency domain phase gradient is packaged to obtain the packaged phase gradient;

[0015] S5. Construct an adaptive weighting function based on the amplitude and noise variance of the frequency domain signal, and obtain the weights of the frequency domain signal;

[0016] S6. Construct a weighted least squares objective function based on the weights of the packaged phase gradient and the frequency domain signal, and solve for the unwrapped phase based on the objective function.

[0017] Furthermore, in step S2, the complex voltage signal is subjected to a Fast Fourier Transform using the following expression:

[0018]

[0019] In the formula, This represents the frequency domain signal obtained after the Fast Fourier Transform; This represents the complex voltage signal at time t; Angular frequency represents the frequency component of the electromagnetic induction signal; The symbol for the imaginary unit.

[0020] Furthermore, the frequency domain phase spectrum of electromagnetic induction is calculated based on the frequency domain signal using the following formula:

[0021]

[0022] in, It is the arctangent function in the four quadrants; Represents the imaginary part of a complex number; This indicates the actual part of the review.

[0023] Furthermore, the frequency domain phase gradient between adjacent frequency points of the complex voltage signal is calculated based on the frequency domain phase spectrum, specifically using the following formula:

[0024]

[0025] In the formula, The frequency domain phase spectrum representing electromagnetic induction; Angular frequency represents the frequency component of the electromagnetic induction signal; This represents the frequency domain spacing between adjacent frequency points of a complex voltage signal.

[0026] Furthermore, in step S4, the calculation of the packaging phase gradient includes the following formula:

[0027]

[0028] in, To package the phase gradient, For the frequency domain phase gradient, .

[0029] Furthermore, prior to packaging, the process also includes:

[0030] Determine the phase gradient in the frequency domain The jump, when satisfy At that time, packaging processing is carried out.

[0031] Furthermore, S5 constructs an adaptive weighting function based on the amplitude and noise variance of the frequency domain signal, specifically including the following functions:

[0032]

[0033] In the formula, This is the noise variance estimate. It represents the amplitude of the frequency domain signal.

[0034] In this step, the weighting function can adaptively adjust the weights at different frequencies to suppress the influence of low signal-to-noise ratio frequencies on the unwrapping results.

[0035] Furthermore, the weighting function noise variance Estimate using the following methods:

[0036] Calculate the power spectrum of the frequency domain signal, construct a power spectral density map, and select the frequency bands with signal power spectral density below a set threshold as noise reference regions;

[0037] The mean power spectral density within the reference region is calculated as an estimate of the noise variance.

[0038] Furthermore, S6 specifically includes:

[0039] Construct a weighted least squares objective function based on the weights of the packaged phase gradient and the frequency domain signal:

[0040] ;

[0041] In the formula, This represents the weight values ​​of the adaptive weight function. To package the phase gradient, The absolute phase to be untangled;

[0042] Based on the aforementioned weighted least squares objective function, construct an optimized objective function for weighted least squares phase unwrapping:

[0043]

[0044] Through the By taking the partial derivatives and setting them to zero, we obtain a system of linear equations.

[0045] The linear equations are solved using the conjugate gradient method or the minimum residual method to obtain the absolute phase after unwrapping.

[0046] Furthermore, the method also includes performing electromagnetic induction inversion based on the unwrapped phase, specifically including:

[0047] Position-sensitive phase features are extracted from the absolute phase of each channel, including phase values ​​at specific frequencies, statistical features of the phase spectrum, and phase difference features.

[0048] Construct a linear mapping relationship between the geometric coordinate matrix and the phase eigenvectors:

[0049] The least squares method is used to fit the linear mapping relationship and solve for the three-dimensional coordinates of the position.

[0050] As can be seen from the above technical solution, compared with the prior art, the present invention discloses a frequency domain gradient-weighted least squares phase unwrapping method, which has the following beneficial effects:

[0051] This invention is the first to systematically introduce frequency domain processing into the field of phase unwrapping, establishing a complete theoretical framework for frequency domain phase unwrapping. This theoretical innovation not only solves the technical problems in practical applications but also provides a new theoretical foundation and technical path for further research in related fields, possessing significant academic value and practical guiding significance. Specifically:

[0052] 1. This invention fundamentally changes the traditional spatial phase unwrapping process by using frequency domain gradient calculation technology. It directly calculates the phase gradient in the frequency domain, avoiding path dependence, eliminating the root cause of error accumulation, and significantly improving the unwrapping accuracy and stability.

[0053] 2. The adaptive weighting function designed in this invention can automatically adjust the weight allocation according to the signal-to-noise ratio characteristics of different frequency points, giving greater weight to frequency points with high signal strength and less weight to frequency points dominated by noise, thereby maximizing the use of effective signal information. This mechanism enables the algorithm to maintain good untangling performance in low signal-to-noise ratio environments and greatly improves the noise resistance of the system.

[0054] 3. This invention utilizes the fast computation characteristics of FFT to reduce the algorithm complexity from the traditional O(N²) to O(NlogN), enabling real-time processing applications of large-scale data.

[0055] 4. This invention adopts a weighted least squares global optimization strategy. By constructing a global objective function, it searches for the optimal solution in the entire frequency domain, ensuring the consistency and optimality of the unwrapping results and significantly improving the stability of the unwrapping quality.

[0056] 5. The frequency domain processing method of this invention makes the algorithm more robust to local missing data and outliers. Even when some data is missing or outliers, the algorithm can still maintain good unwrapping performance through global information in the frequency domain. It is particularly suitable for harsh industrial testing environments and significantly improves the reliability and practicality of the system.

[0057] 6. This invention enables more precise particle displacement detection, material property analysis, and other applications through high-precision phase unwrapping, providing an effective technical solution, especially in industrial testing scenarios that require high precision and high stability.

[0058] 7. The frequency domain processing method of the present invention greatly simplifies the process of parameter adjustment and path selection strategy. Users only need to set a few basic parameters, and the algorithm can automatically complete high-quality phase unwrapping, reducing the complexity of the system and the threshold for use. Attached Figure Description

[0059] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0060] Figure 1 This is the overall flowchart of the frequency domain gradient weighted least squares phase unwrapping algorithm and electromagnetic induction inversion technology of this invention;

[0061] Figure 2 This is a flowchart illustrating the specific application of the frequency domain gradient weighted least squares phase unwrapping algorithm and electromagnetic induction inversion technology in this invention.

[0062] Figure 3 This is a 3D schematic diagram of Fourier transform unwrapping in the frequency domain gradient weighted least squares phase unwrapping algorithm and electromagnetic induction inversion technology of the present invention;

[0063] Figure 4 This is a schematic diagram of the frequency domain phase spectrum extraction principle in the frequency domain gradient weighted least squares phase unwrapping algorithm and electromagnetic induction inversion technology of this invention;

[0064] Figure 5 This is a comparison chart of the frequency domain phase unwrapping processing effects in the frequency domain gradient weighted least squares phase unwrapping algorithm and electromagnetic induction inversion technology of this invention. Detailed Implementation

[0065] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0066] refer to Figure 1 This invention discloses a frequency-domain gradient-weighted least squares phase unwrapping method, which systematically introduces frequency-domain processing into the field of phase unwrapping. The method includes the following steps:

[0067] S1. Acquire complex voltage signals from electromagnetic induction signals. ;

[0068] S2. Obtain the frequency domain signal by performing a fast Fourier transform on the complex voltage signal, and calculate the frequency domain phase spectrum of electromagnetic induction based on the frequency domain signal;

[0069] In this step, the Fast Fourier Transform of the complex voltage signal is performed using the following expression:

[0070]

[0071] In the formula, This represents the frequency domain signal obtained after the Fast Fourier Transform; This represents the complex voltage signal at time t; Angular frequency represents the frequency component of the electromagnetic induction signal; Represents the imaginary number symbol.

[0072] The frequency domain phase spectrum of electromagnetic induction is calculated based on the frequency domain signal using the following formula:

[0073]

[0074] in, It is the arctangent function in the four quadrants; Represents the imaginary part of a complex number; This indicates the actual part of the review.

[0075] S3. Calculate the frequency domain phase gradient between adjacent frequency points of the complex voltage signal based on the obtained frequency domain phase spectrum; specifically, use the following formula:

[0076]

[0077] In the formula, The frequency domain phase spectrum representing electromagnetic induction; Angular frequency represents the frequency component of the electromagnetic induction signal; This represents the frequency domain spacing between adjacent frequency points of a complex voltage signal.

[0078] In this step, the frequency interval for frequency domain gradient calculation is... Determined based on signal sampling rate and number of transformation points:

[0079]

[0080] in, Sampling frequency, This represents the number of points in the FFT transform.

[0081] S4. Wrap the obtained frequency domain phase gradient to obtain the wrapped phase gradient. Packaging phase gradient The calculation includes the following formulas:

[0082]

[0083] in, To package the phase gradient, For the frequency domain phase gradient, .

[0084] Furthermore, before packaging, it is necessary to determine the frequency domain phase gradient. The jump, when satisfy Then, it will be packaged.

[0085] S5. Construct an adaptive weighting function based on the amplitude and noise variance of the frequency domain signal, and obtain the weights of the frequency domain signal; the adaptive weighting function specifically includes the following functions:

[0086]

[0087] In the formula, This is the noise variance estimate. It represents the amplitude of the frequency domain signal.

[0088] The weighting function can adaptively adjust the weights at different frequencies, suppressing the influence of low signal-to-noise ratio frequencies on the unwrapping results. The weighting function... noise variance Estimate using the following methods:

[0089] Select the frequency band with lower signal power spectral density as the noise reference region;

[0090] The mean power spectral density within the reference region is calculated as an estimate of the noise variance.

[0091] S6. Construct a weighted least squares objective function based on the weights of the packaged phase gradient and the frequency domain signal, and solve for the unwrapped phase based on the objective function. Specifically, this includes the following steps:

[0092] Construct a weighted least squares objective function based on the weights of the packaged phase gradient and the frequency domain signal:

[0093] ;

[0094] In the formula, This represents the weight values ​​of the adaptive weight function. To package the phase gradient, The absolute phase to be untangled;

[0095] Constructing an optimized objective function for weighted least squares phase unwrapping based on the weighted least squares objective function:

[0096]

[0097] Through the By taking the partial derivatives and setting them to zero, we obtain a system of linear equations.

[0098] The linear equations are solved using the conjugate gradient method or the minimum residual method to obtain the absolute phase after unwrapping.

[0099] The frequency domain gradient-weighted least squares phase unwrapping method disclosed in this invention can be applied to particle displacement detection. The position coordinates are obtained from the phase after unwrapping. The specific process is as follows:

[0100] Establish a phase-position mapping relationship using the unwrapped absolute phase data;

[0101] The conversion from phase information to particle three-dimensional coordinates is achieved through an inversion algorithm.

[0102] refer to Figures 2-5 In one specific embodiment of the present invention, this method is applied to particle displacement detection. The particle displacement detection system includes: an excitation coil, multiple receiving coils (numbered 1 to M), a signal conditioning circuit, a data acquisition module, and a signal processing unit. The excitation coil generates an alternating electromagnetic field. When a metal particle moves within the detection area, a voltage signal containing position information is induced in the receiving coil. Figure 2 This is a flowchart illustrating the specific application of the frequency domain gradient weighted least squares phase unwrapping algorithm and electromagnetic induction inversion technique in the embodiments. Figure 3 This is a 3D schematic diagram of Fourier transform unwrapping in this specific embodiment; Figure 4 This is a schematic diagram of the frequency domain phase spectrum extraction principle in this specific embodiment; Figure 5 This is a comparison diagram of the frequency domain phase unwrapping processing effect in this specific embodiment.

[0103] like Figure 2 As shown, in the particle displacement detection system, the frequency domain gradient-weighted least squares phase unwrapping algorithm specifically includes the following steps:

[0104] Step 1: Multi-channel signal acquisition:

[0105] M receiving coils are used to simultaneously acquire induced signals to obtain multi-channel (M channels) complex voltage data V1(t), V2(t) ..., V M (t).

[0106] At this point, the complex voltage signal of each channel can be expressed as: , i = 1,2,...,M.

[0107] in, For the first The signal amplitude of each channel at time t Let be the signal phase of the i-th channel at time t, and j represent the imaginary unit.

[0108] Step 2, Frequency Domain Transformation and Phase Spectrum Calculation:

[0109] Complex voltage signal for each channel Perform an N-point Fast Fourier Transform to convert the complex voltage signal from the time domain to the frequency domain:

[0110]

[0111] Where t represents the sampling time in the time domain. In the frequency domain, this corresponds to discrete frequency points. This transformation realizes the transformation from time-domain signals... to frequency domain signal The conversion;

[0112] The phase spectrum is calculated based on the complex voltage signal after frequency domain transformation, using the following formula:

[0113]

[0114] in, It is a four-quadrant arctangent function, ensuring that the phase value is within the interval (-π, π]. Represents the imaginary part of a complex number; This indicates the actual part of the review.

[0115] Step 3: Calculate the frequency domain phase gradient based on the phase spectrum:

[0116] First, calculate the phase difference between adjacent frequency points, with the frequency interval being:

[0117]

[0118] in, Sampling frequency, Number of FFT transform points

[0119] For particle displacement detection system Each channel has a frequency domain phase gradient of:

[0120] k = 0,1,...,N-2;

[0121] in, The frequency domain phase spectrum represents the phase distribution of the signal at different frequencies; ω is the angular frequency, representing the frequency components of the signal. The frequency domain phase gradient represents the rate of change of phase with frequency.

[0122] Step 4, Gradient Packaging Process:

[0123] The frequency domain phase gradient is wrapped first to eliminate the 2π jump. When |∇φᵢ(ωk)| > π, the wrapping process is performed:

[0124] ;at this time, .

[0125] Step 5: Construct an adaptive weighting function based on noise variance and signal amplitude:

[0126] In this step, the frequency band with lower signal power spectral density is preferred as the noise reference region. Typically, a high-frequency band (such as 0.8fs to 1fs) is selected to obtain the noise variance estimate.

[0127] ;

[0128] Then, an adaptive weighting function is constructed based on the noise variance and the frequency domain signal amplitude:

[0129]

[0130] in, This represents an adaptive weighting function used to adjust the importance of different frequencies during the unwrapping process; : Represents the amplitude of the frequency domain signal, reflecting the signal strength at that frequency point; the purpose of constructing the weighting function is to assign more weight to frequency points with high signal strength and less weight to frequency points dominated by noise. Indicates the first The noise variance estimate of the channel is used to measure the _th _th_ ... Noise levels for each channel.

[0131] To further optimize performance, a frequency-related adjustment factor is introduced into the adaptive weighting function. The adaptive weight function at this point is as follows:

[0132]

[0133] in, This is a frequency-dependent adjustment factor, with a value of 1.0 in the main signal frequency band and a value of 0.5 in the noise-dominant frequency band.

[0134] Step 6: Objective function construction and phase solution:

[0135] Establish a weighted least squares objective function for each channel:

[0136]

[0137] Indicates the first The absolute phase after untangling each channel, i.e., the continuous phase after eliminating the 2π jump; Indicates the first Each channel is a packaged phase gradient, i.e., a packaged phase gradient.

[0138] Solving the system of linear equations:

[0139] Through the Taking the partial derivatives and setting them to zero, we obtain the system of linear equations:

[0140] ;

[0141] in, The coefficient matrix, Let be the absolute phase vector to be solved. This is the vector on the right side.

[0142] The obtained linear equation system is solved using the preconditional conjugate gradient method:

[0143] ;

[0144] in, For step size parameters, Indicates the search direction.

[0145] Step 7: Phase Feature Extraction and Position Inversion

[0146] First, position-sensitive features are extracted from the absolute phase after unwrapping of each channel, including: the phase value at a specific frequency point in each channel i. ,in This indicates the system's operating frequency; and the statistical characteristics of the phase spectrum, such as mean, variance, and skewness.

[0147] The phase difference characteristic can be expressed as:

[0148]

[0149] Based on the obtained phase characteristics, a mapping relationship between the phase characteristics and the three-dimensional coordinates of the particles is established:

[0150]

[0151] The least squares fitting method was used to obtain the three-dimensional coordinates of the particles. The objective function for the least squares fitting solution is as follows:

[0152]

[0153] in, For geometric matrices, This is the phase eigenvector.

[0154] This embodiment fully demonstrates the complete application process of the electromagnetic induction phase unwrapping method based on frequency domain gradient in particle displacement detection, verifying the effectiveness and practicality of the method. Compared with the traditional spatial domain method, this method has significant improvements in unwrapping accuracy, computational efficiency, and noise resistance.

[0155] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0156] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A frequency domain gradient-weighted least squares phase unwrapping method, characterized in that, The method comprises the following steps: S1, collecting a complex voltage signal of an electromagnetic induction signal; S2, performing fast Fourier transform on the complex voltage signal to obtain a frequency domain signal, and calculating a frequency domain phase spectrum of the electromagnetic induction according to the frequency domain signal; S3, calculating a frequency domain phase gradient between adjacent frequency points of the complex voltage signal according to the frequency domain phase spectrum; S4, performing packing processing on the obtained frequency domain phase gradient to obtain a packed phase gradient; S5, constructing an adaptive weight function based on the amplitude of the frequency domain signal and the noise variance, and obtaining the weight of the frequency domain signal; S6, constructing a weighted least squares objective function according to the packed phase gradient and the weight of the frequency domain signal, and solving the unwrapped phase based on the objective function.

2. The method of claim 1, wherein, In step S2, the fast Fourier transform on the complex voltage signal adopts the following expression: ； In the formula, denotes a frequency domain signal obtained after a fast Fourier transform; denotes a complex voltage signal at time t; is an angular frequency and denotes a frequency component of the electromagnetic induction signal; denotes the imaginary unit.

3. The method of claim 2, wherein, The frequency domain phase spectrum of the electromagnetic induction is calculated according to the frequency domain signal, and the following calculation formula is adopted: ； wherein is a four-quadrant arctangent function; denotes the imaginary part of a complex number; denotes the real part of a complex number.

4. The method of claim 1, wherein, In step S3, the frequency domain phase gradient between adjacent frequency points of the complex voltage signal is calculated according to the frequency domain phase spectrum, and the following formula is adopted: ； wherein represents the frequency domain phase spectrum of the electromagnetic induction; is the angular frequency, representing the frequency component of the electromagnetic induction signal; represents the frequency domain interval between adjacent frequency points of the complex voltage signal.

5. The method of claim 1, wherein, In step S4, the calculation of the packed phase gradient includes the following formula: ； wherein is a wrapped phase gradient, is a frequency domain phase gradient, .

6. The method of claim 5, wherein, Before the packing processing, it further includes: Determining a jump of a frequency domain phase gradient When satisfies a packing process is performed.

7. The method of claim 1, wherein, S5, constructing an adaptive weight function based on the amplitude of the frequency domain signal and the noise variance, which specifically includes the following function: ； In the formula, is a noise variance estimate, denotes the amplitude of the frequency domain signal.

8. The method of claim 7, wherein, weight function noise variance in is estimated by The power spectrum of the frequency domain signal is calculated, the power spectral density diagram is constructed, and the frequency band with a signal power spectral density lower than a set threshold is selected as a noise reference region; The mean value of the power spectral density in the reference region is calculated as the noise variance estimate.

9. The method of claim 1, wherein, S6 specifically includes: A weighted least squares objective function is constructed according to the packed phase gradient and the weight of the frequency domain signal: ； wherein denotes a weight value of the adaptive weight function, is a wrapped phase gradient, is an absolute phase to be unwrapped; A weighted least squares phase unwrapping optimization objective function is constructed based on the weighted least squares objective function: ； By taking the partial derivative and setting it to zero, a system of linear equations is obtained. By taking the partial derivative and setting it to zero, a system of linear equations is obtained. The conjugate gradient method or the least residual method is adopted to solve the linear equation set to obtain the unwrapped absolute phase.

10. The method of claim 1, wherein, Further, electromagnetic induction inversion is performed according to the unwrapped phase, which specifically includes: The position-sensitive phase feature quantity is extracted from the absolute phase of each channel, including the phase value of a specific frequency point, the statistical feature of the phase spectrum, and the phase difference feature; A linear mapping relationship between the geometric coordinate matrix and the phase feature vector is constructed: The least squares method is adopted to fit the linear mapping relationship, and the three-dimensional coordinates of the position are solved.

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