Phase unwrapping method based on factor graph

Abstract

The invention aims to provide a phase unwrapping method based on a factor graph, which comprises the following steps of: A, cutting an interferogram into a plurality of sub-interferograms with the same size by adopting a coincidence cutting mode, and reserving a plurality of rows or columns of overlapped pixels between adjacent sub-interferograms during cutting; b, constructing a phase unwrapping system equation for each sub-interferogram, then constructing a corresponding phase unwrapping factor graph model formula, and finally converting a factor graph optimization problem into a nonlinear least square optimization problem; C, carrying out iterative solution on the nonlinear least square optimization problem by adopting a Lescenberg-Marquardt method to obtain a nonlinear least square optimization problem; obtaining an unwrapping phase result of each sub-interferogram; and D, combining the unwrapping results of the sub-interferograms by using an unwrapping phase consistency criterion of adjacent regions of the interferograms to obtain an unwrapping phase of the whole interferogram. According to the phase unwrapping method, the consistency and the stability of a phase unwrapping result are improved.

Description

Technical Field

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

[0002] In high-precision InSAR measurements, the interferometric phase characterizing surface elevation or deformation information is typically presented in modulo 2π form, leading to a difference between the wrapped phase and the true phase. Phase unwrapping (PU) is necessary to recover the true phase, a step that directly determines the reliability of surface elevation or deformation parameter inversion based on interferometric phase estimation. To address this, researchers have proposed various PU strategies, including path-following methods, minimum norm methods, and nonlinear filtering optimal estimation methods. Path-following methods (e.g., Quality Guided GPU) are simple to implement, but error propagation is prone to occur when unwrapping interferograms with low signal-to-noise ratios and significant noise interference. Minimum norm methods (e.g., Iterative Least Squares (ILS)) obtain smooth unwrapping results by globally optimizing the phase of the entire unwrapped region, but the accuracy of local interferometric phase details is limited and it is sensitive to noise. As the requirements for interferogram phase unwrapping accuracy increase with the application of InSAR technology, the limitations of traditional PU methods are becoming increasingly apparent. Therefore, it is necessary to explore more efficient and robust strategies to optimize PU performance. Summary of the Invention

[0003] This invention aims to provide a phase unwrapping method based on factor graphs. This method models the phase unwrapping problem as a factor graph optimization problem. By utilizing the constraint relationship between the current unwrapped pixel phase and the phase of the neighboring unwrapped pixels, it performs nonlinear least squares optimization with maximum a posteriori estimation as the objective, thereby improving the consistency and stability of the phase unwrapping results.

[0004] The technical solution of the present invention is as follows: The aforementioned factor graph-based phase unwrapping method includes the following steps: A uses an overlapping cropping method to crop the interferogram into several sub-interferograms of the same size, and leaves several rows or columns of overlapping pixels between adjacent sub-interferograms during the cropping process; B. For each sub-interferogram, construct the phase unwrapping system equations, and set a fixed-step window for each sub-interferogram. Then, for each sub-interferogram, sort the pixels according to the quality-guided strategy based on heap sort, continuously extract the pixel with the best current quality to be unwrapped, and construct two core factors for it, namely the process factor and the observation factor. Then, store the constructed two factors and their related information into the window. As new pixels are continuously added, the window is gradually filled. When the number of pixels in the window reaches a set threshold, integrate all the factors corresponding to all M pixels in the current window, optimize the phase unwrapping system equations, and form a phase unwrapping factor graph model formula containing M pixel factor variables, thereby constructing the following nonlinear least squares objective function for phase unwrapping: (1) In the formula This is the final untangled phase estimation set obtained during the optimization process. for Variance of phase gradient estimation error for pixels Indicates that optimization is needed. Pixel entanglement phase variable, Indicates known The untangling phase of a pixel, express The true phase gradient estimate of a pixel; for Pixel observation noise error variance Interferogram The observation vector of a pixel, Represents the observation function; C. The Lewinberg-Marquardt method is used to iteratively solve the nonlinear least squares objective function to obtain the optimal solution for the M pixels within the window and update the results to obtain the phase unwrapping results for the M pixels. D. Slide the window forward to continue receiving new pixels, and repeat steps B and C until the entire sub-interference pattern is unwrapped. E. Using the unwrapping phase consistency criterion of adjacent regions of the interferogram, the unwrapping results of each sub-interferogram are merged to obtain the unwrapping phase of the entire interferogram.

[0005] In step A, the interferogram is cropped into 4-12 sub-interferograms of the same size.

[0006] In step B, the phase unwrapping system equation is: (2) In the formula, This represents the predictive model of the phase unwrapping system, characterizing Predicted unwrapping phase value of a pixel; This represents the interferogram. Observations of cell state variables without noise contamination; and They represent Like Yuanhe The untangling phase of a pixel; express The true phase gradient of a pixel; express The true phase gradient estimate of a pixel; Interferogram Observed values ​​of pixels; The process noise vector of the system satisfies , Let be the observation noise vector of the system and satisfy... The two are independent of each other, and and These are the error variances of the two, respectively.

[0007] In step B, the process of optimizing the phase unwrapping system equations using nonlinear least squares is as follows: In the factor graph phase unwrapping model, the main factor nodes include predictive factors and observed factors; observed factors: these factors are based on the observation equation. Construction; Process Factors: These factors are based on state transition equations Parameterized representation; Factor nodes define the constraint relationships between variable nodes. These constraints are described by probability density functions to represent this uncertainty. Ultimately, the graph optimization phase unwrapping problem can be transformed into a nonlinear least squares optimization problem, resulting in the following phase unwrapping model: (3) In the formula This is the final untangled phase estimation set obtained during the optimization process. The phase set to be optimized for Variance of phase gradient estimation error for pixels Indicates that optimization is needed. Pixel entanglement phase variable, Indicates known The untangling phase of a pixel, express The true phase gradient estimate of a pixel; for Pixel observation noise error variance Interferogram The observation vector of a pixel, Represents the observation function; This represents the initial unwrapped cell phase value; Due to the initial unwrapped pixel phase mean With covariance It is known that the corresponding constant term has no effect on the optimization result. The formula can be further expressed as a simplified nonlinear least squares objective function, i.e., formula (1).

[0008] The Lewinberg-Marquardt method is used to iteratively solve for each nonlinear least squares objective function. This is accomplished using the lsqnonlin function in the MATLAB software platform, and the specific process is as follows: a. When the sliding window meets the conditions to trigger optimization, the algorithm packages the constraints of all pixels in the window, including process factors and observation factors, into a nonlinear least squares problem in the form of formula (1), and provides a high-quality initial guess value X0. b. Call the lsqnonlin function, whose internal Lewinberg-Marquardt method processes this nonlinear least squares objective function and the initial value guess X0. Through iterative calculation, it finds a set of phase values ​​that minimize the total error, i.e. the sum of squares of all factor errors. c. Update the unwrapping result diagram of the sub-interference diagram with the optimal solution returned by the LM algorithm.

[0009] The convergence condition for the iterative solution is that both the step size tolerance and the objective function tolerance are set to a fixed threshold of 1×10. -8 When the changes in both are less than the threshold, the iteration is considered to have converged and the calculation is terminated.

[0010] This invention also discloses the application of the above-mentioned factor graph-based phase unwrapping method in phase unwrapping for InSAR elevation reconstruction.

[0011] The phase unwrapping method of this invention models the phase unwrapping problem as a factor graph optimization problem. By utilizing the constraint relationship between the phase of the current pixel to be processed and the phases of the unwrapped neighboring pixels, it performs nonlinear least squares optimization with maximum a posteriori estimation as the objective, effectively improving the continuity and consistency of the unwrapping results. Furthermore, the interferogram is first divided into regions, dividing the entire image into several sub-blocks to reduce the optimization scale and computational complexity.

[0012] The phase unwrapping method of the present invention, based on experimental results of simulated and measured data, demonstrates good unwrapping accuracy and robustness in phase unwrapping experiments of different fringe mode interferograms, verifying its effectiveness and application potential. Attached Figure Description

[0013] Figure 1The simulated interferogram of Embodiment 2 of the present invention is shown, wherein (a) and (b) correspond to the actual unfolded phases of fringe mode one and fringe mode two, respectively; (c) and (d) show the noisy entangled phases of (a) and (b), respectively, with signal-to-noise ratios of 7.45dB and 2.18dB, respectively. Figure 2 This is a comparison chart of the untangling effects of different methods in Embodiment 2 of the present invention; wherein the first column, the second column, and the third column represent the processing of ILS, QGPU, and FGPU, respectively. Figure 1 (c) shows the unwrapping results. The first, second, and third rows are the unwrapping phase diagram, unwrapping error diagram, and unwrapping error statistical histogram for each method, respectively. Figure 3 This is a comparison diagram of the untangling effects of different methods in Embodiment 2 of the present invention; wherein, the first to third columns represent the processing of ILS, QGPU, and FGPU, respectively. Figure 1 (d) shows the unwrapping results. 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 of Embodiment 2 of the present invention, wherein (a) and (b) show interferograms of the Etna volcano region on the east coast of Sicily, Italy, and the Yangbi County region of Yunnan Province, China, respectively; Figure 5 The FGPU method of the present invention is used in Embodiment 2 of the present invention for processing. Figure 4 (a) is the result of untangling; Figure 6 The FGPU method of the present invention is used in Embodiment 2 of the present invention for processing. Figure 4 (b) is the result of untangling. Detailed Implementation

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

[0016] This embodiment provides a phase unwrapping method based on factor graphs, which includes the following steps: A uses an overlapping cropping method to crop the interferogram into several sub-interferograms of the same size, and leaves several rows or columns of overlapping pixels between adjacent sub-interferograms during the cropping process; The interferogram is cropped into four sub-interferograms of the same size.

[0017] B. For each sub-interferogram, construct the phase unwrapping system equations, and set a fixed-step window for each sub-interferogram. Then, for each sub-interferogram, sort the pixels according to the quality-guided strategy based on heap sort, continuously extract the pixel with the best current quality to be unwrapped, and construct two core factors for it, namely the process factor and the observation factor. Then, store the constructed two factors and their related information into the window. As new pixels are continuously added, the window is gradually filled. When the number of pixels in the window reaches a set threshold, integrate all the factors corresponding to all M pixels in the current window, optimize the phase unwrapping system equations, and form a phase unwrapping factor graph model formula containing M pixel factor variables, thereby constructing the following nonlinear least squares objective function for phase unwrapping: (1) In the formula This is the final untangled phase estimation set obtained during the optimization process. for Variance of phase gradient estimation error for pixels Indicates that optimization is needed. Pixel entanglement phase variable, Indicates known The untangling phase of a pixel, express The true phase gradient estimate of a pixel; for Pixel observation noise error variance Interferogram The observation vector of a pixel, Represents the observation function; The nonlinear least-squares objective function for phase unwrapping is constructed as follows: a. For each sub-interferogram, construct the phase unwrapping system equations. The phase unwrapping system equations are as follows: (2) In the formula, This represents the predictive model of the phase unwrapping system, characterizing Predicted unwrapping phase value of a pixel; This represents the interferogram. Observations of cell state variables without noise contamination; and They represent Like Yuanhe The untangling phase of a pixel; express The true phase gradient of a pixel; express The true phase gradient estimate of a pixel; Interferogram Observed values ​​of pixels; The process noise vector of the system satisfies , Let be the observation noise vector of the system and satisfy... The two are independent of each other, and and These are the error variances of the two, respectively.

[0018] b. Optimize the phase unwrapping system equations using nonlinear least squares: In the factor graph phase unwrapping model, the main factor nodes include predictive factors and observed factors; observed factors: these factors are based on the observation equation. Construction; Process Factors: These factors are based on state transition equations Parameterized representation; Factor nodes define the constraint relationships between variable nodes. These constraints are described by probability density functions to represent this uncertainty. Ultimately, the graph optimization phase unwrapping problem can be transformed into a nonlinear least squares optimization problem, resulting in the following phase unwrapping model: (3) In the formula This is the final untangled phase estimation set obtained during the optimization process. The phase set to be optimized for Variance of phase gradient estimation error for pixels Indicates that optimization is needed. Pixel entanglement phase variable, Indicates known The untangling phase of a pixel, express The true phase gradient estimate of a pixel; for Pixel observation noise error variance Interferogram The observation vector of a pixel, Represents the observation function; This represents the initial unwrapped cell phase value; Due to the initial unwrapped pixel phase mean With covariance It is known that the corresponding constant term has no effect on the optimization result. The formula can be further expressed as a simplified nonlinear least squares objective function, i.e., formula (1).

[0019] (1) In the formula This is the final untangled phase estimation set obtained during the optimization process. for Variance of phase gradient estimation error for pixels Indicates that optimization is needed. Pixel entanglement phase variable, Indicates known The untangling phase of a pixel, express The true phase gradient estimate of a pixel; for Pixel observation noise error variance Interferogram The observation vector of a pixel, Represents the observation function; C. Using the `lsqnonlin` function in the MATLAB software platform, the Lewinberg-Marquardt method is employed to iteratively solve the nonlinear least squares objective function, obtaining the optimal solution for the M pixels within the window and updating the results. This yields the phase unwrapping result for these M pixels. The convergence condition for the iterative solution is that both the step size tolerance and the objective function tolerance are set to a fixed threshold of 1×10⁻⁶. -8 When the changes in both are less than the threshold, the iteration is considered to have converged and the calculation is terminated.

[0020] The specific process is as follows: a. When the sliding window meets the conditions to trigger optimization, the algorithm packages the constraints of all pixels in the window, including process factors and observation factors, into a nonlinear least squares problem in the form of formula (1), and provides a high-quality initial guess value X0. b. Call the lsqnonlin function, whose internal Lewinberg-Marquardt method processes this nonlinear least squares objective function and the initial value guess X0. Through iterative calculation, it finds a set of phase values ​​that minimize the total error, i.e. the sum of squares of all factor errors. c. Update the unwrapping result diagram of the sub-interference diagram with the optimal solution returned by the LM algorithm; D. Slide the window forward to continue receiving new pixels, and repeat steps B and C until the entire sub-interference pattern is unwrapped. E. Using the unwrapping phase consistency criterion of adjacent regions of the interferogram, the unwrapping results of each sub-interferogram are merged to obtain the unwrapping phase of the entire interferogram.

[0021] Example 2: Phase Unwrapping Comparison Experiment To verify the performance of the phase unwrapping method of this invention, least squares optimization calculations were performed using the lsqnonlin function in MATLAB (which supports various optimization methods, including the Lewenberg-Marquardt (LM) method). During the optimization process, both the step size tolerance and the objective function tolerance were set to fixed thresholds. When the changes in both are less than a threshold, the iteration is considered converged and the calculation is terminated. In the experimental design, ILS, QGPU, and FGPU methods were selected to perform phase unwrapping on the simulated data, and the unwrapping results were compared and analyzed. In the simulation experiment, the unwrapping error map and error histogram were used as evaluation indicators; in the field experiment, the FGPU method was used to perform phase unwrapping on the measured data, and the performance of the method was verified by rewrapping the phase map. The FGPU method is the phase unwrapping method of Embodiment 1 of this invention.

[0022] 1. Simulated data experiment Figure 1 (a) and (b) are two simulated interferograms with different fringe types, showing unwrapped phases (both images are 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 7.45 dB and 2.18 dB, respectively. ILS, QGPU, and FGPU methods were used to analyze... Figure 1 (c) and (d) were subjected to PU experiments, and the untangling results are as follows: Figure 2 and Figure 3 As shown.

[0023] Depend on Figure 2 and Figure 3 It can be seen that the ILS and QGPU methods have a large dynamic range of unwrapping phase error. In contrast, the graph-optimized FGPU method exhibits greater stability and higher unwrapping phase accuracy under different fringe patterns and signal-to-noise ratios.

[0024] 2. Experimental Data To further verify the effectiveness of the FGPU method, two sets of experimental data were selected for experimental analysis. Figure 4 (a) An interferometric map of the Etna volcano region on the east coast of Sicily, Italy, acquired by the SIR-C / XSAR satellite. Figure 4 (b) An interferogram of Yangbi County, Yunnan Province, China, acquired by the ESA-Sentinel-1A satellite. From Figure 5 and 6 The unwrapping results show that the FGPU method of this invention can better preserve the original stripe structure and significantly reduce edge noise, demonstrating higher unwrapping accuracy and stability.

[0025] 3. Conclusion This invention presents a phase unwrapping method based on factor graph optimization. Within the factor graph framework, it simultaneously utilizes observation factors, process factors, and prior factors, transforming phase unwrapping into a nonlinear least squares problem under maximum a posteriori estimation, effectively improving the continuity and consistency of the unwrapping results. Experimental results based on simulated and measured data demonstrate good unwrapping accuracy and robustness in phase unwrapping experiments with different fringe mode interferograms, validating its effectiveness and application potential.

Claims

1. A factor graph based phase un-wrapping method, characterized in that, Includes the following steps: A uses an overlapping cropping method to crop the interferogram into several sub-interferograms of the same size, and leaves several rows or columns of overlapping pixels between adjacent sub-interferograms during the cropping process; B. For each sub-interferogram, construct the phase unwrapping system equations, and set a fixed-step window for each sub-interferogram. Then, for each sub-interferogram, sort the pixels according to the quality-guided strategy based on heap sort, continuously extract the pixel with the best current quality to be unwrapped, and construct two core factors for it, namely the process factor and the observation factor. Then, store the constructed two factors and their related information into the window. As new pixels are continuously added, the window is gradually filled. When the number of pixels in the window reaches a set threshold, integrate all factors corresponding to all M pixels in the current window, optimize the phase unwrapping system equations, and form a phase unwrapping factor graph model formula containing M pixel factor variables, thereby constructing the following nonlinear least squares objective function for phase unwrapping: （1） In the formula This is the final untangled phase estimation set obtained during the optimization process. for Variance of phase gradient estimation error for pixels Indicates that optimization is needed. Pixel entanglement phase variable, Indicates known The untangling phase of a pixel, express The true phase gradient estimate of a pixel; for Pixel observation noise error variance Interferogram The observation vector of a pixel, Represents the observation function; C. The Lewinberg-Marquardt method is used to iteratively solve the nonlinear least squares objective function to obtain the optimal solution for the M pixels within the window and update the results to obtain the phase unwrapping results for the M pixels. D. Slide the window forward to continue receiving new pixels, and repeat steps B and C until the entire sub-interference pattern is unwrapped. E. Using the unwrapping phase consistency criterion of adjacent regions of the interferogram, the unwrapping results of each sub-interferogram are merged to obtain the unwrapping phase of the entire interferogram.

2. The phase unwrapping method based on factor graphs as described in claim 1, characterized in that: In step A, the interferogram is cropped into 4-12 sub-interferograms of the same size.

3. The phase unwrapping method based on factor graphs as described in claim 1, characterized in that: In step B, the phase unwrapping system equation is: （2） In the formula, This represents the predictive model of the phase unwrapping system, characterizing Predicted untangling phase value of a pixel; This represents the interferogram. Observations of cell state variables without noise contamination; and They represent Like Yuanhe The untangling phase of a pixel; express The true phase gradient of a pixel; express The true phase gradient estimate of a pixel; Interferogram Observed values ​​of pixels; The process noise vector of the system satisfies , Let be the observation noise vector of the system and satisfy... The two are independent of each other, and and These are the error variances of the two, respectively.

4. The phase unwrapping method based on factor graphs as described in claim 3, characterized in that: In step B, the process of optimizing the phase unwrapping system equations using nonlinear least squares is as follows: In the factor graph phase unwrapping model, the main factor nodes include predictive factors and observed factors; Observation factors: These factors are based on observation equations Construction; Process factors: These factors are based on state transition equations Perform parameterization expression; Factor nodes define the constraint relationships between variable nodes. These constraints are described by probability density functions to represent this uncertainty. Ultimately, the graph optimization phase unwrapping problem can be transformed into a nonlinear least squares optimization problem, resulting in the following phase unwrapping model: （3） In the formula This is the final untangled phase estimation set obtained during the optimization process. The phase set to be optimized for Variance of phase gradient estimation error for pixels Indicates that optimization is needed. Pixel entanglement phase variable, Indicates known The untangling phase of a pixel, express The true phase gradient estimate of a pixel; for Pixel observation noise error variance Interferogram The observation vector of a pixel, Represents the observation function; This represents the initial unwrapped cell phase value; Due to the initial unwrapped pixel phase mean With covariance It is known that the corresponding constant term has no effect on the optimization result. The formula can be further expressed as a simplified nonlinear least squares objective function, i.e., formula (1).

5. The phase unwrapping method based on factor graphs as described in claim 1, characterized in that: The Lewinberg-Marquardt method is used to iteratively solve for each nonlinear least squares objective function. This is accomplished using the lsqnonlin function in the MATLAB software platform, and the specific process is as follows: a. When the sliding window meets the conditions to trigger optimization, the algorithm packages all the constraints of the pixels in the window, including process factors and observation factors, into a nonlinear least squares problem in the form of formula (1), and provides a high-quality initial guess value X0. b. Call the lsqnonlin function, whose internal Lewinberg-Marquardt method processes this nonlinear least squares objective function and the initial value guess X0. Through iterative calculation, it finds a set of phase values ​​that minimize the total error, i.e. the sum of squares of all factor errors. c. Update the unwrapping result diagram of the sub-interference diagram with the optimal solution returned by the LM algorithm.

6. The phase unwrapping method based on factor graphs as described in claim 1, characterized in that: The convergence condition for the iterative solution is that both the step size tolerance and the objective function tolerance are set to a fixed threshold of 1×10. -8 When the changes in both are less than the threshold, the iteration is considered to have converged and the calculation is terminated.

7. The application of the factor graph-based phase unwrapping method as described in any one of claims 1-6 to phase unwrapping in InSAR elevation reconstruction.

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