Phase unwrapping method based on extended Kalman filtering

Abstract

The invention aims to provide a phase unwrapping method based on extended Kalman filtering. The method comprises the following steps: A, constructing a two-dimensional phase unwrapping program MCC-EKFPU; and B, performing phase unwrapping processing on the interferogram by using a two-dimensional phase unwrapping program MCC-EKFPU to obtain an unwrapped phase of the interferogram. According to the method, the maximum correlation entropy criterion extended Kalman filtering theory is introduced into interferogram wrapping phase unwrapping, and an interferogram phase gradient estimation technology and a robust path tracking strategy are combined, so that the method has higher phase unwrapping precision and better robustness; the method has good application potential in interferogram phase unwrapping application with high accuracy requirements.

Description

Technical Field

[0001] This invention relates to the field of phase unwrapping technology, and more specifically to a phase unwrapping method based on extended Kalman filtering. Background Technology

[0002] In various interferometric techniques, including Interferometric Synthetic Aperture Radar (InSAR), optical interferometry, and digital holographic interferometry, the measured values ​​of the target's physical parameters are often measured using their corresponding interferometric phase. However, the measured phase data obtained from these techniques are typically presented in modulo 2π form, resulting in a 2π-integer multiple difference between the measured phase (commonly known as the wrapped phase) and its true unwrapped phase. Therefore, the measured wrapped phase cannot be directly used for the inversion of the physical parameters of the detected target. The wrapped phase needs to undergo phase unwrapping (PU) processing to restore it to the true unwrapped phase representing the measured target's physical parameters. Thus, as a core step in interferometric techniques, the accuracy of phase unwrapping (PU) directly determines the accuracy of the target's physical parameter estimation. Accurately recovering the unwrapped phase reflecting the target's physical parameters from interferograms of different fringe patterns has become a crucial step in improving the reliability of various interferometric techniques. Summary of the Invention

[0003] This invention aims to provide a phase unwrapping method based on extended Kalman filtering. This method introduces the maximum correlation entropy criterion of extended Kalman filtering theory into interferogram winding phase unwrapping. Combined with interferogram phase gradient estimation technology and robust path tracking strategy, it has higher phase unwrapping accuracy and better robustness, and has good application potential in interferogram phase unwrapping applications with high accuracy requirements.

[0004] The technical solution of the present invention is as follows: The phase unwrapping method based on extended Kalman filtering includes the following steps: A. Construct the two-dimensional phase unwrapping program MCC-EKFPU. The system equations of the two-dimensional phase unwrapping program MCC-EKFPU are as follows: (1) in, For untangled pixels The unwrapped phase is used as a state variable to be estimated by the MCC-EKF phase unwrapping procedure. Represents a pixel Pixels in the neighborhood (eight neighbor pixels) The untangling phase of an untangled pixel; and Representing the neighboring pixels respectively To untangle the pixel The phase gradient estimation and its phase gradient estimation error; Represents a pixel Observations of state variables, Representing interferometric image elements Noise-free observations of state variables. Represents a pixel State variable observation noise vector; B. The interferogram is unwrapped using the two-dimensional phase unwrapping program MCC-EKFPU to obtain the unwrapped phase of the interferogram.

[0005] The solution process of the two-dimensional phase unwrapping program MCC-EKFPU is as follows: a. State prediction, solving for the target pixel Untangling phase prediction value Variance of untangled phase prediction error ; b. State update, based on target pixel Untangling phase prediction value Variance of untangled phase prediction error The final untangling phase estimate is calculated. and the final estimated error variance .

[0006] In step a, the target pixel is solved. Untangling phase prediction value The formula is as follows: (2) (3) (4) in, This represents the two-dimensional label of the pixel to be unwrapped. and These represent the unwrapped pixels. The untangling phase and its estimation error variance; Represents the central pixel The set of coordinates of untangled pixels within the domain; For untangled pixels The weights; Represents the target pixel Unentangled pixels within the domain With the target pixel Phase difference between adjacent phases; Indicates by Unentangled pixels within the pixel domain The initial unwrapping phase prediction value obtained from the phase information; Indicates by Unentangled pixels within the pixel domain Initial unwrapped phase prediction value obtained from phase information The final untangled phase prediction value is obtained after weighted summation.

[0007] In step a, the variance of the unwrapping phase prediction error is calculated. The formula is as follows: (5) In the formula, For target pixel and untangled pixels Variance of phase gradient estimation error between; For target pixel Variance of unwrapping phase prediction error.

[0008] In step b, the target pixel is calculated. Untangled phase estimate and the final estimated error variance The formula is as follows: (6) (7) (8) (9) (10) in, This represents the two-dimensional label of the pixel to be unwrapped. It is a Gaussian kernel function with weighted Mahalanobis distance. Indicates by The target pixel is obtained by using the phase information of all unwrapped pixels within the pixel domain. The predicted value of the untangling phase, Represents a cell Unwrapped phase measurement noise error covariance matrix; Represents the observation function The Jacobian matrix in The value at; Represents a cell The gain matrix; for Pixel unwrapping phase estimate, For pixels Variance of untangled phase estimation error.

[0009] Gaussian kernel function with weighted Mahalanobis distance The expression is: ; In the above formula, The formula for weighted squared Mahalanobis distance is as follows: represent Unwrapping phase estimate of a pixel or observation phase , This can represent the predicted value of the target untangling phase. Or observed predicted value ; Represents the untangling phase prediction error Or observation noise The inverse of the covariance matrix.

[0010] The phase unwrapping method based on extended Kalman filtering of this invention can effectively improve the estimation accuracy of the phase unwrapping procedure under the conditions of strong nonlinearity of the problem and non-Gaussian distribution of noise in the system model, and has greater advantages in terms of stability and accuracy of phase unwrapping results.

[0011] The effectiveness and robustness of the method of the present invention have been fully verified in phase unwrapping experiments using simulated and measured data under different phase distributions and signal-to-noise ratios, and it can be applied to phase unwrapping of different types of interferograms.

[0012] The phase unwrapping method based on extended Kalman filtering of this invention, as demonstrated by simulation and experimental data, shows that the MCC-EKFPU algorithm has higher unwrapping accuracy and better robustness, making it a promising candidate for application in interferogram PU applications where high precision is required. Attached Figure Description

[0013] Figure 1 This is a simulated interferogram from Embodiment 2 of the present invention. Figure 1 (a) and 1(b) are two true unwrapped phases with different phase distributions; Figure 1 (c) and 1(d) are respectively Figure 1 The noise-wound phase diagrams corresponding to (a) and 1(b) have SNRs of 5.86dB and 0.47dB, respectively. Figure 2 The various methods of Embodiment 2 of the present invention are processed Figure 1 In Figure 1 (c) shows the unwrapping results. The first to third columns are the unwrapping results of the QGPU, ILS, and MCC-EKFPU methods, respectively. The first to third rows are the unwrapping phase diagram, unwrapping error diagram, and unwrapping error statistical histogram for each method, respectively. Figure 3 The various methods of Embodiment 2 of the present invention are processed Figure 1(d) shows the unwrapping results. The first to third columns are the unwrapping results of the QGPU, ILS, and MCC-EKFPU methods, respectively. The first to third rows are the unwrapping phase diagram, unwrapping error diagram, and unwrapping error statistical histogram for each method, respectively.

[0014] Figure 4 These are measured interferograms from Embodiment 2 of the present invention: (a) an interferogram of a region in Xi'an; (b) an interferogram of a region in Yangbi County, Yunnan Province. Figure 5 This invention provides an embodiment 2 of the unwrapping method using the MCC-EKFPU method of the present invention. Figure 4 (a) and Figure 4 The results of (b) are shown in columns one through two. Figure 4 (a) and Figure 4 (b) Unwinding results; the first and second rows are the unwinding phase diagram and the corresponding rewinding phase diagram, respectively. Detailed Implementation

[0015] The present invention will now be described in detail with reference to the accompanying drawings and embodiments. Example 1

[0016] The phase unwrapping method based on extended Kalman filtering provided in this embodiment includes the following steps: The phase unwrapping method based on extended Kalman filtering includes the following steps: A. Construct the two-dimensional phase unwrapping program MCC-EKFPU. The system equations of the two-dimensional phase unwrapping program MCC-EKFPU are as follows: (1) in, For untangled pixels The unwrapped phase is used as a state variable to be estimated by the MCC-EKF phase unwrapping procedure. Represents a pixel Pixels in the neighborhood (eight neighbor pixels) The untangling phase of an untangled pixel; and Representing the neighboring pixels respectively To untangle the pixel The phase gradient estimation and its phase gradient estimation error; Represents a pixel Observations of state variables, Representing interferometric image elements Noise-free observations of state variables. Represents a pixel State variable observation noise vector; B. The interferogram is unwrapped using the two-dimensional phase unwrapping program MCC-EKFPU to obtain the unwrapped phase of the interferogram.

[0017] The solution process of the two-dimensional phase unwrapping program MCC-EKFPU is as follows: a. State prediction, solving for the target pixel Untangling phase prediction value Variance of untangled phase prediction error ; b. State update, based on target pixel Untangling phase prediction value Variance of untangled phase prediction error The final untangling phase estimate is calculated. and the final estimated error variance .

[0018] In step a, the target pixel is solved. Untangling phase prediction value The formula is as follows: (2) (3) (4) in, This represents the two-dimensional label of the pixel to be unwrapped. and These represent the unwrapped pixels. The untangling phase and its estimation error variance; Represents the central pixel The set of coordinates of untangled pixels within the domain; For untangled pixels The weights; Represents the target pixel Unentangled pixels within the domain With the target pixel Phase difference between adjacent phases; Indicates by Unentangled pixels within the pixel domain The initial unwrapping phase prediction value obtained from the phase information; Indicates by Unentangled pixels within the pixel domain Initial unwrapped phase prediction value obtained from phase information The final untangled phase prediction value is obtained after weighted summation.

[0019] In step a, the variance of the unwrapping phase prediction error is calculated. The formula is as follows: (5) In the formula, For target pixel and untangled pixels Variance of phase gradient estimation error between; For target pixel Variance of unwrapping phase prediction error.

[0020] In step b, the target pixel is calculated. Untangled phase estimate and the final estimated error variance The formula is as follows: (6) (7) (8) (9) (10) in, in This represents the two-dimensional label of the pixel to be unwrapped. It is a Gaussian kernel function with weighted Mahalanobis distance. Indicates by The target pixel is obtained by using the phase information of all unwrapped pixels within the pixel domain. The predicted value of the untangling phase, Represents a pixel Unwrapped phase measurement noise error covariance matrix; Represents the observation function The Jacobian matrix in The value at; Represents a pixel The gain matrix; for Pixel unwrapping phase estimate, for Variance of pixel unwrapping phase estimation error; Gaussian kernel function with weighted Mahalanobis distance The expression is: ;;

[0021] In the above formula, The formula for weighted squared Mahalanobis distance is as follows: represent Unwrapping phase estimate of a pixel or observation phase , This can represent the predicted value of the target untangling phase. Or observed predicted value ; Represents the untangling phase prediction error Or observation noise The inverse of the covariance matrix.

[0022] Example 2: Phase Unwrapping Comparison Experiment To evaluate the performance of the MCC-EKFPU phase unwrapping method based on extended Kalman filtering in this invention, it was compared with representative methods such as the existing QGPU method and ILS method in the same experimental environment to verify the phase unwrapping accuracy and operating efficiency of each method.

[0023] 1. Simulated data experiment Figure 1 (a and b) are three real unwrapped phases with different phase distributions (image size is 256×256 pixels). Figure 1 (c and d) are respectively Figure 1 The noise-wrapped phase maps corresponding to (a) and (b) have SNRs of 5.86 dB and 0.47 dB, respectively. The QGPU, ILS, and MCC-EKFPU methods were used to analyze the noise-wrapped phase maps. Figure 1 (c) and (d) were subjected to PU experiments, and the untangling results are as follows: Figures 2-3 As shown Figure 2 , 3 They were shown respectively Figure 1 The interferograms shown in (c) and (d) are the results of unwrapping using three methods. From left to right, the columns represent QGPU, ILS, and MCC-EKFPU methods, and the rows show the unwrapping phase diagram, unwrapping error diagram, and error statistical histogram for each method. Figure 2 , 3 As shown, all three methods can be used to... Figure 1 The interferograms shown in (c and d) show good unwrapping results, but the error dynamic range of the first two methods, such as QGPU and ILS, is significantly larger than that of the MCC-EKFPU method. This can be verified by comparing the color scale on the right side of the error map and the horizontal axis range of the error histogram, indicating that the latter has a greater advantage in terms of the stability and accuracy of the phase unwrapping results.

[0024] 2. Experimental Data Figure 4 (a) shows an interferogram slice of a region in Xi’an, China, with an image size of 512 x 512 pixels; Figure 4 (b) shows an interferogram slice of a region in Yangbi County, Yunnan Province, China, with an image size of 512×512 pixels.

[0025] Figure 5Untangling using the MCC-EKFPU method Figure 4 The results in (ab) show that the unwrapping phase is approximately continuous, and the rewrapping phase is consistent with... Figure 4 The interference pattern shown in (ab) has consistent fringes, indicating good unwrapping accuracy and stability.

[0026] 3. Conclusion The effectiveness and robustness of the MCC-EKFPU method have been fully verified in phase unwrapping experiments using simulated and measured data under different phase distributions and signal-to-noise ratios, and it can be applied to phase unwrapping of different types of interferograms.

Claims

1. A phase unwrapping method based on extended Kalman filtering, characterized in that, Includes the following steps: A. Construct the two-dimensional phase unwrapping program MCC-EKFPU. The system equations of the two-dimensional phase unwrapping program MCC-EKFPU are as follows: （1） in, For untangled pixels The unwrapped phase is used as a state variable to be estimated by the MCC-EKF phase unwrapping procedure. Represents a cell Pixels in the neighborhood (eight neighboring pixels) The untangling phase of an untangled pixel; and Representing the neighboring pixels respectively To untangle the pixel The phase gradient estimation and its phase gradient estimation error; Represents a cell Observations of state variables, Representing interferometric image elements Noise-free observations of state variables. Represents a cell State variable observation noise vector; B. The interferogram is unwrapped using the two-dimensional phase unwrapping program MCC-EKFPU to obtain the unwrapped phase of the interferogram.

2. The phase unwrapping method based on extended Kalman filtering as described in claim 1, characterized in that: The solution process of the two-dimensional phase unwrapping program MCC-EKFPU is as follows: a. State prediction, solving for the target pixel Untangling phase prediction value Variance of untangled phase prediction error ; b. State update, based on target pixel Untangling phase prediction value Variance of untangled phase prediction error The final untangling phase estimate is calculated. and the final estimated error variance .

3. The phase unwrapping method based on extended Kalman filtering as described in claim 2, characterized in that: In step a, the target pixel is solved. Untangling phase prediction value The formula is as follows: （2） （3） （4） in, This represents the two-dimensional label of the pixel to be unwrapped. and These represent the unwrapped pixels. The untangling phase and its estimation error variance; Represents the central pixel The set of coordinates of untangled pixels within the domain; For untangled pixels The weights; Represents the target pixel Unentangled pixels within the domain With the target pixel Phase difference between adjacent phases; Indicates by Unentangled pixels within the pixel domain The initial unwrapping phase prediction value obtained from the phase information; Indicates by Unentangled pixels within the pixel domain Initial unwrapped phase prediction value obtained from phase information The untangled phase prediction value obtained after weighted summation.

4. The phase unwrapping method based on extended Kalman filtering as described in claim 2, characterized in that: In step a, the variance of the unwrapping phase prediction error is calculated. The formula is as follows: （5） In the formula, For target pixel and untangled pixels Variance of phase gradient estimation error between; For target pixel Variance of unwrapping phase prediction error.

5. The phase unwrapping method based on extended Kalman filtering as described in claim 2, characterized in that: In step b, the target pixel is calculated. Untangled phase estimate and the final estimated error variance The formula is as follows: （6） （7） （8） （9） （10） in, This represents the two-dimensional label of the pixel to be unwrapped. It is a Gaussian kernel function with weighted Mahalanobis distance. Indicates by The target pixel is obtained by using the phase information of all unwrapped pixels within the pixel domain. The predicted value of the untangling phase, Represents a cell Unwrapped phase measurement noise error covariance matrix; Represents the observation function The Jacobian matrix in The value at; Represents a cell The gain matrix; for Pixel unwrapping phase estimate, for Variance of pixel unwrapping phase estimation error.

6. The phase unwrapping method based on extended Kalman filtering as described in claim 5, characterized in that: Gaussian kernel function with weighted Mahalanobis distance The expression is: ； In the above formula, The formula for weighted squared Mahalanobis distance is as follows: represent Pixel unwrapping phase estimate or observation phase , This can represent the predicted value of the target untangling phase. Or observed predicted value ; Represents the untangling phase prediction error Or observation noise The inverse of the covariance matrix.

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